Teach your children well.
The world of numbers is yes!
Yes heaven, yes hell.
So where do they start?
Why not 0, 1, 2, 3,…
Now go square them — smart!
Place them on a grid,
Axis and diagonal.
Clever you! Done did.
The numbers between —
Subtract a squared from a squared —
Are there to be seen!
A row that ripples
Of squared numbers, gives to you:
Triangle “Triples!”
~ ~ ~
The story of the BIM (BBSISL Matrix) and the connection with the Pythagorean Theorem — TPISC (The Pythagorean  Inverse Square Connection) — is fundamentally simple!
In fact, keep the 3Steps to Nirvana in mind as we parse out some of the details.
Make the BIM:
Locate the PTs (Pythagorean Triples):
Connect the dots to make the ToPPT (Tree of Primitive Pythagorean Triples):
SEE Page 50 (Art Theory101: White Papers) for spreadsheets: BIM: How to Make.
HERE IS THE BIG OUTLINE OF THE CONTENTS:
a. EVENs ÷6—> BIM ÷24
b. EVENs NEVER PPT or PRIME
a. ODDs NOT ÷3 —> BIM ÷24
1.) ALL PPTs
a.) 4 PPT iterations/Row + Column
b.) 4A # ALWAYS on Row/Column
c.) 8A # ALWAYS on "Sister Row" below
d.) Tripartite Tree of Pythagorena Triples
e.) ALL PPTs linked to each other
f.) Forms nodes of asymmetry
g.) Information is: ubiquitous, infinite, and ALWAYS KNOWN
h.) Informs DSEQEC and CaCoST, along with the BIM.
2.) ALL PRIMES
3.) Some Both
4.) Some Neither
b. ODDs ÷3 —> NOT BIM ÷24
1.) NEVER PPT
2.) NEVER PRIME
a. NEVER PPT
b. NEVER PRIME
c. BIM ÷24 (Two alternating sets)
1.) ES_{1}
a.) Outer
b.) Middle
2.) ES_{2}
a.) Outer
b.) Middle
a. ODD NOT ÷3, ALWAYS, after ±1, ÷24 = ARs
1.) ALL PPTs
2.) ALL PRIMES
3.) Some Both
4.) Some Neither
5.) ∆ ALWAYS ÷24
6.) ∆ between AR & NONAR NEVER ÷24
b. ODD ÷3, NEVER, after ± 1, ÷24 = NONARs
1.) NEVER PPTs
2.) NEVER PRIMES
3.) Form INTERMEDIARIES between ARs
4.) Form INTERMEDIARIES between ARs ÷24 Sets
5.) ∆ ALWAYS ÷24
6.) ∆ between AR & NONAR NEVER ÷24
Parsing out the details is also fundamentally simple — but admittedly can appear complex — because it is the nature of the BIM that very easy and simple to understand number differences are the basis for the rich complex content that emerges. The connections and correlations are so vast that one must maintain a high sense of order to corral these into a meaningful, and ultimately useful, pattern arrangement — what we call Number Pattern Sequence(s) or NPS.
The Dickson Method (DM) — an algebraic method for determining ALL PTs — has been found to be inherently a geometric method in that the rvalues become STEPS (rsteps) between geometries on the BIM. Initially, this led to the Extended Dickson Method (EDM) — described in TPISC II: Advanced — and thereafter, the Fully Extended Dickson Method (FEDM) — TPISC_III: Clarity — and the work herein: TPISC IV: Details.
What we have is some form of algebraic geometry. A highly visual geometrybased picture has evolved in which the various Universal Template and Specific variable point locations can all be algebraically determined. The two completely reenforce each other. More than that, this process of seeing the geometry with the algebra, and, seeing the algebra with the geometry, has been extremely fruitful in seeing the larger and larger picture of the TPISC. A picture that just keeps on growing and gaining clarity!
Right now, at this stage, that clarity has resulted in the ToPPT. An infinitely large and expanding structure embedded within the infinitely large and expanding BIM. This has come about by connecting the dots. There are still many, many more dots to connect!
ORGANIZATIONAL NOTE: While it may have made more sense to divide the entire work into three parts — the ➗24 all by itself in the second part, the DoubleSlit Experiment—Quantum Entanglement Conjecture (DSEQEC) , and, the Creation and Conservation of SpaceTime (CaCoST), in the third — preference was given to keeping it all together under "Details" in that each part informs the other and keeping a constant "eye" on the bigger picture outweighs overfocusing on either part alone. In Nature, these "parts" are NOT separated! As before, the "discovery" phase of this work follows a loose chronological order. The storyline actually helps to see how the BIM reveals itself, little by little, by following its leads — bits of information acting like bread crumbs pointing to a path of even greater clarity — via greater detail! The HTML Canvas feature has been extensively utilized to tell this story in a sequential series of interactive images and forms the outline basis of TPISC IV: Details. Along with additional images, animations, tables and charts, reference will be made to this interactive page: Page 42: TPISC III: Clarity & Simplification and TPISC IV: Details: Tree of Primitive Pythagorean Triples (ToPPT) (~ A MathspeedST Supplement~) Interactive Graphics on BIM & BIMtree AREAS.. which the reader may find directly on my website under Portfolio>New Media>Art Theory 101>Page 42. This is an interactive graphics page on the BIM and BIMtree_AREAS — bridging TPISC I, II, and III with TPISC IV: Details. Each canvas may be interactively deconstructed to see how the geometric components are related. Simply refresh the page to start over. A great learning and teaching resource.
Along the way, several new and very exciting findings have popped up! It's just like the BIM to just keep on giving!
The simple — and some of the not so simple — geometries of the BIM and BIMtree have been laid out in a series of interactive canvases. Each canvas opens up on its own webpage. Once open, you just click/touch and drag the geometry around, apart, this way and that way, even rebuilding the initial geometry side by side until you can really see how the parts are so simply related. It's deconstruction!
The entire geometry on canvas series was produced initially without any supporting text to keep it as clean, simple and intuitive as possible. The accompanying text will certainly help fill in the gaps, but it should be noted that the geometry on canvas series sequentially builds from one canvas to another.
The algebraic binomials and their square proofs are widely known. Here the very same proofs overlap with that of the BIM, BIMtree and the ToPPTs! This, too, can be readily seen.
The number 24 — and its factors of 124, 212, 38, and 46 have shown, once again, to be integral to both the BIM in general (remember, the EVEN Inner Grid cell numbers are all evenly divisible by 4), and, the ToPPTs. HOW SO? Well, this is new, really new, so just the surface has been touched, but so far:
Besides intimately tying the ToPPTs to a natural fractal pattern within the BIM:
Thereafter, ALL PTs — primitive “parents” (PPT) and nonprimitive “children” (nPPT) —are found on select Rows of the BIM by simply following the squared numbers on the the Prime Diagonal down to intersecting Rows; ( TPISC I: Basics & TPISC II: Advanced) more…
The mathematical basis relating the PPTs was found, as was the consistent FractalTemplate Profile (FTP) that every PT follows; (TPISC II: Advanced & TPISC III: Clarity) more…
A yet to be explained tiein with the prime numbers and 24: The difference (∆) between the squares of ALL *PRIMES is evenly ➗ by 24!!! (Pattern in Number: from Primes to DNA; GoDNA: The Geometry of DNA; SCoDNA: Structure and Chemistry of DNA; Butterfly Primes: ~let the beauty seep in~; Butterfly Prime Directive: ~metamorphosis ~ ; and, Butterfly Prime Determinate Number Array (DNA): ~conspicuous abstinence ~; and an interactive Butterfly Primes new media net.art project.
The FTP allowed all the PPTs to be sorted out and organized into a definitive Tree of Primitive Pythagorean Triples (ToPPT) that coextends infinitely throughout the infinitely expanding BIM —>BIMtree or BIMToPPT; (TPISC III: Clarity) more… and more…
Rules 178180 introduced the “submatrix grid” briefly as:
** 180 **  BS Rule 180 The SubMatrix: Within the Inner Grid, every progressive number of every column or row is evenly divisible , progressively, by a whole number sequence. Plotting those dividends reveals a truly fundamental "submatrix" grid, underlying the original Inner Grid, of repeating simple whole number sequences, i.e. 1,2,3, ..., horizontally and vertically ...forming the simplest, most basic grid pattern possible, that also includes the axis numbers. All numbers of the original Brooks (Base) Square are predicated on this simple whole number sequence pattern. 
“Note: This amazing submatrix grid pattern explains the entire parent grid matrix and the Inverse Square Law, ISL, relationship that unfolds. See an animated example by CLICKING IMAGE above. Patience. Follow the bold numbers.
Here we find the simplest and most basic pattern of the simple whole number sequence ... 1,2,3,... forming a truly fundamental base layer ... a submatrix ... lying below the original grid. And every number on that original grid is predicated on this simple pattern of 1,2,3,... in both the horizontal and vertical arrays. In fact, we now have complete integration of the axis numbers with the Inner Grid numbers ... together forming the Prime Diagonal (PD) of the Inverse Square Law (ISL). We have come full circle. And it begs the question: Who is the parent and who is the child. It seems the parent has become the child. How many more fractal matrix layers are there?”
Henceforth, we will call this SubMatrix 2.
A new submatrix, SubMatrix 1, results in the Active Row Sets (ARS) grid formed by dividing ALL Inner Grid numbers by 24.
Together SubMatrix 1 and SubMatrix 2 will provide the visualgeometric and algebraicgeometric location of ALL PPTs and PRIMES.
One could simply say that: subtract one from the squared values of any natural, whole integer number (WIN) and if it is evenly ➗ by 24, it is a candidate for being a PPT or 'stepsister' if that same squared value  25 is ALSO ➗ by 24. (And, most interestingly, a PRIME candidate, as well. ALL PRIMES follow this. see the PRIMES Section. *NOTE: ACTIVE Rows may contain NO PTs or PRIMES, PTs but NO PRIMES, PRIMES but NO PTS, or BOTH PTs and/or their 'stepsister' AND PRIMES — but ALL PTs, 'stepsisters' and ALL PRIMES occur — as a NECESSARY prerequisite — on an ACTIVE ➗24 Row!)
If the Row contains BOTH a Factor Pair Set (two squared values that = a^{2} and b^{2}) and the 4A value ONLY, it is a PPT Row.
If it contains BOTH the 4A and 8A values, it is a 'stepsister' Row, i.e. (c^{2}1)/24 and (c^{2}25)/24, if evenly divisible, are PPT and/or PPT 'stepsister' Rows, e.i. Row 17 is the PPT Row of the 81517 PPT, but is also the 'stepsister' Row of the 51213 PPT.
(NOTE: The PPTs intersect with grid cells/24 at: the subtraction of 1,5,7,11,13,17,19,23,25,… from the squared Axis number (c^{2}) and that gives a cell spacing of 42424242… respectively, or a blank step ∆ between of 3,1,3,1,3,1,3,1,… These are the "ACTIVE" Rows. ACTIVE Rows may contain NO PTs or PRIMES, PTs but NO PRIMES, PRIMES but NO PTS, or BOTH PTs and/or their 'stepsister' AND PRIMES — but ALL PTs, 'stepsisters' and ALL PRIMES occur — as a NECESSARY prerequisite — on an ACTIVE ➗24 Row!)
Take the BIM and divide all numbers evenly divisible by 24.
This gives you a crisscross pattern based on 12, i.e. 12, 24, 36 ,48,… from Axis.
Halfway between, are the rows based on 6.
On either side of this 6based and 12based frequency, the rows just before and just after, are ACTIVE Rows. These are ALWAYS ODD # Rows. They form an Active Row Set (ARS). Later these were colored PURPLE.
Their Axis #s are NEVER ➗3.They ALWAYS have their 1st Col value ➗ by 24.
Adding 24 to ANY of the ODD # NOT ➗3 ACTIVE Row Axis values ALWAYS sums to a value NOT ➗3 and thus to another ACTIVE Row Axis value (as adding 2 + 4 = 6, ➗3 added to a value NOT ➗3 = NOT ➗3 sum*).
Another ODD Axis # Row lies before and after each pair of ACTIVE Rows, i.e. between EVERY set of two ACTIVE Rows, is an ODD nonACTIVE Row and their Col 1value is NOT ➗by 24.
Adding 24 to ANY of these ODD # ➗3 Axis values ALWAYS sums to a value also ➗3 (as 2 + 4 = 6, ➗3 added to a value already ➗3 = ➗3 sum *).
While not an exclusive condition, it is a necessary condition, that ALL PPTs and ALL Primes have Col 1 evenly ➗ by 24.
Together, two ACTIVEs + one nonACTIVE form a repetitive pattern down the Axis, i.e. ARS + nonActive Row.
*While 24 seems to define this relationship, any EVEN # ➗3 will pick out much if not all of this pattern, e.i., 6, 12, 18,…
It follows that:
ALL PTs (gray with small black dot) fall on an ACTIVE Row.
ALL PRIMES (RED with faint RED circle) fall on an ACTIVE Row.
The difference, ∆, in the SQUARED Axis #s on any two ACTIVE Rows is ALWAYS divisible by 24.
The difference, ∆, in the SQUARED Axis #s on an nonACTIVE ODD Row and an ACTIVE Row is NEVER divisible by 24.
The difference, ∆, in the SQUARED Axis #s on any nonACTIVE ODD Row and another nonACTIVE ODD Row is ALWAYS divisible by 24.
Going sequentially down the Axis, every ODD number in the series follows this pattern:
nA—AA—nA—AA—nA—AA—
#3—57—9—1113—15—1719—21... Every 3rd ODD # (starting with 3) is ➗by 3 = nA .
#57—9—1113—15—1719 Every 1st & 2nd, 4th & 5th, 7th & 8th,… ODD # is NOT ➗ by 3 = A.
In other words, the two consecutive ODD #s, between the the nA ODD #s, are A ODD #s and are NOT ➗ by 3.
#3,4,5 restated: let A = ACTIVE Row Axis #, nA = nonACTIVE Row Axis #
A_{2}^{2}A_{1}^{2}= ➗ 24 and A ≠ ➗by 3
nA^{2}A^{2}≠ ➗ 24
nA_{2}^{2}nA_{1}^{2}= ➗ 24 and nA = ➗by 3
ODD Axis #s ➗by 3 (every 3rd ODD #) are NEVER ACTIVE Row members — thus never PT/PRIME
ODD Axis #s NOT ➗by 3 (every 1,2 — 4,5 — 7,8….ODD #s ) are ALWAYS ACTIVE Row members and candidates for being PT and/or PRIME.
In brief:
An ACTIVE Row ODD Axis # squared + a multiple of 24 (as 24x) = Another ACTIVE Row ODD Axis # squared , and the Square Root = a PT and/or a PRIME # :
A_{1}^{2}+ 24x = A_{2}^{2} and √A_{2}^{2} = A_{2} = a PT and/or PRIME candidate;
ODD_{1}^{2}+ 24x = ODD_{2}^{2} and √ODD_{2}^{2} = ODD_{2} = a PT and/or PRIME candidate, if and only if, its 1^{st} Col. value is ➗ by 24.
The difference in the squared values of any two PTs/PRIME #s is ALWAYS a multiple of 24!
On the Prime Diagonal, the ODD #s follow the same pattern as on the Axis (see No.7)
BIM➗PPTs and PRIMES: (Latest: as this work was being prepared, a NEW relationship was found.) See below under Why?
((((Table VI series is really the whole evolution of the BIMrow11000+sheets/Primes_sheets+PF, etc here and appendix.)))
referenced as Table VI b
Table VII.referenced as Table VI b with ACTIVE Rows in VioletTable VII.
Table VII closeup
We conclude with the possible tiein of the BIMToPPT with the central question of Quantum Mechanics (Quantum Field Theory): how do you explain the DoubleSlit Experiment and the Quantum Entanglement phenomena — other than simply describing the results? The DSEQEC proposes an explanation.
The original Dickson Method (DM) showed that for r=EVEN numbers, such that r^{2}=st was satisfied, ALL Pythagorean Triples (PTs) could be found. This was a strictly algebraic calculation method.
Previously, we found that ALL PTs could be uniquely located on specific Rows on the BBSISL Matrix (BIM) by finding the a^{2}, b^{2}, c^{2} values.
We also found that our Extended Dickson Method (EDM) provided a Template series — one for each rset value — that more completely characterized a PT and its unique placement on specific Row on the BIM.
The Template upon a Row had a number of key focus points — values — upon, above, and below each PT Row. ALL PTs within a given rset value, strictly follow the respective template for that rset.
The EDM added d, e, f, f^{2}, F, g, h, i, J, √J, k, l, m, n, o, p, q, U, v, W, √W, and d/p to the a, a^{2}, b, b^{2}, c, c^{2}, r, r^{2}, s, and t values found in the original DM.
In this work, we have Fully Extended the EDM — FEDM — to better characterize important profiling focus point values of any given PT. There are now some 40+ focus point values.
The discovery of the Area (A) and Perimeter (P) values within a PT Row Template has provided new insights into the very foundation of the Pythagorean Theorem and its intimate connection with the Inverse Square Law, as fully visual algebraic geometry specifically and unequivocally shown on the BIM.
In this journey, we will methodically lay out how these 40+ values provide a simple, yet unique profile. The expanded proof of the Pythagorean Theorem directly on the BIM is in and of itself proof of The Pythagorean  Inverse Square Connection (TPISC)! Along the way we hope to share some great insights!
Every PT Row contains at least 8 key values directly;
and another 8 values indirectly by counting STEPS;
giving another 8 values, by calculation/STEPS or a mixture of both;
and additionally;
A. The PT's sides, when squared, become SQUARES and have Squared Areas.
B. The PTs, themselves, when combined with their mirror opposite, form RECTANGLES and Areas.
C. As ALL Inner Grid (IG) numbers are simply differences (∆) in PD values, so too, ALL IG numbers are Areas.
D. The key is that those IG Areas that are SQUARES equal the Squared Area sides of Pythagorean Triples!
A. PTs are composed of only WINs.
B. PTs come in two flavors:
Primitive, or Parent, PTs (PPTs) are irreducible;
nonPrimitive, or Child, PTs (nPPTs) are reducible;
a. 6810
b. 91215
c. 102426.
C. PTs are infinite — they grow infinitely large in size as WINs to infinity.
D. PT's sides, designated a & b for the short sides, c for the long, hypotenuse such that:
E. PTs Area (A) times 4 + the difference (∆) in the short sides squared(^{2}) — (ba)^{2} = f^{2} .
TIP: PPTs always have 1EVEN and 1ODD short side.
TIP: nPPTs can have either EVEN & ODD, or, Both EVEN short sides (and long side, too).
A. All proofs are visually present right on the BIM, starting at the origin.
B. PTs and the entire BIM are both composed solely of WINs.
C. PTs and the entire BIM are both ultimately related to the ODDnumber summation series:
D. In that every IG WIN is an Area difference (∆) between two (2) PD perfect SQUARES Areas, it is ONLY those Rows (or Columns) containing the SAME perfect SQUARES Areas as in the PD, that are PT Rows. (see IV next)
E. The building up of sets of two (2) perfect, but asymmetric sized SQUARE Area sides — short sides 1 and 2 —the nonSQUARE RECTANGLE is generated, i.e. by combining the two (2) nonisosceles righttriangle PTs.
TIP: The Tree of Primitive Pythagorean Triples (ToPPT) is also relateable by its SQUARE Areas (TPISC III).
TIP: The Exponentials are also relateable by its SQUARE Areas (TPISC V).
TIP: Here is the BBSISL Matrix (BIM) 10x10 in 5 Easy Steps:
Make 10x10 Template:
Fillin the 1^{st} Parallel Diagonal that runs parallel just below the PD with the ODD WINS: 3579…with a difference (∆) of two (2) between subsequent numbers.
Fillin the 2^{nd} Parallel Diagonal that runs parallel just below the PD with the EVEN WINS: 8121620…with a difference (∆) of four (4) between subsequent numbers.
Fillin the 3^{rd} Parallel Diagonal that runs parallel just below the PD with the ODD WINS: 15212733…with a difference (∆) of six (6) between subsequent numbers.
Continue filling in the rest of the Parallel Diagonals by adding (2)(4)(6)81012…to each number to give the next number with the ∆ appropriate for that Parallel Diagonal:
The ∆ is simply the Axis number x 2;
The first, initial number in the IG is always its PD1 — Prime Diagonal value minus one:
You can doublecheck your work by noticing that every IG number:
A. PTs may be easily found by dropping down from the PD number (WIN>4) to intersect the Row with Paired Sets — a.k.a. Square Paired Sets (SPS)
B. PTs may also be easily found as being on the Row whose PD number has matching numbers on its Row and Column.
C. PTs may also be calculated algebraically and then located on the BIM following the Dickson Method (DM), Extended Dickson Method (EDM) or the Fully Extended Dickson Method (FEDM) as shown in Chapter V.
A. First, it was found that the strictly algebraic Dickson Method (DM) for calculating all PTs coincided with those found on the BIM as presented in TPISC I: Basics:
Let r=EVEN number, such that r^{2}=2st is satisfied;
r^{2}/2=st, where s,t are the Factor Pairs (FPs) of r^{2}/2;
Then, side a=r+s, side b=r+t, and side c=r+s+t;
B. Second, it was found that the parameters described in the DM were actually the grid cell STEPS from one parameter — or focus point character — value to another as this algebrageometry overlay become the Expanded or Extended Dickson Method (EDM), the subject of much of TPISC II: Advanced:
PTs with the EDM, were found to be parsed into r and ssets;
rsets (examples, see Table 1c)
ssets (examples, see Tables 2, 3, 5,...)
C. Third, the PT rsets formed a common Template of EDM relationships and the geometry becomes very visual in the Fully Extended Dickson Method (FEDM). The Tree of Primitive Pythagorean Triples (ToPPT), known in part since Pythagoras' time, became fully realized, and thus essentially defined, by its full presence on the BIM. It provides a large structural framework of forms based on the nonSQUARE RECTANGLES (and Ovals) that are formed from the cojoining of the nonisosceles righttriangle PTs. A new Clarification and Simplification of the TPISC as a whole becomes the subject of TPISC III:.
D. The BBSISL Matrix (BIM) is itself an infinitely expandable — yet highly patterned —Template of perfect SQUARE Areas and perfect CIRCLES into which, the PTs are embedded. Here are a few things to know:
While the actual Area (A) and Perimeter (P) of the PT in question can be shown directly on the Matrix as grid cell areas — as can the PT proof — it is the 4x the A, and, the difference (∆) in the length of the sides, squared, that present the true Pythagorean  Inverse Square Connection:
4A + (ba)^{2} = c^{2} = a^{2} + b^{2};
where, (ba)^{2} = (ts)^{2} = ƒ^{2}, giving 4A + ƒ^{2} = c^{2} = a^{2} + b^{2}.
Finding the A and P on the Matrix necessarily introduced some additional key focus point values — giving rise to the FEDM.
The 40+ key focus points are found consistently in each and every rset Template — defining the PTs.
Every PT Row contains at least 8 key values directly;
And another 8 values indirectly by counting STEPS;
Giving another 8 values, by calculation/STEPS or a mixture of both;
And additionally;
If you plot these out along their rset STEPS (rsteps) spacings, the ∆ between values and their PD for any given Column follows 1r^{2} — 4r^{2} — 9r^{2} — 16r^{2} — 25r^{2} —…;
c^{2}o=1r^{2} 169153=16=1r^{2}, c^{2}d=4r^{2} 169105=64=4r^{2}, c^{2}g=9r^{2} 16925=144=9r^{2}.
Naturally, the PD sequence, up from the Row values, follows the same 1—4—9—16—25—...
The "downward" Diagonal, perpendicular from the PD, back to the Row Axis gives the Ƀ — V — ¥ and cƒ points, and, 8A is always rsteps down the grid from the 4A location on that PT Row.
(Also see Appendix A/Notes20162018/Section 4C/Pages 3236 for more.)
The PT Axis Row: ALL PTs have c — X_{f} — o — b^{2} — 4A — d — a^{2} — c^{2} :
The Main Diagonals (Template) for ALL PTs:
PD — O (origin)—>c^{2} as o—ƒ^{2}—r^{2}—a^{2}—b^{2}—c^{2};
⊥PD — c^{2}—>2c (Axis) follows 4c—8c—12c—16c sequence.
The Secondary Diagonals:
The Tertiary Diagonals:
The Horizontal Axis (TOP) of Columns:
The Vertical Axis (Left SIDE) of Rows:
The 4A—8A—c^{2}—ƒ^{2} Rectangle Connection. (Also see Appendix A/Notes20162018/Section 4C/Pages 2227 for more.)
The Proof a^{2} + b^{2} = c^{2} = 4A + ƒ^{2}.
The Complementary Pair (Square Pairs) Sets of ANY PT.
ALL PTs have matching 4A values on both the PD Row itself and rsteps down that PD Column.
Basic BBSISL Rule 1: All numbers (#s) related by the 149...PD sequence.
Basic BBSISL Rule 2: Every # in the PD sequence is the square of an Axial #.
Basic BBSISL Rule 3: The OddNumber Summation sequence forms the PD sequence.
Basic BBSISL Rule 4: Every EVEN Inner Grid (IG) # is divisible by 4 & all are present.
Basic BBSISL Rule 5: Every IG# is:
A: The difference (∆) between its two PDsequence #s; (Note: A=B=C=D, and, E.)
B: The sum (∑) of the ∆s of each of its PD#s between its two PDsequence #s (as above);
C: The ∆ between the squares of the two Axial #s forming that IG# (as above);
D: The product of the Addition & Subtraction of the two Axial #s forming that IG# (as above);
E: Also, the product of its 2 Axial #s intersected by that IG#'s 90° diagonals;
Basic BBSISL Rule 6: Every ODD IG# is NOT PRIME & all are present;
Basic BBSISL Rule 7: The ODDNumber sequence, and the 149...PD sequence, forms the sequential ∆ between ALL IG#s.
Basic BBSISL Rule 8: The ∆ between #s within the Parallel Diagonals is a constant 2 x its Axial #.
Basic BBSISL Rule 9: The ∆ between #s in the Perpendicular Diagonals follow:
A: From EVEN PD#s, √PD x 4 starts the sequence & follows x1x2x3x4....
B: From ODD PD#s, √PD x 4 starts the sequence & follows x1x2x3x4....
C: From ODD Perpendicular Diagonals between the EVENODD diagonals (above), the sequence starts with the same value as the Axis number ending the diagonal, the sequence following x1x3x5x7..
Basic BBSISL Rule 10: Every #, especially the #s in the ONEs Column, informs both smaller and larger Subset symmetries (much larger grids required to demonstrate).
For any given # X, located on the Axis, its respective x^{2} is, of course, located on the PD, while X^{3}, X^{4}, X^{5},.. (X^{n}), are ALL found on that X Diagonal Parallel ( P∥D) to the PD.
The distance — # of steps diagonally — between successive Exponentials X^{1,2,3..}. for a given X, follows a Number Sequence Pattern (NPS) equal to is X^{n} sequence value.
The Sum (∑) of the Axis Column & Row # values x that Diagonal Axis # X, equals the X^{n} value:
The Sums (∑s) of the ∆s between the PD #s of a given X^{3}_{5}=X^{3} and is simply an expression of the IGGR:
The Area (# of grid cells) of a given X^{x} = Area X^{x}~_{uo}~  Area X^{x}~_{LD}~, and flollows the same NPS progression sequence as X^{3}, X^{4}, X^{5},… in Area and in # of PD steps:
TPISC III, IV and V FEDM Equations (2018)
Everything you need to know is covered here and in the Summary and Conclusion. The Appendix gives you the source of the original Figures,Notes and Tables.
(Also see Appendix A/Notes20162018/Section 4a/Pages 119; Appendix A/Notes20162018/Section 4B/Pages 000010; Appendix A/Notes20162018/Section 4C/Pages 000050c, and thereafter: see Appendix A/Notes20162018/Sections 1 and 2/ ALL Pages for more.)
(complete mappingTemplate of A, P, 4A, 8A, ƒ ƒ2,...)
Because of the bilateral symmetry of the BIM, each PT is represented 4 times (4x) =4A:
Row;
ver1
mirror of ver 1 = ver 2
Column;
Symmetry of ver 1 on Row = ver 3
Mirror of ver 3 = ver 4.
Remember: ALL IG #s, just like ALL PD #s, are Areas!
(Also see Appendix A/Notes20162018/Section 4C/Pages 000050c, and thereafter: see Appendix A/Notes20162018/Sections 1 and 2/ ALL Pages for more. Appendix B will be the goto appendix for BIM/24, PPTs and PRIMES, as well as the DSEQEC.)
(Also see Appendix Figures and Tables for BIM/24, PPTs and PRIMES, as well as the DSEQEC.)
Rules 178180 introduced the “submatrix grid” briefly as:
180  BS Rule 180 The SubMatrix: Within the Inner Grid, every progressive number of every column or row is evenly divisible , progressively, by a whole number sequence. Plotting those dividends reveals a truly fundamental "submatrix" grid, underlying the original Inner Grid, of repeating simple whole number sequences, i.e. 1,2,3, ..., horizontally and vertically ...forming the simplest, most basic grid pattern possible, that also includes the axis numbers. All numbers of the original Brooks (Base) Square are predicated on this simple whole number sequence pattern. 
“Note: This amazing submatrix grid pattern explains the entire parent grid matrix and the Inverse Square Law, ISL, relationship that unfolds. See an animated example by CLICKING IMAGE above. Patience. Follow the bold numbers.
Here we find the simplest and most basic pattern of the simple whole number sequence ... 1,2,3,... forming a truly fundamental base layer ... a submatrix ... lying below the original grid. And every number on that original grid is predicated on this simple pattern of 1,2,3,... in both the horizontal and vertical arrays. In fact, we now have complete integration of the axis numbers with the Inner Grid numbers ... together forming the Prime Diagonal (PD) of the Inverse Square Law (ISL). We have come full circle. And it begs the question: Who is the parent and who is the child. It seems the parent has become the child. How many more fractal matrix layers are there?”
Henceforth, we will call this SubMatrix 2.
A new submatrix, SubMatrix 1, results in the Active Row Sets (ARS) grid formed by dividing ALL Inner Grid numbers by 24.
The simple — and some of the not so simple — geometries of the BIM and BIMtree have been laid out in a series of interactive canvases. Each canvas opens up on its own webpage. Once open, you just click/touch and drag the geometry around, apart, this way and that way, even rebuilding the initial geometry side by side until you can really see how the parts are so simply related. It's deconstruction!
The entire geometry on canvas series was produced initially without any supporting text to keep it as clean, simple and intuitive as possible. The accompanying text will certainly help fill in the gaps, but it should be noted that the geometry on canvas series sequentially builds from one canvas to another.
The algebraic binomials and their square proofs are widely known. Here the very same proofs overlap with that of the BIM, BIMtree and the ToPPTs! This, too, can be readily seen.
The number 24 — and its factors of 124, 212, 38, and 46 have shown, once again, to be integral to both the BIM in general (remember, the EVEN Inner Grid cell numbers are all evenly divisible by 4), and, the ToPPTs. HOW SO? Well, this is new, really new, so just the surface has been touched, but so far:
Besides intimately tying the ToPPTs to a natural fractal pattern within the BIM:
One could simply say that: subtract one from the squared values of any natural, whole integer number (WIN) and if it is evenly ➗ by 24, it is a candidate for being a PPT or 'stepsister' if that same squared value  25 is ALSO ➗ by 24.
If the Row contains BOTH a Factor Pair Set (two squared values that = a^{2} and b^{2}) and the 4A value ONLY, it is a PPT Row.
If it contains BOTH the 4A and 8A values, it is a 'stepsister' Row, i.e. (c^{2}1)/24 and (c^{2}25)/24, if evenly divisible, are PPT and/or PPT 'stepsister' Rows, e.i. Row 17 is the PPT Row of the 81517 PPT, but is also the 'stepsister' Row of the 51213 PPT.
(NOTE: The PPTs intersect with grid cells/24 at: the subtraction of 1,5,7,11,13,17,19,23,25,… from the squared Axis number (c^{2}) and that gives a cell spacing of 42424242… respectively, or a blank step ∆ between of 3,1,3,1,3,1,3,1,... )
Take the BIM and divide all numbers evenly divisible by 24.
This gives you a crisscross pattern based on 12, i.e. 12, 24, 36 ,48,… from Axis.
Halfway between, are the rows based on 6.
On either side of this 6based and 12based frequency, the rows just before and just after, are ACTIVE Rows. These are ALWAYS ODD # Rows. They form an Active Row Set (ARS). Later colored in PURPLE.
Their Axis #s are NEVER ➗3.They ALWAYS have their 1st Col value ➗ by 24.
Adding 24 to ANY of the ODD # NOT ➗3 ACTIVE Row Axis values ALWAYS sums to a value NOT ➗3 and thus to another ACTIVE Row Axis value (as adding 2 + 4 = 6, ➗3 added to a value NOT ➗3 = NOT ➗3 sum*).
Another ODD Axis # Row lies before and after each pair of ACTIVE Rows, i.e. between EVERY set of two ACTIVE Rows, is an ODD nonACTIVE Row and their Col 1value is NOT ➗by 24.
Adding 24 to ANY of these ODD # ➗3 Axis values ALWAYS sums to a value also ➗3 (as 2 + 4 = 6, ➗3 added to a value already ➗3 = ➗3 sum *).
While not an exclusive condition, it is a necessary condition, that ALL PPTs and ALL Primes have Col 1 evenly ➗ by 24.
Together, two ACTIVEs + one nonACTIVE form a repetitive pattern down the Axis, i.e. ARS + nonActive Row.
*While 24 seems to define this relationship, any EVEN # ➗3 will pick out much if not all of this pattern, e.i., 6, 12, 18,…
It follows that:
ALL PTs (gray with small black dot) fall on an ACTIVE Row.
ALL PRIMES (RED with faint RED circle) fall on an ACTIVE Row.
The difference, ∆, in the SQUARED Axis #s on any two ACTIVE Rows is ALWAYS divisible by 24.
The difference, ∆, in the SQUARED Axis #s on an nonACTIVE ODD Row and an ACTIVE Row is NEVER divisible by 24.
The difference, ∆, in the SQUARED Axis #s on any nonACTIVE ODD Row and another nonACTIVE ODD Row is ALWAYS divisible by 24.
Going sequentially down the Axis, every ODD number in the series follows this pattern:
nA—AA—nA—AA—nA—AA—
#3—57—9—1113—15—1719—21... Every 3rd ODD # (starting with 3) is ➗by 3 = nA .
#57—9—1113—15—1719 Every 1st & 2nd, 4th & 5th, 7th & 8th,… ODD # is NOT ➗ by 3 = A.
In other words, the two consecutive ODD #s, between the the nA ODD #s, are A ODD #s and are NOT ➗ by 3.
#3,4,5 restated: let A = ACTIVE Row Axis #, nA = nonACTIVE Row Axis #
A_{2}^{2}A_{1}^{2}= ➗ 24 and A ≠ ➗by 3
nA^{2}A^{2}≠ ➗ 24
nA_{2}^{2}nA_{1}^{2}= ➗ 24 and nA = ➗by 3
ODD Axis #s ➗by 3 (every 3rd ODD #) are NEVER ACTIVE Row members — thus never PT/PRIME
ODD Axis #s NOT ➗by 3 (every 1,2 — 4,5 — 7,8….ODD #s ) are ALWAYS ACTIVE Row members and candidates for being PT and/or PRIME.
In brief:
An ACTIVE Row ODD Axis # squared + a multiple of 24 (as 24x) = Another ACTIVE Row ODD Axis # squared , and the Square Root = a PT and/or a PRIME # :
A_{1}^{2}+ 24x = A_{2}^{2} and √A_{2}^{2} = A_{2} = a PT and/or PRIME candidate;
ODD_{1}^{2}+ 24x = ODD_{2}^{2} and √ODD_{2}^{2} = ODD_{2} = a PT and/or PRIME candidate, if and only if, its 1st Col. value is ➗ by 24.
The difference in the squared values of any two PTs/PRIME #s (>3) is ALWAYS a multiple of 24!
On the Prime Diagonal, the ODD #s follow the same pattern as on the Axis (see No.7).
BIM➗PPTs and PRIMES: (Latest: as this work was being prepared, a NEW relationship was found.) See below under Why?
Now that have overloaded the field with details, let's simply (again)!:
ALL PTs and ALL PRIMES are exclusively on Rows that:
• are referred to as “ACTIVE."
• bookend the Rows that are multiples (evenly ➗ by) 6: i.e. 6,12,18,24,30,36…
• some of these bookend on 6based Rows do NOT have a PT and/or Prime, e.i. Rows 35, 49, 55, 77, 91, 95,… (Certainly, those ➗ by factors other than themselves and one, are NOT Prime.)
• as one progresses across the Matrix Row, the YELLOW and YELLOWORANGE cells on “ACTIVE” Rows follow a pattern under Column #: 1–5—7—11—13—17—19—23—25…i.e., +    +  +    +  +    +  +    and so on (see GrayViolet & White on the Table below).
• the cell value numbers of the YELLOW and YELLOWORANGE cells on “ACTIVE” Rows follow exactly the numbers on Col C of the Table VII: Axis_Sqd_Diff_24x.numbers.
• the Table view gives the number values and relationships in tabular form, the Matrix in a more visual, geometric form.
• remember: ALL candidates for PTs and/or Primes MUST have Col 1/24 = (x2  1)/24 as TRUE.
Distilling the BIM/24 into the underlying "submatrix" of ➗24 Actives, as shown in Figure below. ___ , reveals exactly why the BIM/24 pattern is what it is.
NOTES: Sections 110
Section 1: 2018 (29pgs)
~~ ~~ ~~
In that the discovery that ALL PRIMES land only on ARS — exactly as do the PPTs — yet most emphatically do NOT follow the exact same distribution pattern leads to some open questions:
While a great deal in the way of proof of the PRIMES dependency on the ARS for their distribution will be presented in a number of tables (Tables 123, mostly redirected to the APPENDIX), we must keep these questions in mind as we pursue the PPTs and their relationship to the BIM (Tables 2428).
Now, we introduce
The details are the same:
These (colored inset boxes) SubMatrix 2 values:
Foolproof Steps to Find ALL PPTs:
Axis# must be ODD, NOT ➗3 = Active Row Set (ARS) member;
Only 1 of the 2 ARS can be a PPT;
SubMatrix Col 1 # MUST be ➗4;
SOME may NOT be PPT if ➗Prime Factor (>5);
Remaining Axis # is a PPT. Exceptions:
Squared #s that are PPTs, remain PPTs when x2 or √x:
Squared #s that are NOT, remain NOT when x2 or √x, as above.
Table 29 Exponentials of the first 10 PPTs cvalues to be used in Tables 30ag.
Tables 30ag The SubMatrix 2, when ➗4, and the difference (∆) between this and the next exponential PPT treated this way, is subsequently ➗ by its SubMatrix 2 variable, the PREVIOUS exponential within the series is revealed. Restated as an example: When one subtracts 1 from the exponential values of c (the cvalue of the PPT) you get the SubMatrix 2 value. Divide that by 4 and take the difference (∆) between it and the next. Divide that by 3 to give the PREVIOUS PPT cvalue in the series.
The SubMatrix 2 variable divisor = 3 = SubMatrix 2 value/4 = 12/4. These variables run: 1,3,4,6,7,9,10,...
As to answers to the open questions called above:
what exactly is the relationship between PPTs and PRIMES?
why do they BOTH land on ARS?
why do some ARS have both, neither, or one or the other?
can the PRIMES be used to predict the PPTs?
can the PPTs be used to predict the PRIMES?
SEE: Tables: 33a, 33b and 33c. towards the end of Appendix B for some very NEW INFO on the BIM÷24.
The underlying geometry of the BIM÷24 PRESELECTS the Axis Rows into TWO Groups: ARs and NONARs. The PPTs and PRIMES are EXCLUSIVELY — as a sufficient, but not necessary condition — found on the ARs and NEVER on the NONARs. While both Groups follow (PD^{2}  PD^{2})÷24, they do so ONLY within their own respective Groups. They do NOT crossover. This Grouping divide occurs naturally within the BIM as shown in these images below.
The ISL as presented in the BIM is deeply, intimately structured around the number 24 — and its factors: 4,6, 3,8 2,12, and 1,24.
The interplay between these small sets of Numbers generates an incredible amount of richness and complexity with seemingly simplistic BIM itself. This has led to TPISP: The PythagoreanInverse Square Connection, and the PRIMES.
Open in separate browser tab/window to see all.
A dovetailing of PPTs and PRIMES on the BIM
The discovery of the Active Row Sets (ARS) — the direct result of the BIM ➗24 — in which it has been found that ALL PPTs and ALL PRIMES are exclusively found on, was in and of itself, a slow an arduous journey.
Once found, it has added a great deal of visual graphic clarity! In simple terms, it simply marks out the obvious. Both the PPTs and the PRIMES can not be on Axis Row #s that are EVEN, nor ➗3. This leaves ONLY Rows that are ODD #s and not ➗3.
The BIM ➗24 marked those Active Row Sets indirectly, by being on either side — i.e., +/ 1 — of the Axis Row # intercepted by the ➗24. Directly, the ARS was shown to be picked out by Submatrix 1 and 2 values of the 1st cell Column of those Rows.
So we have the PPTs and the PRIMES occupying the same footprint rows, the ARS Rows. Both as a necessary, but not sufficient requirement, i.e., some ARS Rows do NOT have a PPT or PRIME, or both. ALL PPTs and ALL PRIMES are ALWAYS found on an ARS Row, NEVER on a nonARS Row. Some ARS Rows may have none, either a PPT or a PRIME, or both.
Nevertheless, on this vast matrix array of ISL whole integer numbers, that the PPTs and PRIMES exclusively occupy the same ➗24based footprint points to an underlying connection!
The 1st connection was found and written about in the three white papers of 20056 on PRIMES:
The 2nd connection, as referenced below, has been the latest discovery that Euler’s 6n+1 and 6n1 pick out, as a necessary — but not sufficient for primality — condition ALL the PRIMES.
When you look at the BIM➗24, you can readily see how this theorem simply picks out the very same ARS Rows! (For ARS 5 and above.)
The BIM➗24 becomes a DIRECT GRAPHIC VISUALIZATION of EULER’s PRIMES = 6n+1 and 6n1, where n=1,2,3,..
The same holds true for Fermat's (FermatEuler) 4n + 1 = Sum of Two Squares Theorem, where 4n + 3 ≠ Sum of Two Squares. The 4n + 1 = Sum of Two Squares = Pythagorean Primes (PTs where c = Prime #).
These are simply those ARs that contain BOTH a PRIME (red circle) AND a PPT (black dot) in the figures.
Note: In any given ARS, only one of the two ARs may be a PPT, while both, neither, or one or the other ARs may be PRIME.
Black dot in a Red circle = Pythagorean Triple = 4n + 1 PRIME candidate = Sum of Two Squares.
The 3rd connection is that for those ARS Rows that do NOT contain PRIMES — e.i., 25, 35, 49, … and have been shown to negate the possibility of the # being prime because it is itself prime factorable — divisible by another set of primes — is ALSO DIRECTLY VISUALIZABLE ON THE BIM➗24 AS THE INTERSECTING PRIME COLUMNS!!!
The 4th connection reveals that the BIM Prime Diagonal (PD) — the simple squares of the Axis #s — defines:
The 5th connection is that for those ARS Rows— that may or may not contain PPTs and/or PRIMES — their 1,5,7,11,13,17,19,23,25,.. ODD intersecting Columns NOT ➗3, ARE ALL➗24, and, this is ALSO DIRECTLY VISUALIZABLE ON THE BIM➗24 AS THE INTERSECTING COLUMNS (usually depicted graphically in YELLOWORANGE boxes/cells on the BIM as part of the diamond with centers pattern) !!!
￼ This is revealed in the SubMatrix 1 figures and tables: the BIM ÷24.
Additionally, SubMatrix 2 also selects for ALL ARS as Column 1 ALWAYS ÷4. (See figures.)
In ALL cases, the PRIMES (≧5) are necessary — but not sufficient to insure primality — located on ODD # Rows NOT ÷3. It is as simple as that!
The factors of 24 — 1,24–2,12–3,8–4,6 — when increased or decreased by 1, ultimately pick out ALL ARs. Euler's 6n +/1 is the most direct, Fermat's 4n + 1 gets the Sums of Two Squares = Pythagorean Primes (while 4n + 3 gets the rest).
Fermat's Little Theorem (as opposed to the more familiar "Fermat's Last Theorem") tests for primality.
But now there is a dead simple way to test for primality:
The difference in the squares between ANY 2 PRIMES (≧5) ALWAYS = n24.
For example, take any random ODD #  25 —> it must be ÷24 n times to be PRIME. (n=1,2,3,...)
741
741^{2}5^{2}/24 = n= 22877.3 NOT PRIME
189
189^{2}5^{2}/24 = n = 1487.3 NOT PRIME
289
289^{2}5^{2}/24 = n = 3479 PRIME
￼
"A New Kind of Prime
The twin primes conjecture’s most famous prediction is that there are infinitely many prime pairs with a difference of 2. But the statement is more general than that. It predicts that there are infinitely many pairs of primes with a difference of 4 (such as 3 and 7) or 14 (293 and 307), or with any even gap that you might want."
Quote is from Quanta Mag 9/26/19 article: Big Question About Primes Proved in Small Number Systems
by Kevin Hartnett
~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~
A new and easy visualization!
There is a new game board in town. It has actually been around for awhile, but few know of it. It's basically a matrix grid of natural numbers that defines the Inverse Square Law (ISL). It's called the BIM, short for the BBSISL Matrix. Every grid cell is uniquely occupied by a given number that is simply the difference between the horizontal and vertical intercept values of the main "Prime Diagonal" (not Prime number diagonal) that mirrordivides the whole matrix.
If you show all the matrix values that are evenly divisible by 24, a crisscrossing pattern of diamonddiagonal lines will appear and this ends up giving us a unique visual on the distribuition of ALL PRIMES!
Basically, all ODDs (3 and greater for this presentation) fall into a repetitive pattern of:
ODD ÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3...
The matrix ÷24 described above, nicely picks out these sets of —Not÷3—Not÷3 that we call Active Row Sets (ARS). Each ODD member of the ARS is referred to as an Active Row (AR).
The ÷3 ODD Rows that lie between are called NonActive Rows, or NA.
It is a necessary, but not sufficient condition, that ALL PRIMES are strictly located on ARs. NO exceptions, except for some ARs have NO PRIMES.
Another striking visualization is that ALL Primitive Pythagorean Triples (PPTs) also lie only on the ARs, following the same necessary, but not sufficient, condition. NO exceptions here as well, except for some ARs also have NO PPTs. In addition, only one of the ARs within an ARS can have a PPT.
So any AR within an ARS can have 0/1 PRIMES and/or 0/1 PPTs in any combination, only with the one caveat: that only 1 PPT/ARS is allowed.
Now, not to belabor this BIM game board, let's talk PRIME GAPS!
Let's refer to the lower number value for the ARS as "Lower" and the higher as "Upper." All are ODDs. As will become apparent, you can always tell if your ODD AR is "Lower/Upper" simply by adding 2 and ÷3: if the result evenly ÷3 it is "Upper" and if not, "Lower."
The Even Gap between primes strictly follows this pattern: (all are necessary but not sufficient conditions for primality)
For any given set of twin primes, or any single prime, that is the smaller "Lower" of an AR set:
For any given set of twin primes, or any single prime, that is the larger "Upper" of an AR set:
One can readily see that all this is simply the natural result of two ARs alternating with a NA row:
AR—AR—NA—AR—AR—NA—AR—AR—NA—AR—AR—NA—…
“Lower” AR number
AR—5
AR—7
NA—9
AR—11
AR—13
NA—15
AR—17
AR—19
NA—21
AR—23
AR—25
NA—27…
It is 6 steps, i.e. +6, from any NA to the next NA, from any “Lower” AR to the next “Lower” AR, from any “Upper” AR to the next “Upper” AR. ALL of the various EVEN GAPS may be shown to be a direct consequence of the natural number sequence and easily visualized on the BIM.
To any “Lower” Active Axis Row ODD:
“Upper” AR number
AR—7
NA—9
AR—11
AR—13
NA—15
AR—17
AR—19
NA—21
AR—23
AR—25
NA—27…
To any “Upper” Active Axis Row ODD:
Again, It is 6 steps, i.e. +6, from any NA to the next NA, from any “Lower” AR to the next “Lower” AR, from any “Upper” AR to the next “Upper” AR. ALL of the various EVEN GAPS may be shown to be a direct consequence of the natural number sequence and easily visualized on the BIM.
The ARS pattern on the BIM clearly shows the above patterns and may be extrapolated to infinity:
A number of PRIME conjectures have been shown to be easily visualized on the BIM:
Reference: TPISC IV: Details: BIM + PTs + PRIMES
The factors of 24 — 1,24–2,12–3,8–4,6 — when increased or decreased by 1, ultimately pick out ALL ARs. Euler's 6n +/1 is the most direct, Fermat's 4n + 1 gets the Sums of Two Squares = Pythagorean Primes(while 4n + 3 gets the rest).
Fermat's Little Theorem (as opposed to the more familiar "Fermat's Last Theorem") tests for primality.
But now there is a dead simple way to test for primality:
The difference in the squares between ANY 2 PRIMES (≧5) ALWAYS = n24.
For example, take any random ODD #  25 —> it must be ÷24 n times to be PRIME. (n=1,2,3,...)
741
741252/24 = n= 22877.3 NOT PRIME
189
189252/24 = n = 1487.3 NOT PRIME
289
289252/24 = n = 3479 PRIME
I would be remiss if I did not mention that throughout this long journey that began with the Inverse Square Law and the Primtive Pythagorean Triples that the PRIMES kinda of just fell out. They just kept popping up. Not the least, but certainly not without effort, the BIM actually directly visualizes ALL of the NOPRIMES. In doing so, one is left with information that is simply the inverse of the PRIMES. Subtract the NOPRIMES information from the list of ODDS (disregarding 2) and what remains are ALL the PRIMES! A simple algebraic expression falls out from that:
NP = 6yx +/ y
letting x=1,2,3,… and y=ODDs 3,5,7,… with the only caveat is that if you don't first eliminate all the ÷3 ODDS, you must include exponentials of 3 (3^{x}) in the NP tally.
In 200910, a solution to Euler's "strong" form of the Goldbach Conjecture "that every even positive integer greater than or equal to 4 can be written as a sum of two primes" was presented as the BBSISL Matrix Rule 169 and 170. This work generated a Periodic Table of Primes (PTOP) in which Prime Pair Sets (PPsets) that sequentially formed the EVEN numbers were laid out. It is highly patterned table. A recently annotated version is included with the original below. (See above link for details.)
It turned out this PTOP was actually embedded — albeit hidden — within the BIM itself as shown in Rule 170 and here, too, a recently annotated version is included below. (See above link for details.)
Rule 169: Periodic Table of Primes.
Rule 169: annotated
Rule 170: Periodic Table of Primes (PTOP): embedded within Brooks (Base) Square.
Rule 170: annotated
With that review of the ongoing work, the full presentation of this work starts below. It will cover everything above plus the NEW work leading to the findings and proof of the Goldbach Conjecture. This will be the basis for the ebook: PTOP: Periodic Table of PRIMES & the Goldbach Conjecture
PTOP Goldbach Conjecture from Reginald Brooks on Vimeo.
PTOP: Periodic Table Of PRIMES & theGoldbach Conjecture
Here’s the thing. Amongst a myriad of other connections, there exists an intimate connection between three number systems on the BBSISL Matrix (BIM):
1 The Inverse Square Law (ISL) as laid out in the BIM;
2 PTs — and most especially PPTs — as laid out on the BIM;
3 The PRIME numbers — PRIMES — as laid out on the BIM.
The BIM is the FIXED GRID numerical array of the ISL.
PTs are the Pythagorean Triples and PPTs are the Primitive Pythagorean Triples. They have been extensively covered in the TPISC (The PythagoreanInverse Square Connection) series as: TPISC I Basics, TPISC II Advanced, TPISC III Clarity, TPISC IV Details. (See links at bottom.)
The PRIMES vs NOPRIMES (2019) was covered earlier.
The original MathspeedST (200914) work, Brooks (Base) Square (200911), that started this journey, was divided in to two sections:
I. TAOST (The Architecture Of SpaceTime);
II. TCAOP (The Conspicuous Absence Of PRIMES).
You see, other than the natural whole numbers that form the BIM Axis’ and the standard 1^{st} Parallel Diagonal (containing ALL the ODDs (≥3), there are NO PRIMES on the BIM.
This is the basis of the PRIMES vs NOPRIMES work.
Yet, convert that same 1^{st} Parallel Diagonal to ALL EVENS (≥4, by adding 1 to each former ODD), and now the BIM reveals the stealthily hidden PRIMES relationship in forming symmetrical pairs of PRIMES that are ALWAYS equal distance (STEPS) from ANY given EVEN # divided by two.
All this occurs on the BIM Axis. The apex of the 90° Rightangled, isosceles triangle so formed lies on a straight line path from the EVEN/2 to the given EVEN # located on that converted 1^{st} Parallel Diagonal. This relationship is geometrically true and easily seen on the BIM.
Extracting those PRIMEPair sets (PPsets) for each given EVEN forms the basis for the PTOP — Periodic Table Of PRIMES.
While “hidden” on the BIM, it clearly forms a definitive pattern on the PTOP: for EVERY 3+PRIME PPset that forms the 2^{nd} column on the PTOP — acting like a bifurcation point — a “Trail” of PPsets forms a zigzagging diagonal pointing down and to the right.
These are ALWAYS — much like a fractal — added with the SAME PRIME Sequence (3,5,7,11,13,17,19,23,29,31,37,...).
When you read a given horizontal line from left to right across the PTOP, you see that each given PPset that was contributed by a PPset Trail, adds up — composes — its respective EVEN. The sum (∑) number of PPsets that form its EVEN is totaled in the last column.
The fractallike addition of one new, additional PRIME Sequence PPset to each subsequent “Trail” formed results in the overlapping trails growing at a rate that far, far exceeds the growth and incidence of the PRIME Gaps. This ensures that there will always be at least one PPset that will be present to compose ANY EVEN.
The Goldbach Conjecture has been satisfied and a new PRIME pattern has been found.
MathspeedSTTPISC Resource Media Center Intro
MathspeedSTTPISC Resource Media Center
PTOP  Goldbach Conjecture Video
While the (strong) Goldbach Conjecture has been verified up to 4x10^{18}, it remains unproven.
A number of attempts have demonstrated substantial, provocative and often beautiful patterns and graphics, none have proven the conjecture.
Proof of the conjecture must not rely solely on the notion that extension of a pattern to infinity will automatically remain valid.
No, instead, a proof must, in its very nature, reveal something new about the distribution and behavior of PRIMES that it is absolutely inevitable that such pattern extension will automatically remain valid. The proof is in the pudding!
Proof offered herein is just such a proof. It offers very new insights, graphical tables and algebraic geometry visualizations into the distribution and behavior of PRIMES.
In doing so, the Proof of the Euler Strong form of the Goldbach Conjecture becomes a natural outcome of revealing the stealthy hidden Number Pattern Sequence (NPS) of the PRIMES.
Proof of the Goldbach Conjecture (strong form, ≥6)
Natural (n), Whole Integer Numbers (WIN) — 0,1,2,3,…infinity — form horizontal and vertical Axis of a simple matrix grid.
The squares of such WINs — n^{2}=1^{2}=1, 2^{2}=4, 3^{2}=9,…infinity — forms the central Diagonal of said grid — dividing it into two bilaterally symmetric triangular halves.
Every Inner Grid (IG) cell within is simply the difference (∆) between its horizontal and vertical Diagonal intercept values. They extend to infinity. The Diagonal WINs form the base of a 90° Rangled isosceles triangle with said IG cell value at the apex.
Every IG cell within is also the product of two Axis WINs (Either horizontal or vertical, not both), that form the base of a 90° Rangled isosceles triangle with said IG cell value at the apex.
The complete matrix grid extends to infinity and is referred to as the BIM (BBSISL Matrix).The BIM forms — and informs — a ubiquitous map (algebraic geometry) to:
The Inverse Square Law (ISL);
The 1^{st} Diagonal that runs parallel to either side of the main Prime Diagonal (PD, not of PRIME numbers, but primary), is composed of the ODD WINs: 1,3,5,…infinity.
If we add +1 to each value, that 1^{st} Diagonal now becomes a sequence of ALL the EVEN WINs (≥4): 4, 6, 8,…infinity. NOTE: this is why the PTOP is hidden, in the normal, base BIM these remain ODDs.
Select ANY EVEN WIN and plot a line straight back to its Axis WIN — that Axis WIN = EVEN/2 = core Axis #.
Upon that same Axis, PRIME Pair sets (PPsets) — whose sum (∑) equals the EVEN WIN (on the 1^{st} Diagonal) — will be found that form the base of 90° Rangled isosceles triangle(s) whose apex lie(s) on that straight line between the EVEN and its 1/2 Axis WINs. PPsets with identical PRIMES = 1/2 Axis value.
The proof that every EVEN WIN has ≥1 PPsets can be seen in the Periodic Table Of PRIMES (PTOP) that stealthily informs the BIM of how each and every EVEN WIN is geometrically related to one or more PPsets.
These PPsets are NOT randomly contributing their ∑s to equal the EVEN WINs, rather they come about as the consequence of a strict NPS: the sequential — combining, linking, concatenation — addition of the PRIMES Sequence (PS) — 3,5,7,11,13,17,19,…—to a base PS — 3,5,7,11,13,17,19,….
The NPS of this addition forms the PTOP: for each vertical PS — the 1^{st} PRIME (P_{1}) remains constant (3), the 2^{nd} PRIME (P_{2}) sequentially advances one (1) PS WIN — is matched diagonally with the 2^{nd} PS, but now the 1^{st} PRIME sequentially advances, while the 2^{nd} PRIME remains constant within a given PPset.
This matching addition of the 2^{nd} PS at the bifurcation point of the common 2^{nd} PRIMES, forms the zigzag diagonal PPset Trails that are the hallmark of the PTOP.
For every subsequent vertical PPset match, the Trail increases by one (1) PPset.
The rate of such PPset Trail growth far exceeds the PRIME Gap rate.
The zigzag diagonal PPset Trails combine horizontally on the PTOP to give the ∑# of PPsets whose ∑s = The EVEN WIN.
More than simply proving the Goldbach Conjecture, the PTOP hidden within the BIM reveals a new NPS connection of the PRIMES: PRIMES + PRIMES = 90° Rangle isosceles triangles.
The entire BIM, including the ISL—Pythagorean Triples—and, PRIMES, is based on 90° Rtriangles!
Similar to how the grid cell values of the Axis, PD and IG of the Pythagorean Triples reveal additional, intimate connections within the BIM, so too do the PPsets: the 1^{st} PRIME values of each set points to the # of STEPS from the PD that intersects the given EVEN WIN (Axis^{2}), on a straight line path back to its Axis, at the apex where its other PPset member intersects — this is no mere coincidence — and that apex is, of course, the 90° Rangle isosceles triangle that results. For example: EVEN = 24, Axis = 12, PD = 144, # of STEPS from PD towards Axis = 5 for the 5+19 PPset and 7 for the 7+17 PPset, and 11 for the 11+13 PPset that each forms the EVEN 24.
As the bifurcation concatenation of the PS — 3,5,7,11,13,17,19,..— with the same base PS — 3,5,7,11,13,17,19,…— of those EVEN WIN — that when “3” is subtracted, the remainder is a nextinthesequence PRIME — remains one of a similar split with the Pythagorean Triples: for every “Primitive” parent PT, there are multiple “NonPrimitive” child PTs and it is the PPTs (Primitive Pythagorean Triples) that ultimately form the interconnectedness of ALL PTs back to the original PPT — the 345. With the PRIMES, one raises the question: why are these “EVENS” preselected to be the “parent” EVENS forming the “beginning” or “start” of every PPset Trail with all other EVENS hitching on to that Trail further down the sequence?
Another set of STEPS (S) from the core Axis value , directly on the Axis, identifies each symmetrical pair of a given PPset that forms that given EVEN. These STEPS may by Universally calculated from the EVENS, P_{1} and P_{2} values. Examples are given further down.
As every new discovery unlocks many more questions, it follows that the details of the PTOP and BIM visualizations should both satisfy the proof offered and, more importantly, provide provocative data that will advance the field for the next researcher!
BIM (BBSISL Matrix): grid visualizations that overview the entire work
PTOP: the actual Table
PTOP: Analysis
Reference
PRIME PPset Trails
The DATA has been grouped in its own section down below. It is highly recommended that you, the reader, preview first — review, thereafter.
~~~~
The definitive proof depends on demonstrating that the PPset Trails grow and extend to cover the “next” EVENS at a rate that exceeds the Prime Gap rate. PRIMES Sequence, PS is the key.
The PS is well established. How many primes are there?
As every beginning PPset — of 3,P_{2} — forms from successively increasing the 2^{nd} PRIME by the next number in the PS (3,5,7,11,13,17,…), the Trail formed by bifurcating off from that point increases the total number sum (∑) of PPsets by one, i.e. Trail, Trail+1, (Trail+1)+1, ((Trail+1)+1)+1,…
A necessary, and sufficient, condition is that the Trail lengths — i.e. the total number of PPsets, in their overlapping aggregate, always exceed both the number of PRIMES and their Gaps for any and all numbers.
Specifically from Table 46 and Fig.__, locate a P_{2} PRIME with a large Gap, e.i. P_{2}= 23, Gap=6.
line#———PRIMES, P_{2}—∑#set/Trail—PRIME Gap—∆TrailGap— EVEN:∑#sets—EC——E_{ending}——EVEN, E
8  23  8  6  2  26: 3  11  46  26 

We see that 23 has 8 PPsets in its Trail, it has a Gap of 6 to the next PRIME (for a ∆=2). It’s EVEN=26 and there are 3 PPsets (going horizontally across the PTOP) that will make 26. The 11 is the number of EVENS Covered (EC) as the PPsets zigzag diagonally down the PTOP. If we add double this EC  1 to the EVEN 26, we get the EVEN Ending of 46 as: 2(111) + 26 = 46. So this Trail alone inclusively covers the EVENS 2646, although their will be some holes. To fill the holes, we look at the Trails that started up above this 23 Trail.
For the Specific set of equations exclusively for the 3,P_{2} sets, where ∆ = P_{2}  3 (see Table 45: Equations):
2EC = P_{2} 1
Ending EVEN Covered = E_{e} = 2(EC  1) + EVEN
5  13  5  4  1  16: 2  6  26  16 

6  17  6  2  4  20: 2  8  34  20 
7  19  7  4  3  22: 3  9  38  22 
We can readily see that the 17 and 19 Trails, when their EVENS are added to their 2EC1, respectively, will equal or exceed the EVEN of the NEXT PRIME up by the GAP=6.
Take Trail 17: EVEN = 20 with EC = 8. 2(81) + 20 = 34.
Take Trail 19: EVEN = 22 with EC = 9. 2(91) + 22 = 38.
And these Trails 17 and 19 will overlap the Trail 23 EC span, filling in any and all holes between it and the next Trail 29.
9  29  9  2  7  32: 2  14  58  32 

For completeness, we can see that Trail 13, while reaching Trail 23 at it’s start, does not overlap any further.
Take Trail 13: EVEN = 16 with EC = 6. 2(61) + 16 = 26. We also can see that 13 + 13 = 26.
~~~
The PPset Trails for EVEN 26 gives 3 PPsets via the overlapping Trails 17, 19 and 23, AND, their overlapping Trails extend past the next 3 EVENS (28, 30 and 32), ensuring that they are “covered” with PPsets in the Gap jump to the next PRIME (29). All very neat and clean.
The actual PPsets ∑s equal the EVEN 26:
which comes about more simply as the core Axis value ± steps away:
But what about a larger Gap? How about Trail 113, EVEN = 116 and a Gap of 14 to the next Trail 127? (See image from Table 36.)
Trail 113 is overlapped by Trails 73, 79, 97, 103, and 109 that account for the 6 PPsets that will form the EVEN = 116 (the 3,113 PPset is included in the tally and you can follow it in Table 43). Again, from Table 46:
1  3  1  2  1  6: 1  1  6  6 

2  5  2  2  0  8: 1  2  10  8 
3  7  3  4  1  10: 2  3  14  10 
4  11  4  2  2  14: 2  5  22  14 
5  13  5  4  1  16: 2  6  26  16 
6  17  6  2  4  20: 2  8  34  20 
7  19  7  4  3  22: 3  9  38  22 
8  23  8  6  2  26: 3  11  46  26 
9  29  9  2  7  32: 2  14  58  32 
10  31  10  6  4  34: 4  15  62  34 
11  37  11  4  7  40: 3  18  74  40 
12  41  12  2  10  44: 3  20  82  44 
13  43  13  4  9  46: 4  21  86  46 
14  47  14  6  8  50: 4  23  94  50 
15  53  15  6  9  56: 3  26  106  56 
16  59  16  2  14  62: 3  29  118  62 
17  61  17  6  11  64: 5  30  122  64 
18  67  18  4  14  70: 5  33  134  70 
19  71  19  2  17  74: 5  35  142  74 
20  73  20  6  14  76: 5  36  146  76 
21  79  21  4  17  82: 5  39  158  82 
22  83  22  6  16  86: 5  41  166  86 
23  89  23  8  15  92: 4  44  178  92 
24  97  24  4  20  100: 6  48  194  100 
25  101  25  2  23  104: 5  50  202  104 
26  103  26  4  22  106: 6  51  206  106 
27  107  27  2  25  110: 6  53  214  110 
28  109  28  4  24  112: 7  54  218  112 
29  113  29  14  15  116: 6  56  226  116 
30  127  30  4  26  130: 7  63  254  130 
What about the Gap = 14 going to the Trail 127, EVEN = 130? Overlapping Trails 71, 83, 89, 101,107 and 113 combine with the 127 to account for the 7 PPsets that will form EVEN = 130.
Of course to account for all the EVENS between Trail 113 and 127 — EVENS 116 to 130, we have Trails 59 to and including 127 in various combinations to account for that. We can easily see this on Table 46, lines 1630, where each has an E_{e} (118–254) that meets or exceeds those EVENS (116–130). Altogether, there are some 80 PPsets that account for these 15 EVENS 116130. Easily done.
What if you want to know specifically who covers, say EVEN 128? Looking at the full table, line 18, we see that Trail 67 and above have E_{e} > 128. How can we determine that the 19+109, 31+97 and 61+67 PPsets are the ones we are looking for?
Without even knowing the values for the Column 6 (Table 46): EVEN, ∑# of PPsets, one can:
look at an expanded PTOP
look at an expanded BIM
calculate using this trick:
on Table 46, find where EVEN 128 would be located between line 17 and 18 as 1/2 of 128 = 64
see that PRIMES 61 & 67 on either side = 128 when added together and both are PRIMES
knowing from the BIM that ALL PPsets are symmetrical about the center Axis core that points to the 90° Rangled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 64 slot like the 61 & 67
the next pair out would be 31 & 97, followed by 19 & 109
this is easily calculated as the core Axis value ± steps away:
their ∑s equal the EVEN 128:
61 + 67 = 128
31 + 97 = 128
19 + 109 = 128.
We can confirm this on the PTOP (Table 35), and, of course, directly on the BIM. Tables 43 and 44 can also confirm ALL EVENS up to 2000. One can also use an online calculator to get the results.
it can be quite helpful to follow these examples directly on the BIM. The visualizations of how the symmetrical PPsets contributed from the separate, individual PRIME Trails line up their 90° Rangled isosceles triangles over the central core with each of their respective apexes inline and pointing towards the EVEN that they are forming. On the Axis, one can easily see the symmetrical steps from the core to each PPset.
(See the PDF versions for clear details.)
What if you want to know specifically who covers, say EVEN 126? Looking at the full table, line 18, we see that Trail 67 and above have E_{e} > 126. How can we determine that the
13+113
17+109
19+107
23+103
29+97
37+89
43+83
47+79
53+73
59+67
PPsets are the ones we are looking for?
Without even knowing the values for the Column 6 (Table 46): EVEN, ∑# of PPsets, one can:
look at an expanded PTOP
look at an expanded BIM
calculate using this trick:
on Table 46, find where EVEN 128 would be located between line 17 and 18 as 1/2 of 126 = 63
see that PRIMES 59 & 67 on either side = 126 when added together and both are PRIMES
knowing from the BIM that ALL PPsets are symmetrical about the center Axis core that points to the 90° Rangled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 63 slot like the 59 & 67
the next pair out would be 53 & 73, followed by 47 & 79.
this is easily calculated as the core Axis value ± steps away:
63 ± 4 = 59 and 67
63 ± 10 = 53 and 73
63 ± 16 = 47 and 79
63 ± 20 = 43 and 83
63 ± 26 = 37 and 89
63 ± 34 = 29 and 97
63 ± 40 = 23 and 103
63 ± 44 = 19 and 107
63 ± 46 = 17 and 109
63 ± 50 = 13 and 113
their ∑s equal the EVEN 126:
Towards the bottom of Table 46, lines 1092  1107, there is a good run of PRIME Gaps in close proximity. Below is the table in plain form with the same column headings:
line #  PRIMES, P**2≥3**  ∑# of 3+P**2PPsets/Trail**  Prime Gap  ∆ **TrailGap**  EVEN: ∑# of**PPsets**  EC=# of EVENs covered  E**e = Ending EVEN covered**  EVEN 

1080  8681  1080  8  1072  94  4340  17362  8684 

1081  8689  1081  4  1077  100  4344  17378  8692 
1082  8693  1082  6  1076  89  4346  17386  8696 
1083  8699  1083  8  1075  96  4349  17398  8702 
1084  8707  1084  6  1078  132  4353  17414  8710 
1085  8713  1085  6  1079  92  4356  17426  8716 
1086  8719  1086  12  1074  112  4359  17438  8722 
1087  8731  1087  6  1081  101  4365  17462  8734 
1088  8737  1088  4  1084  136  4368  17474  8740 
1089  8741  1089  6  1083  93  4370  17482  8744 
1090  8747  1090  6  1084  138  4373  17494  8750 
1091  8753  1091  8  1083  104  4376  17506  8756 
1092  8761  1092  18  1074  111  4380  17522  8764 
1093  8779  1093  4  1089  91  4389  17558  8782 
1094  8783  1094  20  1074  95  4391  17566  8786 
1095  8803  1095  4  1091  125  4401  17606  8806 
1096  8807  1096  12  1084  124  4403  17614  8810 
1097  8819  1097  2  1095  102  4409  17638  8822 
1098  8821  1098  10  1088  90  4410  17642  8824 
1099  8831  1099  6  1093  93  4415  17662  8834 
1100  8837  1100  2  1098  141  4418  17674  8840 
1101  8839  1101  10  1091  93  4419  17678  8842 
1102  8849  1102  12  1090  91  4424  17698  8852 
1103  8861  1103  2  1101  93  4430  17722  8864 
1104  8863  1104  4  1100  118  4431  17726  8866 
1105  8867  1105  20  1085  121  4433  17734  8870 
1106  8887  1106  6  1100  148  4443  17774  8890 
1107  8893  1107  30  1077  94  4446  17786  8896 
1108  8923  1108  6  1102  95  4461  17846  8926 
1109  8929  1109  4  1105  125  4464  17858  8932 
1110  8933  1110  8  1102  95  4466  17866  8936 
1111  8941  1111  10  1101  101  4470  17882  8944 
1112  8951  1112  12  1100  104  4475  17902  8954 
1113  8963  1113  6  1107  96  4481  17926  8966 
1114  8969  1114  2  1112  84  4484  17938  8972 
1115  8971  1115  28  1087  104  4485  17942  8974 
1116  8999  1116  2  1114  110  4499  17998  9002 
1117  9001  1117  6  1111  95  4500  18002  9004 
1118  9007  1118  4  1114  133  4503  18014  9010 
1119  9011  1119  2  1117  96  4505  18022  9014 
1120  9013  1120  16  1104  119  4506  18026  9016 
Without even knowing the values for the Column 6 (Table 46) which have been inserted here: EVEN, ∑# of PPsets, one can take EVEN 8872:
look at an expanded PTOP
look at an expanded BIM
calculate using this trick: (see *below for a simple Universal calculation method as shown in Table 45: Equations.)
on Table 46, find where EVEN 8872 would be located as the core Axis value between lines 601 and 602 as 1/2 of 8872 = 4436
see that PRIMES 4423 & 4441 on either side = 8864 when added together and both are PRIMES we are close
knowing from the BIM that ALL PPsets are symmetrical about the center core that points to the 90° Rangled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 4436 slot like the 93 PPsets that equal EVEN 8872:
as before, these can be simplified as:
We can further simplify by applying these Universal (vs Specific, see Table 45:) equations:
let S = steps, E = EVEN = 2(core Axis value) = 2(A_{x}), as E/2 = A_{x}
S = P_{2}  E/2 = P_{2}  A_{x}
the PPset for a given EVEN:
P_{2} = S + E/2 = S + A_{x}
P_{1} = P_{2}  (2S)
2S = P_{2}  P_{1}
S = A_{X}  P_{1}
E = P_{1} + P_{2}
Example: 3,5 PPset for EVEN = 8: S = P_{2}  E/2 = P_{2}  A_{x} 1 = 5 4
P_{2} = S + E/2 = S + A_{x}
5 = 1 + 4
P_{1} = P_{2}  (2S) 3 = 5  (2*1)
And, as P_{1} = E  P_{2}, or E = P_{1} + P_{2}
P_{1} = P_{2}  (2S) = E  P_{2}
2P_{2}  E = 2S 2(5)  8 = 2(1) S = 1
As S = (2P_{2}  E)/2
S = (2P_{2}  E)/2 = P_{2}  E/2
2P_{2}/2  E/2 = P_{2}  E/2
P_{2}  E/2 = P_{2}  E/2 = S
P_{2}  A_{x} = P_{2}  A_{x} = S
The builtin symmetry of the PPsets around the core Axis value is easily calculated as these four examples show:
EVEN 24 with core Axis value (A_{x}) = 24 / 2 = 12 with 3 PPsets:
P_{1} = P_{2}  (2S)
as S = P_{2}  E/2 = P_{2}  A_{x}
11,13 1312 = 1 13  (2x1) = 11
7,17 1712 = 5 17  (2x5) = 7
5,19 1912 = 7 19  (2x7) = 5
EVEN 26 with core Axis value = 26 / 2 = 13 with 3 PPsets:
P_{1} = P_{2}  (2S)
as S = P_{2}  E/2 = P_{2}  A_{x}
13,13 1313 = 0 13  (2x0) = 13
7,19 1913 = 6 19  (2x6) = 7
3,23 2313 = 10 23  (2x10) = 3
EVEN 22 with core Axis value = 22/2 = 11 with 3 PPsets:
P_{1} = P_{2}  (2S)
as S = P_{2}  E/2 = P_{2}  A_{x}
11,11 1111 = 0 11  (2x0) = 11
5,17 1711 = 6 17  (2x6) = 5
3,19 1911 = 8 19  (2x8) = 3
EVEN 100 with core Axis value = 100 / 2 = 50 with 6 PPsets:
P_{1} = P_{2}  (2S)
as S = P_{2}  E/2 = P_{2}  A_{x}
47,53 5350 = 3 53  (2x3) = 47
43,57 5750 = 7 57  (2x7) = 43
41,59 5950 = 9 59  (2x9) = 41
29,71 7150 = 21 71  (2x21) = 29
17,83 8350 = 33 83  (2x33) = 17
11,89 8950 = 39 89  (2x39) = 11
~~~~~~~_{~} ⁃
As one moves successively along the EVENS, the Trails get longer and longer, adding one new member for each successive PS (See Tables 4446.). This rate of increase far exceeds the size and incidence rate of the PRIME Gaps (See Tables 42 and 46.), ensuring that for every EVEN ≥6, there is at least one PPset of ODD PRIMES that will form it. Actually, as we have shown, there are always a minimum of 2 sets of PPsets that form the EVENS (≥14, including 10).
Table 44 shows how ∑s of the number of PPsets/ EVEN grows. It increases such that for every 60 successive EVENS on average, the ∑ increases by 3 (as calculated from 41080).
For example:
Columns generally have 60 entries ending in multiples of 120.
∑of PPsets per column Ave/Col
Col A 262 / 59 = 4.4 = 4
Col B 516 / 60 = 8.6 = 9
Col C 718 / 60 = 12 = 12
Col D 930 / 60 = 15.5 = 16
Col E 1076 / 60 = 17.9 = 18
Col F 1267 / 60 = 21.1 = 21
Col G 1302 / 60 = 21.7 = 22
Col H 1534 / 60 = 25.6 = 26
Col I 1687 / 60 = 28.1 = 28
It is worth repeating here from above:
As seen in Table 44, EVEN 68 it the last EVEN to only have 2 PPsets, EVEN 128 is the last to have 3, 332 the last 6, 398 the last 7, 488 the last 9, 632 the last 10, 878 the last 14,…as the table shows the progression steadily goes forth never regressing by more than 1 or 2 to a previous lower value. The number of PPsets for:
If we take a sampling (from Table 46), the ∑# of PPsets and and divide by their respective EVENS, we get the percentage incidence of:
This builtin geometry provides the necessary coverage that ALL EVENS (≥14) will easily be covered by two or more PPsets. The geometry defines the algebra = algebraic geometry.
The Goldbach Conjecture is proven. With the EVENS having been actually tested for compliance with the conjecture up to 4x10**18, one can say that it is statistically impossible for any of the holes not to be covered by the PPset Trails no matter what size EVEN is considered! The largest confirmed PRIME Gap of 1550 would have —————
Here is the index of the five sets of data to consider (click links to go):
BIM (BBSISL Matrix): grid visualizations that overview the entire work
PTOP: the actual Table
PTOP: Analysis
Reference
PRIME PPset Trails
BIM (BBSISL Matrix): grid visualizations that overview the entire work
PTOP: the actual Table
PTOP: Analysis
Reference
PRIME PPset Trails
Fractal — Symmetry — Inverse Square Law (ISL). These are not the usual descriptive terms associated with the Prime numbers (PRIMES) Yet that is exactly what best describes the PRIMES!
The PRIMES, when brought together as P_{1} & P_{2} members of PRIME Pair Sets (PPsets) demonstrate a robust symmetry and an intimate relationship with the ISL when shown on the BIM (BBSISL Matrix grid of the Inverse Square Law). This symmetry is brought out in the geometric relationship between the PPsets and the EVEN numbers (EVENS) that they inform consistent with Euler’s Strong Form of the Goldbach Conjecture.
This come about as the well established PRIMES Sequence (PS) — 35711131719232931... acts in a fractallike manner, i.e., it demonstrates redundant, repetitive and reiterative behavior in presenting selfsimilar reflection of itself as it constructs PPsets and PPset “TRAILS.”
The TRAILS are formed from the concatenation of PS’s progressively onto each successive PRIME of a given PS — forming a series of PPsets along the way. The TRAILS can also be seen to be formed directly as individual PPsets on the BIM. Here, each PPset is easily seen to be intersection of 1 PPset member from the Horizontal AXIS and 1 PPset member from the Vertical AXIS, together forming the P_{1}, P_{2} members. The TRAIL is simply all those PPsets thus formed across a Row — or down a Column — on the bilaterally symmetrical BIM. All this is easily visible on the BIM.
That leads us to our story about the PRIMES—Fractals—Symmetry—and the intimate relationship they have with the ISL as seen on the BIM.
The “PTOP (Periodic Table Of PRIMES) & the Goldbach Conjecture” (2019), also referred to as BIM: Part I, updated and clarified the PTOP, PPsets and their PPset TRAILS that were originally presented in MathspeedST (2010). BIM = BBSISL Matrix.
In this work, a major update and refinement has been made. Each of the three parts — BIM: II, III and IV presents new findings that visibly demonstrate the PRIMES on the BIM.
Each and every PRIME, when treated as part of a PPset (P_{1}, P_{2}), can be found and individually profiled DIRECTLY on the BIM as seen in BIM: Part II.
Plotting all the Lower Diagonal P_{2} PRIMES on a table and then replotting those results back onto the BIM opened up a new vista. By substituting the AXIS values for each of those P_{2 }PRIMES from the table, one has now formed a new SubMatrix of the BIM with ALL the PPsets in place — BIM: Part III.
In BIM: Part IIII, we have mostly used just half of the bilaterally symmetric BIM to reveal and describe the geometry.
In BIM: Part IV, we now look at the whole BIM with the PPset SubMatrix in place.
Treating these PPsets as objects and counting them within progressively larger square areas — forming what is called “Object AREAS” — a pure ISL Number Pattern Sequence (NPS) is found. As the PSFractal series of each AXIS joins to form the PPsets, their actual numbers — as PPset TRAILS — progressively grows and sums up to quantities with Object AREAS that directly mirror the fundamental ISL NPS: 1—4—9—16—25–…
And while this intimate relationship of the PRIMES to the ISL can not predict the next largest PRIME, it can ABSOLUTELY account for each and every PRIME at, and below, any given PRIME, regardless of size.
While Euler’s Strong Form of the Goldbach Conjecture is proved along the way, the real significance is what we have seen unfolding in BIM: Parts IIV. The role of Symmetry and Fractal underlie everything about the PRIMES.
The PRIMES on the BIM is all about how the fractal nature of the PS becomes expressed as symmetry on the BIM as isosceles and equilateral triangles, forming the PPsets that ultimately form ALL the EVEN numbers!
What is marvelous, incredible, mindblowing in every way, is that this same symmetry—fractal—isosceles/equilateral triangle relationship is found — indeed, is part and parcel — throughout the BIM.
The ISL seems to reflect the most basic and fundamental relationship(s) between quantity and the numbers that account for it.
SYMMETRY, STEPS, EVENS, EVENS/2, PPsets, PRIME SEQUENCE FRACTALS, ISOSCELES & EQUILATERAL TRIANGLES, INVERSE SQUARE LAW, PYTHAGOREAN TRIPLES; ALL ON THE BIM (BBSISL Matrix).
In MathspeedST, the PRIMES were found to be stealthily hidden within the BIM, revealing their presence ONLY when the BIM itself had its ODD #s 1st Parallel Diagonal modified to EVENS by the addition of 1 to each ODD.
This resulted in the original PTOP (Periodic Table Of PRIMES) on the BIM. It was refined by pulling the values off the BIM and making a separate PTOP. All this became the subject of “PTOP (Periodic Table Of PRIMES) & the Goldbach Conjecture” ebook and white paper (2019). It is also referred to as BIM: Part I. The PPsets and their TRAILS were introduced.
Each and every PRIME, when treated as part of a PPset, can be found and individually profiled DIRECTLY on the BIM. BIM: Part II was formed.
Plotting all the Lower Diagonal P_{2} PRIMES on a table and then replotting those results back onto the BIM opened up a new vista. By substituting the AXIS values for each of those P_{2} PRIMES from the table, one has now formed a new SubMatrix of the BIM with ALL the PPsets in place.
This is a direct consequence of locating the PPsets as the grid space resulting from the intersection of the PRIMES Sequence (PS) of the Horizontal AXIS with the PS of the Vertical AXIS. This becomes the basis for the BIM: Part III.
In BIM: Part IIII, we have mostly used just half of the bilaterally symmetric BIM to reveal and describe the geometry.
In BIM: Part IV, we now look at the whole BIM with the PPset SubMatrix in place.
What we find is that by treating these PPsets as objects and counting them within progressively larger square areas — forming what is called “Object AREAS” — a pure ISL Number Pattern Sequence (NPS) is found. Yes, as the PSFractal series of each AXIS joins to form the PPsets, their actual numbers — as PPset TRAILS — progressively grows and sums up to quantities with Object AREAS that directly mirror the fundamental ISL NPS: 1—4—9—16—25–…
And while this relationship of the PRIMES to the ISL can not predict the next largest PRIME, it can ABSOLUTELY account for each and every PRIME at, and below, any given PRIME, regardless of size.
While Euler’s Strong Form of the Goldbach Conjecture is proved along the way on this journey, the real significance is what we have seen unfolding in BIM: Parts IIV. The role of Symmetry and Fractal underlie everything about the PRIMES.
The PRIMES on the BIM is all about how the fractal nature of the PS becomes expressed as symmetry on the BIM as isosceles and equilateral triangles, forming the PPsets that ultimately form ALL the EVEN numbers!
What is marvelous, incredible, mindblowing in every way, is that this same symmetry—fractal—isosceles/equilateral triangle relationship is found — indeed, is part and parcel — throughout the BIM.
The ISL seems to reflect the most basic and fundamental relationship(s) between quantity and the numbers that account for it.
The PRIMES have been found on the BIM despite there being NO PRIMES in the Strict Inner Grid (SIG). The SIG is simply the BIM without the AXIS, Prime Diagonal (PD) that runs diagonally down the middle, dividing the grid into two bilaterally symmetrical triangular halves, and, the 1st Parallel Diagonal of ALL ODD numbers.
So where are they?
Initially, they were found in a quasistealth mode on the BIM. Why quasistealth? The PRIMES were naturally found on the AXIS (n=1,2,3,…). It was only when they were paired symmetrically to either side of another AXIS #—a number that turned out to be an EVEN ÷ 2— that the lines of the triangular sides (diagonals) meet on a line from said EVEN ÷2 # on the AXIS, back to the 1st Parallel Diagonal. Yes, here we had to use the entire BIM, and, that 1st Parallel Diagonal had to have ALL its ODD #s advanced (+1) to make them EVEN.
From this, it was found that every EVEN # on the adjusted 1st Parallel Diagonal, had, indeed, one or more sets, or pairs, of PRIMES on either side of the EVEN/2, found on the AXIS.
Visually, this demonstrated that every EVEN was flanked by one or more pair sets of PRIMES. We call these PPsets and each contains a P_{1} and P_{2} value that resides symmetrically on either side of the EVEN/2 AXIS #. The only exception being when P_{1} = P_{2}, then the symmetry is one, i.e., EVEN = 10 has EVEN/2=5 and there are two PPsets that inform EVEN = 10 are 3,7 and 5,5.
Visually, this also demonstrated both the symmetrical nature of the PPsets and the isosceles, righttriangle each made pointing back to the line connecting the EVEN with its EVEN/2 AXIS #.
This early (discovered in 2009–10) “PRIMES on the BIM” will be referred to as BIM: Part I.
A simplification of BIM: Part I formed the basis of the PTOP (Periodic Table Of PRIMES, as first described in MathspeedST, 2010).
The PTOP extracts the PPsets and EVENS from the BIM and relates them to each other in a straightforward and graphical manner.
A distinct PPset “TRAIL” is seen for each PPset starting with the P_{1}=3 and P_{2} = each successive number in the natural PRIMES sequence—(2),3,5,7,11,13,17,19,23,29,31,37,…
In fact, this natural PRIMES Sequence (PS) acts in every way just like a fractal, whereby a fractal set consisting of the PS is “added” —i.e. concatenated—to each successive P_{1} value down the Column 3 on the PTOP.
This fractal PS property ensures that there are more overlapping “TRAILS” of PPsets than PRIME Gaps, ensuring that ALL EVENS = the sum of two PRIMES. Euler’s Strong Form of the Goldbach Conjecture is proven. More importantly, a definitive relationship between PRIMES and the Inverse Square Law is defined. This joins the ISL, Primitive Pythagorean Triples and PRIMES together under the umbrella of the Inverse Square Law as presented on the BIM. Part I and the original PTOP is fully described in both the white paper and ebook: PTOP (Periodic Table Of PRIMES) and the Goldbach Conjecture (2019). PRIMES Index.
In BIM: Part II, the focus was shifted to looking at individual PPsets on the BIM—those making one EVEN number at a time. It is quite interesting how once again those stealthy PRIMES only appear on the AXIS and 1st Parallel Diagonal, yet make a clear, distinctive Number Pattern Sequence (NPS) by their interactions.
In this case, for every EVEN/2 on the AXIS, an isosceles triangle is formed with the base along the AXIS, one side along the PD to the apex of the (EVEN/2)^{2} and the other side back to the base on the AXIS.
The PPset P_{1}s are found on the Upper Diagonal PD as the √ of where the PS # on the AXIS intersects the PD—there the square root of that PD #=P_{1}, or more easily, just take its Axial value.
The Lower Diagonal (LD) ALWAYS points back to the EVEN on the AXIS. Along the way, it intercepts with the PS values on the AXIS above, to give the P_{2} values. Dividing the various grid values along the LD by 4x(Even/2) will select out and confirm which values house the P_{2} PRIMES.
Incredibly, it turns out the number of STEPS from one apex to another around each of the triangles formed, will show them to ALWAYS be equal, thus equilateral. The larger base:PD:LD isosceles triangle is actually made of two equilateral triangles positioned backtoback with the centerline being the line from the EVEN/2 back to it squared value on the PD.
Table 51 was formed by plotting all the LD values for each EVEN as seen in BIM: Part II. These values were then plotted back on to the BIM and BIM: Part III was formed.
In Part III, we combine and simplify everything from the Part I & II including:
SYMMETRY, STEPS, EVENS, EVENS/2, PPsets, PRIME SEQUENCE FRACTALS, ISOSCELES & EQUILATERAL TRIANGLES, INVERSE SQUARE LAW, PYTHAGOREAN TRIPLES; ALL ON THE BIM.
~~~~~~
These PPsets are formed from the Horizontal and Vertical AXIS numbers that intercept those BLUE circle values. The PRIMES Sequence (PS)—3,5,7,11,13,17,19,23,…— determines the pattern. Notice that the PPsets ALSO fall exactly on the same BIM/24 Active ROWS (PURPLE Bands) as do the PPTs and PRIMES
Those BLUE values from Table 51 are now BLUE CIRCLE values on the BIM. And when those values are overlaid with the PPsets it now becomes Table 52 and the NEW PTOP (Periodic Table Of PRIMES) on the BIM! Here they are plain and simple. Let the story begin.
Like shining a light at a sculpture in the dark, each angle reveals a new “look,” yet it is just one sculpture. That’s what we are doing here..
BIM: Part III becomes significant when one finds that the NPS of those PPsets falls on exactly the SAME Active Rows that previously ALL PRIMES and ALL PPTs were found to occupy on the BIM! No small coincidence and quite easily seen when, as in Table 52, the actual PPset values are overlaid on those LD values. The PS fractals along the Horizontal and Vertical AXIS intercept at these points on the BIM. They form the NPS of the PTOP directly on the BIM! It is no longer hidden!
The PRIMES Sequence Fractal—3–5–7–11–13–17–19–23–29–31…—found along each AXIS, together forms the PPsets = EVENS = Goldbach Conjecture Proof.
The PPset TRAILS (Rows) ALWAYS INCREASE FASTER than the PRIME Gaps ensuring that the PPsets = EVENS = Goldbach Conjecture Proof.
One could say, that by definition, ALL EVENS are indeed formed from the OVERLAPPING PS Fractal TRAILS of PPsets = EVENS = Goldbach Conjecture Proof.
~~~
EVEN = 30
EVEN/2 = 15
7, 23 = 7 + 23 = 30
23  7 = 16
16/2 = 8
8 = STEPS to either side of 15 as:
8 + 15 = 23
15  8 = 7
11,19 = 11 + 19 = 30
19 –11 = 8
8/2 = 4
4 = STEPS to either side of 15 as:
4 + 15 = 19
15  4 = 11
13, 17 = 13 + 17 = 30
17  13 = 4
4/2 = 2
2 = STEPS to either side of 15 as:
2 + 15 = 17
15  2 = 13
EVEN = 30 = 3 PPsets = (7, 23), (11, 19), and (13, 17)
The 3 PPsets—(7, 23), (11, 19), and (13, 17)—result from the overlap of
3 PPset TRAILS: starting at (3, 17), (3, 19) and (3, 23) PS Rows.
The TRAIL overlap occurs under the
7, 11 and 13 PS Columns, respectively.
One can see that combining the P_{2} PS Row values with the P_{1} PS Column values gives the
PPsets = (7, 23), (11, 19), and (13, 17).
And, of course, the symmetrical STEPS between the P_{2}  P_{1} values follows as shown.
It also follows that taking half of the difference in sums of the P_{2}  P_{1} = the sums of STEPS.
[(17 + 19 + 23)  (7 + 11 + 13)] ÷ 2 = STEPS = 14 = 2 + 4 + 8 STEPS.
These STEPS are BOTH the number of steps from the PD to each P_{1} or P_{2}, and,
the number of steps from the P_{1} or P_{2} to the center Core Axis Value (EVEN/2) line — the line, or ROW, that connects the EVEN/2 on the AXIS to the square of the EVEN/2 on the PD.
EVEN/2 = 15 and Row 15 line intersects and terminates on the PD at (EVEN/2)2.
The NPS holds true for ALL PPsets that together define any given EVEN!
Summary of the math:
let E = EVENS, V= Value of grid cell
Like shining a light at a sculpture in the dark, each angle reveals a new “look,” yet it is just one sculpture. That’s what we are doing here. Hopefully, the BIM: Part III look will give us the simplest, most direct and informative view of just how EVERY EVEN IS MADE OF SYMMETRY AND PS FRACTAL ENDOWED PPsets IN THE FORM OF THEIR OVERLAID PPset TRAILS. Euler’s Strong Form of the Goldbach Conjecture simply falls out from this builtin relationship between the ISL, the PPTs and PRIMES as seen on the BIM.
In closing out the “PTOP (Periodic Table Of PRIMES) on the BIM: Parts I, II and III,” one may ask if that is it it?
The hidden PTOP (Part I) was visibly profiled as individual PPsets (Part II), and the two combined to be fully revealed in plain sight in Part III. But is that it?
Besides showing that the Primitive Pythagorean Triangles, PRIMES and the PPsets and their PPset TRAILS were ALL similarly located on specific Active Rows on the BIM—Rows identified by dividing the BIM Inner Grid cell values evenly by 24—all the EVENS were shown to be informed by the PPsets. These PPsets were themselves informed geometrically by the symmetry and PRIMES Sequence (PS)fractal nature of the PRIMES patterning in their very formation. The PS is a fractal that interacts dynamically with other PSfractal iterations that together form the PTOP.
But is that it? Is that all? Is there something more these PRIMES can tell us about the basic numbers that inform the Universe?
Most of the graphics on the PTOP on the BIM have focused on looking at just one side of the bilaterally symmetric BIM square—and often that “square” has been stretched into a rectangular format to accommodate both the everincreasing larger numbers and some just plain, simplified spaceshapes to present info and comments on.
But what about looking at the whole BIM? In doing so, we find that the AREAS enclosed by looking at both of the bilaterally symmetric sides of the BIM gives us some NEW INSIGHTS into just what is the relationship of the PRIMES to the ISL (Inverse Square Law) ?
In “PTOP (Periodic Table Of PRIMES) on the BIM: Part IV” we will be doing just that.
Here is an older graphic, now overlayed with the NEW PRIMESPPsets. Take all the Inner Grid cell values on the BIM that divide evenly by 24, color them YELLOWORANGE, and this is what you get. It showed that the Primitive Pythagorean Triangles (PPTs) always and exclusively fell on the Active ROWs—where the 1st Col. values were either YELLOW or ORANGE. Turns out the SAME is true for the PRIMES. They, too, only land on the Active ROWs. Note: some Active ROWs have neither PPT or PRIME, some have both, and some one but not the other.
Notice the pattern unfolding when the PRIMES Sequence (PS)Fractals—3–5–7–11–13–17–19–23–29…—from the horizontal and vertical AXIS combine on the BIM. Soon we will be looking at this pattern across the entire BIM grid.
Clarifications: The BIM is NOT a traditional multiplication table. NO WAY. While the central Prime Diagonal (PD) does represent the squares of the AXIS #s, ALL the Inner Grid (IG) values are determined by the difference (∆) in their horizontal and vertical PD intercept values, ei., 16 is PD 25  PD 9 (25–9=16).
That said, in the PPsets as presented in the PTOP on the BIM, we will go from this standard approach, to overlaying key matrix values with the coordinates of their AXIS, i.e., 9 is overlaid with AXIS 3, AXIS 3 or simply 3,3 and 16 is overlaid with 3,5 and so on. This forms a “SubMatrix” on the BIM of overlaid values that will be quite useful. Once you see how simple this overlay is, a great simplification overall will reveal itself.
One other note: the PPsets as 3,3–3,5–5,5–… form the EVENS by adding the set members, i.e. 3,3 = 3 + 3 = 6, and, a diagonal line from a given PPset back to the AXIS will reveal that EVEN.
AREAS — as Object Areas — are introduced here. Object Areas simply are the # of objects within an “area.” Here that AREA is a square on the BIM. Counting the # of objects within each AREA will give the number. More surprises await.
The PPset Areas are NOT traditional areas. Instead, they represent the number (#) of PPsets captured within the boundaries of the squared AXIS #s, i.e., the squared values of the PSfractal 3–5–7–11–13–17–19–23–29… are 9–25–49–121–169–289–361–529–841… but they are represented, as a “SubMatrix” on the BIM, as 3,3–5,5–7,7–11,11–13,13–17,17–19,19–23,23–29,29—…PPsets.
It is the # of these PPsets within the squared PSfractal AXIS value treated as area. One is really capturing the # of “objects” or Object AREAS.
This IS a different way of capturing the way the PRIMES as PPsets are part of the BIM ISL 1–4–9–16–25… pattern!
This way of capturing the # of objects is NOT limited to the PPsets, but rather is a natural characteristic of the BIM. Other nonprime or prime with nonprime mixtures of objects can be shown to also demonstrate this ISL pattern. It is simply the result of progressively adding 1,2,3 or more objects — in any interval — regularly to the squared AREA. It will naturally generate an ISL pattern. This points, again, to the larger role that the ISL relationship of numbers—as shown in the BIM—encompasses the expression of “other” number systems like the PPTs, PRIMES, and, any regular, progressive enlargement of the number of objects within a class of objects whose initial size = 1, singularly or as a group acting as one.
From the above, it may seem all too easy to just disregard the PPsets as simple “objects” not unlike those of any other random selection. But there is more here, than just that.
We have already seen in the BIM: Parts IIII that the PPsets are intimately related to and described by the ISL as seen on the BIM.
In Parts II & III, we saw directly how each and every PPset – set as factors on the BIM – related diagonally back to the AXIS EVEN number.
We also saw how the PPset TRAILS – that were extensively examined in Part I – are now easily seen as the PPset values occupying any Active Row on the BIM.
Now, in Part IV, we open the BIM up to see both symmetrical sides. In doing so, we see that underlining those PPset TRAILS Rows gives us the progressively larger squares: the Object AREAS of PPset TRAIL numbers.
That’s all in well, but significantly, those PPset TRAIL “objects” are not just any objects. They are a very specific collection of objects – specific PPsets that are part of specific PPset TRAILS, or groups, that ONLY exist and occupy their specific location on the BIM. A location that both allows them to individually relate back diagonally to the EVEN # on the AXIS, and, to relate ACROSS the Row as the collective set, or group, of ALL PPsets whose members consist of one constant PRIME and one PRIME Sequence PRIME, e.i., 3,11–5,11–7,11–11,11. The latter become the PPset TRAILS.
Ultimately, it is the PPset TRAILS on the Rows that interact and inform the diagonallyrelated EVENS. Their rate of growth – (increasing by 2 for each PS Object AREA) – far exceeds the Prime Gaps growth. This is built into the geometry of the BIM. It ensures that Euler’s Strong Form of the Goldbach Conjecture is fully satisfied!
The 1–4–9–16–25… ISL relationship of the Object AREAS: PPset TRAILS provides a definitive and intimate connection of the PRIMES to the ISL as seen on the BIM.
~~~~*Each square AREA = sum of the ODD#s leading to it, i.e. the ODD number summation series. Thus AREA = 9 = the sum of the ODDs 1 – 3 – 5. The next AREA = 16 = the sum of the ODDs 1 – 3 – 5 – 7. The next AREA = 25 = the sum of the ODDS 1 – 3 – 5 – 7 – 9, and so on. One can calculate the tally of any given square AREA by subtracting the two previous square AREAS to get their ODDs summation value, then add 2 to that and increase the larger square AREA by its sum, e.i., 25  16 = 9, add 9 + 2 = ll, add 11 + 25 = 36. The next square AREA is 36 and it contains 11 PPsets in its TRAIL.
As the TRAILS follow the ISL NPS — 1—4—9—–16—25—36—49—64—81—100—…—the actual number of unique PPsets within a trail = √(square), e.i., √36 = 6. There are 6 unique PPsets: Since the TRAIL falls on Row 17, those PPsets are: 3,17—5,17—7,17—11,17—13,17—17,17.
We know Object AREA 36 = 6*6, i.e., it is the 6th in the ISL NPS. Likewise, we can take any whole √ of a number and know where it is at in the ISL NPS. Another example, √100 = 10, so on Row 31, we have 19 PPsets, with 10 of them unique: 3,31—5,31—7,31—11,31—13,31—17,31—19,31—23,31—29,31—31,31. How do we know there are 19 PPsets? The ISL NPS is, of course, based on the squares of the numbers 1,2,3,…. The difference (∆) in the AREAS being the ODD # summation series: 1+3+5+7+…difference, or simply adding 2 to the previous ODD. Adding the √ of the current and previous AREAS will give the ∆, e.i. √100 + √ 81 = 10 + 9 = 17.
So once any Row in the ISL NPS is known, we can readily determine:
1. AXIS and squared AXIS — PD — numbers;
2. The ∆ between the current ISL NPS AREA and the one previous;
3. The total number of PPsets within the current AREA;
4. The net number of unique PPsets forming the PPset TRAIL;
5. The numbers, both total and unique, of ALL other AREAS before and after.For example, Row 31:
1. AXIS = 31 and squared AXIS = 961 on AREA = 100;
2. ∆ 100  81 = 19;
3. PPsets total = 19;
4. Net PPsets = (19–1)/2 = 10, or simply the √100;
5. Previous AREA = 100–19 = 81, 81 –17 = 64, 64 –15 = 49, … and Next AREA = 100 + 21 = 121, 121 + 23 = 144, 144 + 25 = 169,… or simply follow the 1^{2}—2^{2}—3^{2}—4^{2}—5^{2}—6^{2}—7^{2}—8^{2}—9^{2}—10^{2}—11^{2}—12^{2}—13^{2}—…The question of how do we know Row 31 is where the ISL NPS AREA = 100 lands? Or any other PPset TRAIL Object AREA?
The rigidly regular ISL NPS — 1—4—9—16—25—36—49—64—81—100—…, — itself built on 1^{2}—2^{2}—3^{2}—4^{2}—5^{2}—6^{2}—7^{2}—8^{2}—9^{2}—10^{2} —…, the ∆ being the ODD # summation series of 1+3+5+7+9 +11+.. with a constant ∆ of 2 — must be reconciled with the PRIMES Sequence (PS) with its variable Prime Gap.
What Row does ISL NPS AREA, say, PS 41, occupy?
We could think, well, it must be AREA 121, up 21 from AREA 100, but on what Row?
Wrong we would be, as the NEXT PS after 31 is 37, followed by 41. So Row 37 is AREA 121 and Row 41 is actually AREA 144, i.e., 10^{2}—11^{2}—12^{2}, respectively, for Rows 31—37—41.
Currently, one must know one to get the other and that one must be the PS Row. From that, one can calculate the ISL NPS AREAS and the number and configuration of ALL the PPsets within the PPset TRAIL.
If we know Row 41 — as PS 41 — we can calculate back to a known RowAREA like Row 31—AREA 100, count the number of PS jumps from 31 to 41 — jump one to 37, jump two to 41 — and know that we have moved two positions up the ISL NPS — 100 to 121 to 144 — giving 144 as the respective AREA for Row 41.
From (12^{2} = 144)  (11^{2} = 121) we see there is a ∆ of 23 between the AREA 144 and the previous AREA 121.
Since we know from the PS, the previous to 41 is 37, thus AREA 121 is on Row 37. The total number of PPsets equals the AREA difference from the previous AREA, thus for 144 it is 23 and for 121 it is 21. The net unique number of 12 PPsets within the TRAIL for Row 41—AREA 144 = √144 = 12. For the Row 37—AREA 121 it equals √121 = 11.
Let’s go back to Row 41—Area 144. What lies ahead on ISL NPS AREA 441. What Row is it on? Aside from simply looking at the BIM:
• The √441 = 21.
• There are 21 unique PPsets in the PPset TRAIL.
• There are 9 jump steps from AREA 144 to AREA 441 — 144—>169—>196—>225—>256—>289—>324—>361—>400—>441, corresponding to 12^{2}—>13^{2}—>14^{2}—>15^{2}—>16^{2}—>17^{2}—>18^{2}—>19^{2}—>20^{2}—>21^{2}.
• There must be 9 corresponding jump steps for the PS: 41—>43—>47—>53—>59—>61—>67—>71—>73—>79.
• AREA 441 = Row 79
• Take 441  400 = 41.
• There are 41 total PPsets in the AREA 441.
• Another way to get the total 41 is to add 20 + 21.
• All the Row—AREAS between Row 41—AREA 144 and Row 79—AREA 441 are easily determined in a like manner.One more example: Starting with PS 113, what ISL NPS AREA does it occupy and what are the numbers?
• We could start from scratch with PS 3 and count the jump steps to 113 = 28.
• Now match that with 28 jump steps from ISL NPS 1 + 28 = 29.
• 29^{2} = 841.
• There are 29 unique PPsets in the PPset TRAIL on Row 113—AREA 841.
• Take (29^{2} = 841)  (28^{2} = 784) = 57.
• There 57 total PPsets in the AREA 841.
• Another way to get the total 57 is to add 28 + 29.
• All the Row—AREAS between Row 3—AREA 1 and Row 113—AREA 841 are easily determined in a like manner.There are several ways to calculate the PPsets and their associated ISLNPS AREAS. A tip to keep in mind: as the Object AREAS increase as 1^{2},2^{2},3^{2},...the ∆ between them being the ODD # Summation series — 3579... — the ∆ of the ∆, if you will, is 2. ALL total PPsets within an AREA will be ODD. Another visual way to look at this: as each AREA progressively enlarges, one new PPset is added to the Horizontal and one new PPset is added to the Vertical Row/Col, the sum equals 2, so starting with 1, 1 + 2 = 3, 3 + 2 = 5, 5 + 2 = 7,... The unique PPsets may be either.
With optimum computing power, one could determine ALL PS Row—ISL NPS AREAs up to and including the largest PRIME.
One can use the NOPRIMES (NP) = 6yx ± y (covered extensively in PRIMES vs NOPRIMES, 2019) to determine ALL the nonPRIMES between any run of numbers.Example: what are the NPs from 3–59?Consulting NP=6yx+y_with3^x.jpg, you could simply pick out the NPs.Calculating the NPs:NP = 6yx ± y. Let x = 1,2,3,… and y = ODD # ≥3, and add 3 exponential, 3^{x}:1. let x = 1, y = 32. NP = 6yx ± y = 6(3)(1)±3 = 18 ± 3 = 15, 213. let x =1, y = 54. NP = 6yx ± y = 6(5)(1)±5 = 30 ± 5 = 25, 355. let x =1, y = 76. NP = 6yx ± y = 6(7)(1)±7 = 42 ± 7 = 35, 497. let x =1, y = 98. NP = 6yx ± y = 6(9)(1)±9 = 54 ± 9 = 45, 639. let x =1, y = 1110. NP = 6yx ± y = 6(11)(1)±1 = 66 ± 11 = 55, 7711. let x =2 y = 312. NP = 6yx ± y = 6(3)(2)±3 = 36 ± 3 = 33, 3913. let x =3 y = 314. NP = 6yx ± y = 6(3)(3)±3 = 54 ± 3 = 51, 5715. add 3 exponential, 3^{x} = 3^{2} = 9, 3^{3} = 2716. NPs 3—59 = 9,15,21,25,27,33,35,39,45,49,51,55, and 57.Knowing the NPs allows one to make the jumps from one known PS and its associated ISL NPS AREA to another.The value of determining the net unique number of PPsets within a TRAIL is that each TRAIL member will sequentially contribute to the pool of PPsets whose members add up to form a given EVEN. This constant and regular growth of the PPsets outpaces the rate of growth of the Prime Gaps ensuring the Goldbach Conjecture is always fulfilled.
Another valuable outcome of determining ALL the unique PPsets within ALL the encompassing AREAS is that ALL PRIMES are accounted for. Should a new and/or larger PRIME be discovered, its PS Row—ISL NPS AREA can be calculated and the net unique PPsets determined. There should NOT be any holes!
From PRIMES vs NOPRIMES we know that we can testvalidate that any ODD (≥5) is PRIME by seeing if the difference between its squared value and that of any other squared PRIME value is an even multiple of 24, e.i. 7^{2}  5^{2} = 24, 31^{2}  11^{2} = 840 and 840/24=35 and so on.
Note that it is the PS PRIMES from the AXIS on the BIM that is squared on the PD that follows this PRIMES validation, not the ISL NPS Object AREAS, per se, i.e., the 31^{2}  11^{2} example above works as expected, but the overlaid ISL NPS Object AREAS of 100 and 16 values, respectively, do not work this way.
The latter only works when the values are from squared PS values like 121 and 49, i.e., 121 = 11^{2} and 49 = 7^{2}, the ∆ being 121  49 = 72 and 72/24 = 3.
SYMMETRY, STEPS, EVENS, EVENS/2, PPsets, PRIME SEQUENCE FRACTALS, ISOSCELES & EQUILATERAL TRIANGLES, INVERSE SQUARE LAW, PYTHAGOREAN TRIPLES; ALL ON THE BIM.
In the first publication of the BIM (BBSISL Matrix — originally referred to as the Brooks (Base) Square—Inverse Square Law Matrix) in MathspeedST (2010), the work was divided in to two main sections:
In MathspeedST, the PRIMES were found to be stealthily hidden within the BIM, revealing there presence ONLY when the BIM itself had its ODD #s 1st Parallel Diagonal modified to EVENS by the addition of 1 to each ODD.
This resulted in the original PTOP (Periodic Table Of PRIMES) on the BIM. It was refined by pulling the values off the BIM and making a separate PTOP. All this became the subject of “PTOP (Periodic Table Of PRIMES) & the Goldbach Conjecture” ebook and white paper (2019). It is also referred to as BIM: Part I. The PPsets and their TRAILS were introduced.
Not good enough just by itself, further research found that, indeed, each and every PRIME, when treated as part of a PPset, could be found and individually profiled DIRECTLY on the BIM. BIM: Part II was formed.
Still not good enough, plotting all the Lower Diagonal P_{2} PRIMES on a table and then replotting those results back onto the BIM opened up a new vista. By substituting the AXIS values for each of those P_{2} PRIMES from the table, one has now formed a new SubMatrix of the BIM with ALL the PPsets in place.
One immediately sees that they occupy EXACTLY the same footprint on the BIM as do the individual PRIMES (and the PPTs): namely, falling only on the BIM/24 Active Rows.
In the case of the PPsets, one can also see that this is a direct consequence of locating the PPsets as the grid space resulting from the intersection of the PRIMES Sequence (PS) of the Horizontal AXIS with the PS of the Vertical AXIS. This becomes the basis for the BIM: Part III.
In BIM: Part IIII, we have mostly used just half of the bilaterally symmetric BIM to reveal and describe the geometry.
In BIM: Part IV, we now look at the whole BIM with the PPset SubMatrix in place.
What we find is that by treating these PPsets as objects and counting them within progressively larger square areas — forming what is called “Object AREAS” — a pure ISL NPS is found. Yes, as the PSFractal series of each AXIS joins to form the PPsets, their actual numbers — as PPset TRAILS — progressively grows and sums up to quantities with Object AREAS that directly mirror the fundamental ISL NPS:
1—4—9—16—25–… And while this relationship of the PRIMES to the ISL can not predict the next largest PRIME, it can ABSOLUTELY account for each and every PRIME at, and below, any given PRIME, regardless of size.
While Euler’s Strong Form of the Goldbach Conjecture is proved along the way on this journey, the real significance is what we have seen unfolding in BIM: Parts IIV. The role of Symmetry and Fractal underlie everything about the PRIMES.
The PRIMES on the BIM is all about how the fractal nature of the PS becomes expressed as symmetry on the BIM as isosceles and equilateral triangles, forming the PPsets that ultimately form ALL the EVEN numbers!
What is marvelous, incredible, mindblowing in every way, is that this same symmetry—fractal—isosceles/equilateral triangle relationship is found — indeed, is part and parcel — throughout the BIM.
The ISL seems to reflect the most basic and fundamental relationship(s) between quantity and the numbers that account for it.
ISL—> BIM
BIM—> PPT
BIM—> PRIMES
BIM—> PPset Object AREAS
BIM—> other Object AREAS
As in showing that ALL EVENS (≥6) can be formed from the sum of two ODD PRIMES (Euler’s Strong Form of the Goldbach Conjecture), one would like to see — in the context of the PRIMES being the fundamental “atoms” — the PRIMES being the source of the ISL and everything thereafter!
PRIMES—> ISL—>BIM—>PPT & PRIMES patterns on the BIM, etc, etc.
We are not there yet.
PRIMES Index
PTOP on the BIM Part II from Reginald Brooks on Vimeo.
PRIMES on the BIM Part III from Reginald Brooks on Vimeo.
MathspeedST: TPISC Media Center
Artist Link in iTunes Apple Books Store: Reginald Brooks
Back to Part I of the BIMGoldbach_Conjecture.
Back to Part II of the BIMGoldbach_Conjecture.
Back to Part III of the BIMGoldbach_Conjecture.
Reginald Brooks
Brooks Design
Portland, OR
brooksdesignps.net
Unexpected Beauty in Primes  Cantor’s Paradise  Medium
https://medium.com/cantorsparadise/unexpectedbeautyinprimesb347fe0511b2
File:Goldbach conjecture with Excel.pdf  Wikimedia Commons
https://commons.wikimedia.org/wiki/File:Goldbach_conjecture_with_Excel.pdf
Goldbach's conjecture  letsprovegoldbach Blog
http://letsprovegoldbach.altervista.org/lacongetturadigoldbach/
5  The Goldbach's conjecture with Excel  By reading Archimedes
https://sites.google.com/site/byreadingarchimedes/thegoldbachsconjecturewithexcel
Google Translate of the original Italian webpage above
A table of prime counts pi(x) to 1e16
http://www.trnicely.net/pi/pix_0000.htm Prime Number Theorem
numbers.computation.free.fr/Constants/Primes/pixtable.html
http://numbers.computation.free.fr/Constants/Primes/pixtable.html
PTOP: Periodic Table Of PRIMES & theGoldbach ConjectureINTRODUCTIONSTATEMENT: Layout & EssentialsThere are five sets of data to consider:FINDINGS & PROOFSUMMARY and CONCLUSIONDATA: Images and TablesDATA: Images and TablesREFERENCES
MathspeedST: TPISC Media Center
Back to Part I of the BIMGoldbach_Conjecture.
Back to Part II of the BIMGoldbach_Conjecture.
Back to Part III of the BIMGoldbach_Conjecture.
BACK: > Simple Path BIM to PRIMES on a separate White Paper BACK: > PRIMES vs NOPRIMES on a separate White Paper BACK: > TPISC_IV: Details_BIM+PTs+PRIMES on a separate White Paper BACK: > PRIME GAPS on a separate White Paper (This work.) BACK: > PeriodicTableOfPrimes(PTOP)_GoldbachConjecture on a separate White Paper BACK: > Make the PTOP with Fractals on a separate White PaperTPISC, The PythagoreanInverse Square Connection, has evolved into a connection with the PRIMES (TPISCP).
REFERENCES:
https://primes.utm.edu/notes/faq/six.html
“Are all primes (past 2 and 3) of the forms 6n+1 and 6n1?
"Perhaps the most rediscovered result about primes numbers is the following:
"I found that every prime number over 3 lies next to a number divisible by six. Using Matlab with the help of a friend, we wrote a program to test this theory and found that at least within the first 1,000,000 primes this holds true.
"Checking a million primes is certainly energetic, but it is not necessary (and just looking at examples can be misleading in mathematics). Here is how to prove your observation: take any integer n greater than 3, and divide it by 6. That is, write
n = 6q + r
where q is a nonnegative integer and the remainder r is one of 0, 1, 2, 3, 4, or 5.
"So if n is prime, then the remainder r is either
"Remember that being one more or less than a multiple of six does not make a number prime. We have only shown that all primes other than 2 and 3 (which divide 6) have this form.”
From Another prime page by Chris K. Caldwell caldwell@utm.edu
~~~
"Euler's 6n+1 Theorem
"Every Prime of the form can be written in the form ."
https://archive.lib.msu.edu/crcmath/math/math/e/e282.htm
~~~
~~~
http://www2.mae.ufl.edu/~uhk/sixnplusone.pdf.
~~~
~~~
https://en.m.wikipedia.org/wiki/Formula_for_primes
~_{~}
http://www2.mae.ufledu/~uhk/PRIMETEST.pdf **
http://www2.mae.ufl.edu/~uhk/NUMBER FRACTION.pdf
~~~
http://mathworld.wolfram.com/Eulers6nPlus1Theorem.html
~_{~}
http://eulerarchive.maa.org See Number Theory Section
E744en.pdf **!!!!!!*!!!!! MOST IMPORTANT READ
http://eulerarchive.maa.org/docs/translations/E744en.pdf
NOTE: These article links from the Euler Archive/Number Theory/ INDEX are IMPORTANT READS for PRIMES & Squares. You can link directly from the Index as well.
http://eulerarchive.maa.org/pages/E026.html
http://eulerarchive.maa.org/pages/E054.html
http://eulerarchive.maa.org/pages/E134.html
http://eulerarchive.maa.org/pages/E175.html
http://eulerarchive.maa.org/pages/E191.html
http://eulerarchive.maa.org/pages/E241.html
http://eulerarchive.maa.org/pages/E242.html
http://eulerarchive.maa.org/pages/E243.html
http://eulerarchive.maa.org/pages/E244.html
http://eulerarchive.maa.org/pages/E256.html
http://eulerarchive.maa.org/pages/E262.html
http://eulerarchive.maa.org/pages/E270.html
http://eulerarchive.maa.org/pages/E279.html
http://eulerarchive.maa.org/pages/E283.html
http://eulerarchive.maa.org/pages/E369.html
http://eulerarchive.maa.org/pages/E394.html
http://eulerarchive.maa.org/pages/E405.html
http://eulerarchive.maa.org/pages/E445.html
http://eulerarchive.maa.org/pages/E449.html
http://eulerarchive.maa.org/pages/E467.html
http://eulerarchive.maa.org/pages/E523.html
http://eulerarchive.maa.org/pages/E542.html
http://eulerarchive.maa.org/pages/E552.html
http://eulerarchive.maa.org/pages/E554.html
http://eulerarchive.maa.org/pages/E564.html
http://eulerarchive.maa.org/pages/E596.html
http://eulerarchive.maa.org/pages/E610.html
http://eulerarchive.maa.org/pages/E699.html
http://eulerarchive.maa.org/pages/E708.html
http://eulerarchive.maa.org/pages/E715.html
http://eulerarchive.maa.org/pages/E718.html
http://eulerarchive.maa.org/pages/E719.html
744  On divisors of numbers of the form mxx + nyy 

http://eulerarchive.maa.org original 744 Euler article
https://www.britannica.com/science/Fermatstheorem
http://mathworld.wolfram.com/FermatsLittleTheorem.html
http://mathworld.wolfram.com/EulersTotientTheorem.html
http://mathworld.wolfram.com/TotientFunction.html
https://www.geeksforgeeks.org/fermatslittletheorem/
https://www.geeksforgeeks.org/eulerstotientfunction/
http://mathworld.wolfram.com/Eulers6nPlus1Theorem.html
https://brilliant.org/wiki/fermatslittletheorem/
https://brilliant.org/wiki/eulerstheorem/
https://primes.utm.edu/notes/proofs/FermatsLittleTheorem.html
https://primes.utm.edu/notes/conjectures/ Conjectures
https://primes.utm.edu/notes/faq/ FAQ Index
https://primes.utm.edu/notes/faq/one.html One
https://primes.utm.edu/notes/faq/six.html Six —>Important Page
http://unsolvedproblems.org/index.htm Unsolved Problems Index
https://en.m.wikipedia.org/wiki/Proofs_of_Fermat%27s_little_theorem
https://en.m.wikipedia.org/wiki/Fermat%27s_little_theorem
https://en.m.wikipedia.org/wiki/Euler%27s_theorem
~~NEW REFERENCE LIST 11082018_{~}Fermat’s Little Theorem ++++Euler
https://web.math.princeton.edu/swim/SWIM%202010/ShiXie%20Presentation%20SWIM%202010.pdf
http://mathworld.wolfram.com/Fermats4nPlus1Theorem.html
http://nonagon.org/ExLibris/eulerprovesfermatstheoremsumtwosquares
https://storyofmathematics.com/17th_fermat.html
https://en.wikipedia.org/wiki/Prime_number
http://mathworld.wolfram.com/PrimeNumberTheorem.html
https://www.britannica.com/science/primenumbertheorem
https://goingpostal.com/2018/02/fermatslittletheorem/
https://ibmathsresources.com/2014/03/15/fermatstheoremonthesumoftwosquares/
https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares *
https://en.wikipedia.org/wiki/Pythagorean_prime *
https://oeis.org/A002144 Pythagorean Primes of the form 4n + 1
https://oeis.org/A002144/list List of Pythagorean Primes of the form 4n + 1
http://nonagon.org/ExLibris/fermatsumtwosquarescalculator *
https://www.britannica.com/science/Mersenneprime Mersenne Primes
https://primes.utm.edu/mersenne/
https://en.wikipedia.org/wiki/Mersenne_prime#About_Mersenne_primes
http://mathworld.wolfram.com/MersenneNumber.html
http://mathworld.wolfram.com/MersennePrime.html
https://www.encyclopediaofmath.org/index.php/Mersenne_number
https://www.mersenne.org GIMPS
https://www.mersenne.org/primes/
https://primes.utm.edu/lists/small/millions/ Primes LIST by section
http://compoasso.free.fr/primelistweb/page/prime/liste_online_en.php Primes LIST by section
http://primerecords.dk/primegaps/gaps20.htm Prime Gaps
http://primerecords.dk/primegaps/maximal.htm
https://en.wikipedia.org/wiki/Prime_gap
https://primes.utm.edu/notes/GapsTable.html
https://en.wikipedia.org/wiki/Prime_ktuple
https://primes.utm.edu/glossary/page.php?sort=ktuple
https://en.wikipedia.org/wiki/Twin_prime
http://mathworld.wolfram.com/TwinPrimes.html
https://primes.utm.edu/notes/faq/
~~
https://oeis.org/wiki/Pythagorean_primes Pythagorean vs nonPythagorean Primes
http://oeis.org/A002145 nonPythagorean Primes of the form 4n + 3
http://oeis.org/A002144 Pythagorean Primes of the form 4n + 1
https://en.wikipedia.org/wiki/Double_Mersenne_number Double Mersenne Primes
https://primes.utm.edu/mersenne/index.html#unknown Questions remain
~~~~~~~~~~NEW REFERENCE LIST 12132018~~~videos——must watch~~~~
See: Animated math: videos by 3Blue1Brown (Grant Sanderson, et al.) https://www.3blue1brown.com/videos/
Feynman's Lost Lecture (ft. 3Blue1Brown) https://www.youtube.com/watch?v=xdIjYBtnvZU
But WHY is a sphere's surface area four times its shadow?: video by 3Blue1Brown https://youtu.be/GNcFjFmqEc8
Why is pi here? And why is it squared? A geometric answer to the Basel problem: video by 3Blue1Brown https://www.youtube.com/watch?v=do3eB9sfls&frags=pl%2Cwn
Pi hiding in prime regularities: video by 3Blue1Brown https://youtu.be/NaL_Cb42WyY
Visualizing the Riemann hypothesis and analytic continuation: video by 3Blue1Brown https://youtu.be/sD0NjbwqlYw
All possible pythagorean triples, visualized: video by 3Blue1Brown https://www.youtube.com/watch?v=QJYmyhnaaek
((((Table VI series is really the whole evolution of the BIMrow11000+sheets/Primes_sheets+PF, etc here and appendix.)))
referenced as Table VI a
referenced as Table VI b
referenced as Table VI b
Table VII.
Thomas Young established the wave nature of light in the Young DoubleSlit Experiment.
Max Planck established the quantum nature of light energy mathematically (E=hⱱ).
Albert Einstein established that ALL energy is quantized (E=hⱱ), energy and matter are interconvertible ((E=mc^{2}), and that the speed of light is always constant (c=ƴⱱ) — and later, the curvature of ST as the driving dynamic informing gravitation (General Relativity).
Niels Bohr established the quantization of the atomic orbitals within the atom (Bohr Model of the Atom). Along with Heisenberg, the leading voices in the Copenhagen Interpretation of Quantum Mechanics.
Louis de Broglie established that all objects express the waveparticle duality as defined by its momentum being its energy per wavelength (p=h/ƴ).
Werner Heisenberg established the Uncertainty Principle that reveals the limitation of precisely and simultaneously defining both an objects position and its momentum (∆x∆p≥h/2𝛑).
Arnold Schrödinger established the timedependent Schrödinger (Quantum) Wave Equation.
Max Born established the statistical basis of the same Quantum Wave function.
Wolfgang Pauli established the Pauli Exclusion Principle that disallows any quantum particle to occupy the same quantum state (parameters) within the same ST.
Paul Dirac established the Dirac Equation, a relativistic generalization of the Schrödinger equation
wave equation for fermionic spin ½ particles, bringing together the two pillars of special relativity (3 dimensions of space+1 dimension of time, and the constancy of the speed of light, c=ƴⱱ) and quantum mechanics (h/2𝛑=E/v) leading to Quantum Field Theory and the Standard Model.
Quantum Entanglement , initially raised in opposition in 1935 in the Einstein, Podolsky, Rosen paper (EPD Paradox) and papers by Schrödinger, was brought to the forefront of debate in 1964 when John Stewart Bell's Inequality Theorem ruled out the "local hidden variables" of the opposition. Since then the quantum entanglement — i.e., particles bound in entanglement cannot be factored out into separate states that would explain their entanglement, the particles act together as a whole, not the sum or product of their parts — has been successfully tested and appears to be profoundly consistent with the Quantum Field theory. It is in direct opposition to Einstein's nothing — no information — can travel faster than the speed of light. And yet, here we have proofs stacking up that show, for example, two entangled particles with equal but opposite spin will simultaneously settle into one of the other spin directions opposite its entangled partner instantly once its partner's spin is determined — faster than the speed of light. Einstein's "local hidden variables" may not be completely ruled out if, indeed, one reveals a different sort of underlying geometric order that sidesteps being ruled out by the Inequality Theorem — a geometric order that precedes the very formation of ST itself (BIMBIMtree).
DoubleSlit Experiment:
In essence, even a single photon or electron passing through a doubleslit will display interference, collapsing upon any interacting measurement upon the initiating wave particle.
Fig.73. DoubleSlit Experiment
Quantum Entanglement:
In essence, any two (and possibly more) waveparticle ST units born of and thus assigned the same, equal but opposite quantum number state, i.e. spin angular momentum, will maintain and function as a single waveparticle system having a single quantum number state regardless of distance or time separation, in the absence of external negating influences. Transference is possible.
If c, the speed of light, is fundamentally buit into the formation of ST itself — moving ƴ distance with each pulsepropagating ⱱ frequency — as described in LightspeedST — and as combined with MathspeedST, i.e. through the BIM informing that same ST pulsepropagating unit — all potential ST information, as to density and disposition of space per time, is inherently known!
The BIM itself is defined as a Universal Relationship. Its numerical value of the quantity of parts (the value in each grid cell), in relation to its whole (the square values of the PD — that, by the way, directly inform the parts) is instantly and ubiquitously known for every value throughout BIM as extended to infinity.
There is no propagation of information faster than c because all its information is simultaneously known. We thus have a basis for Quantum Entanglement: two or more particles born of the same quantum number state wholy remain connected regardless of distance of separation precisely because they are informed by the SAME BIMderived ST pulse. No information is exchanged — no violation of fasterthanlight signals because both (all) particles have the SAME BIM information map informing their quantum state parameters at any and all ST unit pulses (ⱱ).
The Quantum Entanglement example most referenced is that of spin, a vector momentum parameter.
Two entangled particles (e.i., two photons or an electronpositron pair) are simultaneously born, each existing with both possible equal but opposite spin states.
Upon any distance (or time) separation, determining the spin quantum state of one particle — collapsing its wave potential of both into that of one polemic — immediately determines the other as equal but opposite.
Equal and opposite what?
The spin quantum state is a vector whose magnitude and expression is dependent on orientation (direction).
So, for whatever context that vector exists within, a polemic is established, i.e., +/, ↑/↓, yes/no, that inherently establishes a positive and negative state (Dirac Equation) — e.i., matter/antimatter, +charge/charge, spin+/spin.
The Quantum Entanglement State (QES) is just such a context — a container, if you will — holding the full potential of either/or vector states within its potential wave form.
Measurement (Heisenberg Uncertainty Principle) —that is interacting with such potential state — necessarily collapses the potential wavestate into a definitive, particulatelike state — call it +/, ↑/↓, yes/no, or Even/Odd.
In effect, one has induced the vector state. In doing so, the other entangled member has its vector state induced to that equal, but opposite polemic value to its partner.
Now, let the QES=BIM. Let collapsing the wavepotential (the entire BIM) = √BIM. Since BIM is built on squaring the 1,2,3… Axis numbers to give the PD — and all the Inner Grid cell values as its children — taking the √BIM is akin to collapsing BIM back to the equal and symmetrical opposite Axial numbers along the horizontal and vertical Axises.
Alternatively, the BIM may indeed project into the imaginary number space, i, giving the equal but opposite value a more hidden meaning.
Yet another alternative — and within the context of the current mathematical space the BIM occupies — one may induce the ubiquitous and infinity expanding Pythagorean Triples to provide the necessary polemical expression of a vector from the multipotential of its waveform. In this case — and exactly in tune with the BIM layout — there is a 90°right angle relationship between the sides. In the BIM it is within a square, within the PTs it is within a rectangle (formed from the two orientations of the triangle). Both have a symmetry involved. The BIM has an absolute, bilateral mirrorreflection symmetry. The latter an algebraic symmetry in that various whole integer combinations of a^{2} + b^{2} =c^{2}.
In fact, there are, as we mentioned in the Intro, 4 geometric manifestations of each algebraically described PT. Two for each side of the bilaterally symmetric BIM, and two orientations (180° flips along the row/column). Perhaps the polemic we are looking for is simply the wavecollapse selection of one of the 4 possible orientations of the PT, with its opposite simply its mirror reflection on the other side of the BIM.
Fig 74. The Quantum Entanglement Conjecture. Every PT has 4 potential presentations: each has a mirror symmetry, and because BIM has itself reflective, mirror bilateral symmetry, there are two more.
The Quantum Entanglement Conjecture:
Every PT has 4 potential presentations: each has a mirror symmetry, and because BIM is internally itself reflective, mirror bilateral symmetry, there are two more.
For simplicity, the TPISC painting image (Fig.74.) has been overlaid with three sets of PTs — each having 4 potential presentations. Exactly the same is true for the strictly Primitive PTs within the ToPPT.
As ST formation is informed most certainly by the ISL (BIM), the PTs, and most especially the Primitive PTs within the ToPPT, are viable candidates for laying down a bit of slightly asymmetrical bilateral symmetry framework embedded within the perfect rotational and bilateral symmetry the perfect squares and circles generated by the BIM.
While the BIM itself is composed of alternating ODD and EVEN whole integer numbers, the PTs, especially the PPTs, pick these ODD and EVEN numbers out as part of their natural structure. Every PPT has one short side made of an ODD number and the other is always the opposite — an EVEN number.
If one pictures the 4 potential presentations of the PT as existing as the wavepotential state of a forming particle, the naturally builtin polemics of these ubiquitous triangles comes into play: selecting to "measure" any one parameter as say, the spin ½ momentum vector, could collapse the wave function such that the other, opposite side of the polemic — which is always known as part of the Entangled Quantum State — could only express that opposite value.
It could be something as simple as solving the Pythagorean Theorem. If you measure (solve) for one short side — an let's say it is an EVEN whole integer number — you instantaneously know the other side is ODD. You also can know its value if you know the hypotenuse.
If you measure (solve) for one short side — an let's say it is an ODD whole integer number — you instantaneously know the other side is EVEN. You also can know its value if you know the hypotenuse.
The geometry of ST formation generates descriptive parameters referred to as quantum numbers or their quantum number state — unique identifiers for each and every subatomic particle. These waveparticle ST units exhibit unusual and nonintuitive behaviors that reveal a connection to each other below the physical reality we observe. The Conjecture is that that underlying connection is the BIM and BIMtree geometry that informs the creation and disposition of ALL ST — including their quantum number state identifiers.
Back to the DoubleSlit Experiment Conjecture:
In the simplest terms, the ST formation of the waveparticle pulsepropagates into and out of existence according to its energy.
Each pulse expands form its own singularity outward via the BIM.
The STBIM expands outward in all directions as a sphere, despite its forward motion as a vector. Thus it travels through both slits and interferes with itself on the other side.
It is the BIM geometry information that is traveling through both slits, i.e. it doesn’t actually travel, rather it is known.
If you block one of the slits or otherwise collapse the geometry wave by measuring or interacting with it, no interference occurs.
How does it work?
There are several possibilities.
It may be as simple as positive interference patterns may appear when the same Square/Circle WIN values of the BIM resonate and negative when they do not — giving the classical lightdarklightdark pattern.
Or, it may be the same resonance interference positive/negative superpositions for the PTs. As ST expands, a positive interference pattern would result when two (or more) 345 PPTs harmoniously arrive at the same ST location on the screen. The same for the 51213, 81517, etc. In either case, if you block one slit or collapse the expanding BIM geometry before entering the slit, interference will also collapse.
A third, and perhaps more relevant possibility includes the interaction with the slit material itself. Remember, ALL ST pulsepropagates into and out of existence. All matter and all energy, including the material that the slits are made of, pulse. When only a single slit is open, the BIM geometry of the incoming waveparticle interacting with the slit ST does NOT generate an interference pattern.
However, when two slits are open, the BIMST geometry of the slit material favorably interacts with the BIMST geometry of the waveparticle to generate interference!
One cannot ignore the contribution of the slit BIMST geometry in forming the net pattern. Having more than a single waveparticle ST unit passing through the slit(s) is simply magnifying the fundamental interaction between the slit material and waveparticle BIMST geometries!
As Heisenberg so profoundly pointed out, all contributions to any observationmeasurement scenario must be accounted for.
Together, the DoubleSlit Experiment and Quantum Entanglement Conjecture (DSEQEC) are but two sides of the same coin. Details to follow in TPISC IV and V.
The absolut key notion in the DSEQEC is that the geometry of the BIM and BIMtree both simultaneously and automatically precede and inform the subsequent ST unit expression, and, that this geometry is ubiquitous — expanding to infinity!
Since the initial DSEQEC Conjecture, presented as quoted above in TPISC III: Clarity (2017), a barrage of new articles confirming Quantum Entanglement have appeared. Perhaps the most intriguing, and relevant, has to be the series of new articles relating back to earlier work by Leonard Susskind, Juan Maldacena and Mark Van Raamsdonk — some of which are referenced below. In a brief, open letter to their ideas, I wrote:
Open letter addressing the "ER=EPR" works of Leonard Susskind, Juan Maldacena, and Mark Van Raamsdonk, …
The ER = EPR conjecture is a very interesting insight into the joining of quantum mechanics with quantum gravity through the geometry of spacetime.
What is needed is the other, complementary side that seals the deal: any entanglement of spacetime (ST) and ST units/particles, i.e. photons, gravitons, Higgs Bosons,… must reckon with:
 ST itself is formed in such a way that it obeys the Inverse Square Law (ISL), is always consistent with the Conservation of Energy, and its very formation must be one and the same with the formation of its progeny ST units and the builtin velocity of light, c;
 ST itself, and thus its ST progeny, must be conserved;
 ST itself, and thus it ST progeny, must be fully accountable for all intricacies of the DoubleSlit Experiment.
One can satisfy all three constraints by allowing ST to be pulsepropagated from its singularity (S) into and back out of our ordinary view, forming the photon with its builtin velocity, c, with each pulse. A graviton is nothing more than the positive interference of two similarspin photons, while the Higgs Boson, in the pure state, is two photons with oppositespins interfering. The Higgs, of course, gains mass as it decouples into ST units with 1/2 spin, be it quarks or electronsneutrinos, or their composites. Note that one could start with the Higgs and decouple, first to the photon and thereafter to its progeny. (LightspeedST)
Key to the ordered formation of ST is the ISL. A fully, selfconsistent matrix grid array numerically depicting the ISL in all its glory is easily formed by filling in all the Inner Grid cell values as the difference between squares of the x and yaxis whole integer numbers (MathspeedST). It’s called the BIM, or BBSISL Matrix.
Here’s the thing. Amongst a myriad of other connections, there exist an intimate connection between three number systems:
 The ISL as laid out in the BIM;
 PTs — and most especially PPTs — as laid out on the BIM;
 The PRIME numbers — PRIMES — as laid out on the BIM.
The BIM is the FIXED GRID numerical array of the ISL.
The PTs are the Pythagorean Triples, and more specifically, the PPTs are the Primitive Pythagorean Triples (parent, nonreproducible). All PTs — primitive and nonprimitive — are found on the BIM ([TPISC I: Basics, TPISC II: Advanced, TPISC III: Clarity [Tree of Primitive Pythagorean Triples, ToPPT], and TPISC IV: Details](http://www.brooksdesignps.net/Reginald_Brooks/Code/Html/MSST/MSSTTPISC_resources/MSSTTPISC_resources.html)).
Amongst its vast array of interconnecting Number Pattern Sequences (NPS) — i.e., number systems — two such systems stick out and do so in such an overtly visual — as well as mathematical — way that their connection to each other is more than implied.
You see, both the PPTs and PRIMES strictly align themselves on the SAME paths within the BIM.
But we are not here to discuss the PRIMES, just the BIM and the PPTs. And why is that? How do they shed light on the DoubleSlit Experiment, Quantum Entanglement and the whole Geometry of Spacetime scenario that, I believe, underlies ER = EPR?
Briefly — very briefly — the PTs introduce rectangles, ovals and nonisosceles right triangles as articulation nodes within the architectural framework of the pure squares, circles and isosceles triangles that the pure ISL builds, as clearly revealed — through a highly visible algebraic geometry — on the BIM.
Visually, right on the BIM grid, one can see — literally draw — the PTs on each active Row (or Column since the BIM is bilaterally symmetric).
Now, two points of emphasis before carrying on: both the BIM and the embedded PTs are fixed, nonvariable mathematical entities, and, they both extend to infinity. Every ST unit pulse is informed by this grid information at the start (from its S), and is known throughout its pulsepropagation history. It is “information” that exists with no need to travel at or faster than the speed of light. If one knows “any” value and location on the grid one inherently knows “all” the information that defines that history. Not only that, but the BIM and PTs are so intimately linked throughout the grid that one can be used to define the other.
Because the BIM is bilaterally symmetrical, and, because every PT can be inherently represented on the BIM in both “left” and “right” hand depictions — e.i. on an active Row, a 345 PPT can be depicted equally with the hypotenuse along the axis Row (lower left BIM) and its short side to the “left” and to the “right” and if you turn to the Columns side of the BIM (upper right), again the “left” and “right” versions of the same 345 PPT are depicted. What is even more amazing, the Area of the PPT x 4 is always present on that same Row (or Column) and is integral to defining the relationships to the next PPT within the ToPPTs. There is so much interconnectedness between the BIM, the PTs and the next PPTs built right in to the grid that one feels a blueprint for the fractalbased holographic ST universe is right before one’s eyes!
Back to the point: let the 4 iterations of the PT — any PT — structurally represent the quantum state parameter of spin (spin angular momentum). When one measures the vector spin value and orientation, one is measuring one of the 4 PT iterations. Knowing one value, one always instantaneously knows the value of the “other” entangled ST unit coherently existing as part of a complementary pair sharing the same total quantum state. If your axis of measurement defines “left” (or ) on one ST unit moving in one direction, you will automatically register the equal but opposite “right” (or +) measurement on the other entangled ST unit moving in the opposite direction. No information is transmitted or signaled. It is inherently known and the very process of which axis is used to make the measurement preselects the two opposite spin orientations. Sorry, but the hidden information is neither hidden nor transferred at faster than lightspeed — it is simply known! This is the Quantum Entanglement Conjecture.
The DoubleSlit Experiment Conjecture suggests that the very same known information is what is ultimately revealed on the screen as an interference pattern of even a single photon. It appears that the photon splits itself into two (or more) parts, each going through one of the doubleslits and thereafter recombining as positive and negative interference patterns, and while I do believe this is very possibly part of the story here, what is really being exposed — most emphatically not transferred — is the known information of the BIM/ISL as the collapse of the all possible wave probabilities of a given photon traveling through one or the other of the two slits. Information can demonstrate positive and negative interference!
Together, the DoubleSlit Experiment Conjecture and the Quantum Entanglement Conjecture are the result of the one and same information expression: DSEC = QEC, or DSEQEC for short (DoubleSlit Experiment Quantum Entanglement Conjecture).
The ER = EPR conjecture brilliantly ties the fractal, holographic Universe(s) into a selfconsistent whole by turning the geometry of spacetime in on itself. It implies that connections between disparate “space” and disparate “time” realizations may be fruitfully manifested by the rejoining the one with the other by tunneling (ER wormhole). The DSEQEC provides for the very “mechanics” or “architecture” of that possibility. The statistical, probability of the allpossible paths quantum wave is simply this: every “possible” path has embedded within it the known information of the BIM — including the PPTs and PRIMES — and the collapse of that waveform of possibilities into a manifest reality selects for that known information set of that particular quantum state identity at the time and place of collapse. Perhaps the mostpositive interference of the known information leans that selection to the most statistically likely “choice.”
The information briefly presented here is freely available in its full form on my website and is included in several of my free iBooks. A few of the more advanced iBooks have a nominal charge, but I have free redeemable coupons available upon request.
NOTE: The decoupling of strict symmetry of the Higgs Boson generates mass expression in forming ST units with mass, i.e. fermions, baryons and mass+ bosons.
NOTE: Every PT is on the BIM, along with its proof. In addition, the Axis Row containing the PT always contains the squared value of its sides and hypotenuse, as well as its Area value x 4.
NOTE: Every PT is represented on the BIM 4x, a "left" and "right" version on both the Axis Row and its symmetrical Axis Column.
The ramifications of Susskind, Maldacena and Van Raamsdonk's works are most profound, offering a fresh, imaginative and farreaching overview of the Geometry of Spacetime. The DSEQEC fully resonates with the ER = EPR conjecture — providing the complementary geometry from the "ground floor" — if you will — on up. The very formation of ST must crucially contain the constraining parameters of the large scale Universe(s) from the get go. These were alluded to in TPISC III: Clarity and reaffirmed here, now, in TPISC IV: Details.
DSEQEC — 1 from Reginald Brooks on Vimeo.
DSEQEC — 2 from Reginald Brooks on Vimeo.
DSEQEC3 from Reginald Brooks on Vimeo.
Note: The hypothetical photon is a model of my own making (see LightspeedST, LUFE) and has no experimental confirmation. Despite its highly visualized nature within the proposed quantum, pulsepropagating context in which it is described, it still remains just a model. The proposed connection between the two Left and two Righthanded PPTs — as shown above — as the fundamental, underlying ST source of the phenomena of "spin", as depicted on the BIM, remains! It is this information as to the vector parameter of spin — magnitude and direction — that exists for each ST unit particle. Information that provides equal, yet opposite, spin vector direction simultaneously. It is known, does not need to be transported, and becomes the in situ basis of entanglement.
DSEQEC: DoubleSlit Experiment — Quantum Entanglement Conjecture
On the BIM (BBSISL Matrix), ALL Pythagorean Triples — parent, Primitive Pythagorean Triples (PPTs) and child, nonPrimitive PTs (nPPTs) — are found.
Because the BIM extends all Inverse Square Law (ISL) information to infinity, and, knowing any one matrix grid cell value and location, one (Nature) ubiquitously knows ALL the BIM values at ALL grid cell locations, and because the PTs, as part of the BIM, are also ubiquitously known, NO FASTER THAN LIGHT signal is required to extend the information to its ENTANGLED cohort particle regardless of the degree of space or time separation, i.e. SpaceTime (ST) separation.
Entangled ST unit particles, like the photon, coherently share the same quantum state, only with equal but opposite values. The Conservation Laws (energy, charge, spin angular momentum) require the combined coherent state of the entangled ST unit particles to be the net value of its individual units when combined. Spin is often the quantum state value measured to label two equal but opposite values contributed. Spin may be designated as updown, leftright, + or , …
It is proposed in the DSEQEC that the PTs represent the quantum state values of spin, with the direction of travel of the spin vector that is measured being one of the sides of the PT.
Since the BIM itself is bilaterally symmetrical, and, because every PT on the BIM is represented on a Row (or Column), and can be depicted in either “left” or “right“handed iterations, there are 4 total iterations of any given PT.
Ones particular axial frame of reference when measuring the spin vector as updown, leftright, + or ,…. ALWAYS results in the measurement reading of the opposite, but coherently entangled particle spin vector to be INSTANTLY equal but opposite precisely because the spin information is ALWAYS known. The movement in opposite direction ensures that, given the same axial frame of reference, the opposite will always be chosen.
One may argue that the measurement is collapsing the waveprobability function in which both spin vector directions simultaneously exist and it is the act of measuring that collapses the wave function to select one or the other spin vector direction. And in one sense, that is true. But NOT in the sense that some hidden information traveling faster than light is then communicating, via a signal, to the other particle to register the opposite spin vector direction.
NO. There are is NO hidden variable information and there is NO faster than light signal pushing information to the other entangled particle. What is happening is that the choice of measurement to utilize a specific and consistent axial frame of reference to register the spin vector angular momentum direction crucially preselects the COMPLETELY, SIMULTANEOUSLY KNOWN INFORMATION of the other entangled ST unit particle, giving the equal, but opposite spin vector direction when measured with the same axial frame of reference.
The DSEQEC is consistent with and provides a complementary theoretical basis for the ER = EPR work of Leonard Susskind, Juan Maldacena and Mark Van Raamsdonk. The ER = EPR conjecture is a very interesting insight into the joining of quantum mechanics with quantum gravity through the geometry of spacetime.
But what of the DoubleSlit Experiment? The same KNOWN INFORMATION of the PTs on the BIM means that even a single ST unit particle — be it a photon or an electron — can interfere with itself in going through the two slits because the INFORMATION of BOTH possible paths is KNOWN from the start. Now, it is the process of detecting (measuring) which slit the ST particle went through that destroys the heretofore entangled INFORMATION, resulting in the loss of the interference pattern.
A tip of the hat to Jacob Kafka’s “Rough Animator” app.
Original soundtrack.
http://www.brooksdesignps.net/Reginald_Brooks/Code/Html/netarti5.htm
Thanks for viewing!
The quantum universe begs the question: what is the quantum?
Perhaps we can say the basic pulsepropagation of ST from its singularity — a singularity connected to the larger pool of all singularities by the Conservation of Energy — out into full spacetime extension, and back, is the fundamental quantum. The ST so formed has articulation points, nodes of structural formation made by the embedded Pythagorean Triples. These nodes are also quantized as they both are dependent upon the unfolding Inverse Square Lawbased ST, and they have distinct vector parameters inherently built into their asymmetric forms. They have both magnitude and direction — and as vectors their net expression can be the result of the total added or subtracted interfering ST units. The fact that every primitive PT comes in both its parent PPT form as well as its child nPPT form provides a built in fractallike structure, that, when combined with the fact that each and every PT — PPT and nPPT — has 4 iterations of expression within the unfolding ST unit pulsepropagation, and we we now have a fractalbased, holographic quantum universe(s)!
The Conservation Laws, and specifically the Conservation of Energy — the Conservation of Lightspeed — the Conservation of SpaceTime — has been at the center of the discovery of the BIM. The ISL is at the heart of ST formation and its expression as matter and fields. At the “end of the day” it must be conserved!
And while discovering how intimately and profoundly the Pythagorean Triples — especially in the form of the ToPPTs — are so incredibly Number Pattern Sequence interrelated to each other and their placement on the BIM — as also referred to as the BIMtree — is a giant step forward, it is its structural implications that give it meaning. The PTs give structural nodes and forms to the generalized BIM structure in the unfolding of ST.
As ST forms, i.e. expresses itself as, matter and waves, it must address the underlying imperative that in total, its Conservation Laws must be upheld.
If the ST parameters that give identity to waveparticle forms of matter and/or to waveparticle forms of an energy field — parameters such as energy, momentum, charge, baryon number, and lepton number — are a manifestation of their underlying quantum number state, then these too must be intimately involved in the Conservation Laws.
When a waveparticle ST unit seems to either separate into two parts that later recombine as they interfere with each other (DoubleSlit Experiment), and/or, when they are part of a system of two (or more) ST units entangled (Quantum Entanglement) via sharing the same quantum number state, they BOTH MUST DO SO IN A WAY THAT SATISFIES THE CONSERVATION LAWS!
The total angular spin momentum of any system must be such that it does NOT violate the Conservation Laws. Naturally, if the entangled ST units have spin measured UP on one axis of one of the entangled pairs, it MUST measure DOWN on the same axis of measurement on the other entangled unit. This is the Conservation Law in action.
If the ST that informs the waveparticle and its parameters ALWAYS obeys the Conservation Laws, then the geometry that informs that ST formation must in and of itself ALWAYS obey the same laws.
The BIM and BIMtree, being the geometry of the BIM and BIMtree expression,both simultaneously and automatically precede and inform any and all subsequent ST unit expression, and, that this geometry is ubiquitous — expanding to infinity!
~~ ~~ ~~
From the introduction to The Pythagorean—Inverse Square Connection (TPISC I: Basics) to its proof and distribution (TPISC II: Advanced) on the BBSISL Matrix (BIM), we have now come full circle to TPISC III: Clarity & Simplification: Tree of Primitive Pythagorean Triples (ToPPT).
Here we “see” the pattern: the arrangement of PPTs into triplet branches (Cluster of Tertiary Branches/Tiers) as organized into the ToPPT — with each and every PPT related within a Cluster, Cluster to Cluster, Cluster to Branch and Branch to Branch, back to the initiating 345 PPT Trunk.
The devilish details proving the linkage nevertheless — in the end — serve the clarity and simplification!
That a SpaceTime (ST)forming Matrix patterned on simple Squares (& Circles) — a.k.a. the ISL — should pairup their Squares to become and reveal in their sum, a larger Square — i.e. a^{2}+b^{2}=c^{2} — forming fractal iterations of Pythagorean Triples (PTs), themselves patterned throughout the BIM, is a sublime beauty to be realized with a small effort.
The fractal within a fractal!
Such profound harmony of form naturally begs the question: to what effect?
Several conjectures come to mind:
 Would not the fractalnodes of the PTs lead to structural articulation points along the expanding ST that would allow the geometry of the Squares to be enhanced by the embedded geometry of the nonSquare Rectangles holding these PTs?
 Would not these same fractalnodes of the PTs be internal, integral and identifying structural parameters (Quantum State Numbers) of any waveparticle ST unit formed?
 Would not the immensely interconnecting linkage of the PTs, between the PTs and between the PTs and the BIM itself, provide the very basis of the waveparticle duality and entanglement phenomenon we see in the DoubleSlit Experiment and the Quantum Entanglement (DSEQEConjecture) that lies at the core of our understanding the Quantum Universe?
Form follows function — function follows Form!
~~ ~~ ~~BACK: > Part I of II CaCoSTDSEQEC on a separate White Paper ~~ ~~ ~~~~ ~~ ~~BACK: > Part II of II CaCoSTDSEQEC on a separate White Paper ~~ ~~ ~~CaCoST — Creation and Conservation of SpaceTime
DSEQEC (Part I) SUMMARY or *"Where did all the antimatter go?"
Top 10 FAQ and their Answers
What? How are these two even related?
(Note: Based on earlier work as fully referenced at the bottom of this Section under "My References.")
CaCoSTDSEQEC from Reginald Brooks on Vimeo.
Q. Where is all the antimatter?A. Right next to all the matter.
Q. Why can't we see it?
A. Ordinary matter exists (primarily) in Our View (OV). We can see and measure it directly. Antimatter exists (primarily) in the Alternate View (AV). We can not see it, only infer it by indirect measurements like the DSE and QE.
Q. What is the difference (∆) between the two?
A. Just as we've been taught: antimatter is matter with Equal And Opposite (EAO) charge and spin (spin angular momentum) direction.
Q. How does one characterize the two in their respective OV and AV SpaceTime (ST)?
A. Same form with EAO charge and EAO spin direction. Each quantum pulsepropagation (pp) from the Singularity (Ⓢ) — that dividing OV/AV — out into its view at and forming light speed (c)based ST extension, and back to the Ⓢ.
All ST is formed from pp of the virtualpairs of ST units: one matter in OV and one antimatter in AV, most likely in an alternating sequence.
(Note: when you have an actual antimatter ST unit in OV, its virtualpair partner is the EAO matter ST unit in the AV.)
Q. Does this virtualpair pp apply to all ST unit particles, and if so, can you explain?
A. Yes. Picture a line whose forward direction represents time. This is the Ⓢ. (The Planck Event Horizon [PEH] is the border interface of that line.) It actually contains ALL the energy, mass, momentum, charge and angular momentum of the Universe. The line is contiguous. It is a constant: neither created or destroyed.
The energy is the total field. The ST field.
All the other are parameters that define the expression of that ST energy field as it manifests itslef in quantum pp as virtualpairs, alternating EAO values first in OV, the AV, OV, AV,...
They do so in a manner in which whole discrete units of specific parameter expression — e.i., charge and spin — do so in discrete, quantum units of whole integer amounts (directly) or indirectly as simple ½, ⅓, ⅔ fractional amounts that add up to a net whole integer amount.
For example, electric charge is 1 WIN (Whole Integer Number), 2 WIN,... or, ⅓ + ⅔ = 1, or ⅔  ⅓  ⅓ = 0 WIN.
For example, spin, by mathematical convention, is in ½ "integral" amounts and may be +/ ½, or ½ + ½ = 1, or ½ + ½ = 0, or ½ + ½ + ½ + ½ = 2, or ½ + ½ + ½ + ½ = 0, the fermions having a net fractional spin ½ integral and the bosons a net 0, 1, 2 whole integer number.
The point is, that whatever the matter ST unit has, its antimatter virtualpair partner will always have exactly the EAO entangled values.
(Note: think of charge as so much up/down relative to the Ⓢ line and spin as so much rotation at the 90° horizontal position to the charge vector.)
Q. Can you give a little more of an actual example, please?
A. The Higgs Boson (ℋ), in its purest form, is really the massless, chargeless, and spinless ST unit/field that is parent to ALL ST units. It is composed of the two EAO virtualpairs in OV/AV.
The first child is the photon (𝛾). Two EAO photons together make the ℋ.
(Note: interestingly — for future reference — two Not Equal And Opposite photons, i.e., two Equal And Samespin photons in OV, will temporarily combine as gravitons[𝓖] as the two constructively interfere with each other. Temporary, in the sense that their point of interference is not static, but travels as in any ordinary waveinterference pattern, away from its source, following the Inverse Square Law [ISL].
The 𝓖 is actually the epitome of the waveparticle duality in that it really is a crosswave interference expression that can collapse to its 2photon waveparticle ST unit constituents.
The 𝓖 is also directly related — an even derived from — the ℋ in that 2Higgs, composed of four photon spin1 states, can reconfigure such that the 4 sets of spin recombine into two sets of EQUAL spin, i.e, spin 2 gravitons: as c+cc and cc+c reconfigure to c+c and cc+cc as shown in the image below.)
The photon (𝛾), itself a boson, gives birth to ALL other ST units — as can the ℋ — by deconstructing (i.e., rearranging) its charge and spin vectors to form ST units with mass. Mass being a parameter expression in which some form of asymmetry results in the ST unit now unable to generate full lightspeed ST extension with each pp.
In other words, it partly spins on itself and in doing so it is less than lightspeed extension gives what we measure as mass, e.i., the 𝛾 forms an electron/positron or positron/electron (e^{+}/e^{}) virtualpair with EAO charge (+1/1), and, EAO spin vector (e^{+}~_{R}~/e^{}_{L} or e^{+}~_{L}~/e^{}_{R}). Each having lost the near perfect symmetry of the 𝛾 (spin 1, charge 0, mass 0), now express mass.
Q. Where does the ISL come in to play?
A. Everywhere! ALL energy — all influence — dissipates from its source inverse squarely with distance.
Those gravitons (𝓖) mentioned earlier, dissipate as an expanding sphere of ST curvature — a diminishing ST curvature as the area (surface) of constructive interference quadruples in size for every doubling of the distance from the source, while the actual number of interfering nodes remains constant.
Because the Conservation of Energy ultimately dictates the virtualpair expression, we effectively have a Conservation of SpaceTime (ST) in effect. The Ⓢ must ultimately know and balance ALL ST curvature back to its constant — be it "0" or "1" depending on how you like to look at it.
Therefore, it is precluded that the ISL drives the amount and balance of ST curvature.
ST curvature so dilute that it effectively scores as negative will have an expansive, antigravity expression, a.k.a. Dark Energy.
Intense, concentrated ST curvature the opposite: dense constructive interference will express as Dark Matter upon the surrounding masses.
Thus Dark Energy = Dark Matter/the inverse of. The Conservation of ST is maintained.
(Note: The Sun's corona represents a local highST curvatureDark Matter effect. The intense concentration of constructive interference results in a temperature expression of millions of degrees versus that of the actual surface of the sun at 56000 degrees C. See the final Summary.)
(Note: In 2008, "The Conservation of SpaceTime~ The role of the Higgs, graviton and photon bosons in defining Dark Matter = Dark Energy (the inverse of) in the fractal Universe ~" white paper presented in great detail much of what we are presenting here.
This was built on the 2003 "Dark Matter = Dark Energy (the inverse of): The Conservation of Spacetime by The Conservation of Force.
The History of the Universe in Scalar Graphics and The History of the Universe  update  The Big Void round out this early series, presenting a consolidated image and summary view. Well worth a read!
The BIM was discovered and published in 2009, adding the underlying mathematical foundation and validation of the ST concepts put forth then, and now! Dynamic ST has waveparticle interference manifestations that directly accounts for the current "mystery" of Dark Energy and Dark Matter. One simply must look in the right spot(s) to see the ripples!)
Here is an exerpt from "DarkDarkLight: Dark Matter = Dark Energy (the inverse of:"
Similar to the real world irony, here too, smaller size (area) represents a greater focus of energy. Thus the smaller the λ = the greater the energy, as shown earlier,
E = pc = hc/λ .
Therefore, smaller areas in the rectangle represent greater concentration of energy. And because all energy is gravitationally active, in that higher energy area (higher than boring spacetime energy of 1 · 1 units) the gravitational acceleration is relatively enhanced. Opposite to this are the lower energy areas (again lower than the ideal, boring spacetime energy), where the spacetime energy density is so relatively unattractive that it appears to be repulsive when viewed from the higher energy position. Those smaller rectangles will be progressively more attractive the smaller they get and vice versa.
Effectively, this describes a force in general, and here, the gravitational force in particular. Geometrically, the higher spacetime curvature of the higher massenergy density gravitationally attracts all other energy to itself. The concentration of h units here, at higher frequencies than their surrounding spacetime, ST, leads to a net attractive force here.
Because dividing up the symmetrical 25 unit^{2} Total Energy, E_{T}, Rectangle "B" (or "A") into an asymmetrical, weighted Rectangle "C", still at E _{T} = 25 units^{2}, the lighter, airy portion in the upper right of Rectangle "C" is equal and opposite to the heavy, denser portion in the lower left.
Equal and opposite to what? Equal and opposite in force, F. Conservation of Force. The degree, or amount of F that is gravitationally attractive, relative to boring old ideal ST of 1 · 1 = 1 unit^{2}, is equal, and opposite, to the amount of F that is less attractive (i.e. repulsive) relative to that same boring ST.
F_{attraction = }F_{repulsion} (Conservation of Force)
Picture a crosssection through the ST continuum where the curvature of ST is represented by vertical lines. In the middle, which also would represent ideal, boring ST the lines would be perfectly vertical and evenly spaced. Towards the increasing massenergy density on our left, the lines would increasingly curve towards the left as the spacing between the lines also decreases in progression, depicting what effectively becomes the gravitationally positive "missing mass", or dark matter. Towards the right where the massenergy density progressively thins out, the vertical lines become spaced further and further apart, and, become increasingly curved to the right, depicting what effectively becomes the negative or antigravity of the dark energy. The image is not far removed from the common image of iron filings in the presence of the opposite poles of a magnet, i.e. a magnetic field.
Expansion and contraction are relative concepts…because of the relativity of ST. If ST consists of all the Planck h units and all the Planck h units times all their frequencies of presentation equals a constant Total Energy, E_{T}, then there can be no net expansion or contraction of the universe…only that of one occurs at the expense of the other. The more contraction force is expressed here, the more the expansion force is expressed there. E = hν will not allow it any other way. The universe…and ST…is expanding at an accelerated rate if viewed from a relatively high ST contraction position.
It is very helpful to think of ST as space/time, or simply as distance (displacement) per unit of time (or frequency):
λ /t = λ ν .
All ST manifests itself as mass or massenergy (matter or fields). All ST exists as pulses, at a given frequency related to their energy, of Planck's unit of h, and the energy of any such ST existence is
E = hν .
Mass, as matter, represents an asymmetry in a given ST parameter (an attribute which generally limits its mobility, but also provides for net expressions of the charge vector, whether +, , or 0; and, the spin, or intrinsic angular momentum, vector).
Massenergy, as a field, represents the relatively symmetrical expression of the ST (an attribute which allows for its lack of mass, lightspeed mobility, neutrality of the charge vector, and gauge boson behavior, i.e. the spin vector is always paired, resulting in only net integral spins of 0, 1 or 2). It is relatively symmetrical, and not absolutely symmetrical. The Higgs Boson represents the absolute symmetry, the progenitor of the ST field. Breaking the symmetry of the Higgs Boson generates the relatively symmetrical ST field mentioned above, which can itself form, through further symmetry breaking, the matter field, a.k.a. the mass expression.
This relatively symmetrical ST field mimics in every way the massless, chargeless, spin = 1 photon, whose λ and ν are always reciprocally related to insure that
c = λ ν ,
the velocity of light, equals the fundamental constant of nature.
In 201213, "LightspeedST" was published, bringing together in a single work all of the previous 30 years. Both as a freely available web project and as an ebook. Here is a movie from a page excerpt: "LightSpeed = ST: The Movie Script" (navigate to The Movie Script>Plot>Simplification>6) or this Vimeo summary movie below.
LightspeedST: The Movie (short) from Reginald Brooks on Vimeo. A fulllength version is also on Vimeo.
Q. What is the BIM?
A. The BIM (BBSISL Matrix = Brooks Base SquareInverse Square Law Matrix) describes the ISL in a simple fashion using nothing but WINs.
With the Axis of 0, 1, 2, 3,... and the central Prime Diagonal (PD) of their squares, every BIM cell is simply the ∆ between the horizontal (Row) and vertical (Column) PD numbers.
Every BIM cell value is known, fixed, and expands to infinity.
The BIM enjoys a 90° bilateral symmetry. Combining both triangular sides gives one the perfect square areas (and their circles).
Looking at just the Rows, one finds that dividing all cell values by 24 selects out ALL Active Rows: Rows of ODD numbers, not ➗3, that contain ALL the Primitive Pythagorean Triples (PPTs), as well as ALL (>2) PRIMES.
Q. What is the significance of ➗the BIM by 24?
A. There are 3 main number theory systems contained within the BIM. Each extends to infinity:
 ISL is synonymous with the BIM.
 The PTs are distributed throughout the BIM, the PPTs are ALWAYS on Active Rows.
 The PRIMES — while not on the strict Inner Grid — ALWAYS have Active Axis Rows, occupying the same, but not necessarily identical Active Rows.
(Note: TPISC_IV:_Details presents the BIM, Pythagorean Triples and Primes in great — wait for it — detail.)
Q. What is the significance of the PTs, especially the PPTs?
A. As the PTs are both ubiquitously and infinitely embedded within the BIM, they provide key structural framework nodes for the expanding, pp ST unit.
Key in that the nonisosceles, 90° righttriangle PTs are present with 4 iterations for each PT on the BIM.
Not only do these asymmetrical triangles form rectangles (and ovals) versus squares (and circles) when combined in pairs, individually each of their triangular iterations may actually be representative of a quantum state number (parameter).
Horizontally, on a Row, the PT has a Left and Right (mirror) iteration, and the same — when rotated 90° — on its bilaterally symmetrical Vertical Column.
If we assign the spin vector to an iteration, we have 2 sets of EAO spin vectors for every PT. As the iterations are known — ubiquitously and infinitely — we have the basis of the DSEQEC.
(*From the Big Bang genesis, "inflation" was both the result of and the solution to potential annihilation — by separating and sequencing matter and antimatter — giving the initial birth of ST.)
 Big Bang
 *INFLATION
 Matter = Antimatter
 Dark Matter = 1/Dark Energy
 DSE = QEC
 ER = EPR
The DSEQEC ~ One more time
The Quantum Mechanics that ultimately explains the DoubleSlit Experiment (DSE, a.k.a. the waveparticle duality conundrum) must, ultimately, also explain and inform Quantum Entanglement (QE).
DSE —> QE
1st Level: The photon (𝛾) consists of EAO ST units whose mass, charge and spin vectors give a net mass 0, charge 0 and spin 1.
On a simplistic level, the 𝛾 splits into its EAO parts, each going through, in a synchronous fashion, one of the slits.
2nd Level: The photon — and ALL other ST units — is but a child of the Higgs Boson (ℋ) and exists as part of a virtualpair — part time in Our View (OV) and part in the Alternate View (AV), not entirely unlike an AC electric current, yet we only perceive OV (½ the time it is in OV< the other ½ in the AV with EAO parameters).
The ℋ, in its ideal, perfect field state, is really 2 EAO photons, so this is not that difficult to envision, only in this scenario, its EAO photon STs occupy two separate views — OV and AV — and most likely in a serial, timeseparated manner, not unlike a sine wave.
3rd Level: The Higgs=photon+photon ST unit forms ST — and its accompanying velocity of light (c) parameters — with each pulsepropagation (pp) from its Singularity (Ⓢ) out past its Planck Event Horizon (PEH) into OV (and sequentially, alternatively and in syncronicity in AV).
This all occurs under the constraints and directives of the Inverse Square Law (ISL), as portrayed on the BIM (BBSISL Matrix).
The BIM is bilaterally symmetric with ½ mirror reflected 90° to the other, divided by the Prime Diagonal (PD).
4th Level: Within the BIM, ALL possible Pythagorean Triples (PTs) are present and manifest their nonisosceles, 90° righttriangle directionvectorspecific presence — and automatically, the mirror of that and the 90° mirror symmetry of both — giving 4 iterations for each PT.
graphic)
5th Level: If we let the vectorspecificdirection of the 4 PT triangle presentations present the respective horizontal and vertical spin axis of the individual 𝛾 ST, we can account for ALL possible manifestations at any moment in time over any extension in space as the 𝛾 ST unit is itself the unfolding, expansion of the BIM.
For example using the 345 PPT and letting the short side "a" — or, if you prefer, simply the "outward" area projection from the hypotenuse — represent the spin angular momentum vector (spin) on the horizontal plane. Turn it 90° to represent it — with the other two PT representations — on the vertical plane.
Thus 2L's and 2 R's.
6th Level: The superposition of both spin states — spin ½_{L} and spin ½_{R} — is present simultaneously as two virtual states, but only one state is revealed upon measurement in OV — instantaneously, simultaneously and without transferring any information — the other EAO spin state is revealed in the AV.
Likewise, two entangled photons — with EAO spin states — do exactly the same thing, in effect 4 spin states are effected anytime any 1 spin state is measured.
7th Level: Important to notice is the maintenance of symmetry in service to the overall Conservation Laws: Conservation of Spin Angular Momentum, Conservation of Charge, Conservation of Energy and, of course, Conservation of ST.
8th Level: The Higgs Boson (ℋ) represents the perfect symmetry — and Conservation — of ST. composed of 2 EAO entangled photons.
Let's say that again: The Higgs Boson = the state of 2 EAO entangled photons (𝛾_{E}).
The Higgs Boson, in its ideal, perfect field state, is simply two EAO entangled photons.
(The Higgs in the transformative state of composing other ST units, will, of course, have various masses. )
When deconstructed into its 2 "independent" photons in OV, it simultaneously reflects as and EAO photon if AV.
9th Level: In other words, every photon (𝛾) comes as a virtualpair, entangled set — ½ in OV, ½ in AV — together forming the Higgs (ℋ).
When two real OV photons are entangled (EAO) in real time OV, there are, indeed, 2 sets = 4 entangled photons = 2 Higgs Bosons. One ℋ in OV, the other in AV.
(Note: not to be confused with the graviton (𝓖), that is simply the constructive interference of two similar — Not EAO — photons, i.e.. 2 clockwiseR photons or 2counterclockwiseL photons: 𝛾_{R} + 𝛾_{R} = 𝓖_{R} and 𝛾_{L} + 𝛾_{L} = 𝓖_{L} as spin 1 + spin 1 = spin 2 of the 𝓖.
Naturally, two EAO gravitons will form 2 Higgs Bosons: 𝓖_{R} + 𝓖_{L} = ℋ as spin 2_{c} + spin 2_{cc} = spin 0
)
Thus, even a single, real OV photon can demonstrate interference in the DSE.
10th Level: Everything about the ℋ —>𝛾 + 𝛾 virtualpair scenario is true for ALL ST units from leptons  baryons, fermions  bosons, matter  antimatter.
ℋ —>𝛾 + 𝛾 —>Matter + Antimatter as described in the 1st Level.
The electron (e^{})—positron (e^{+}) are the virtualpair components of the photon, directly, and the ℋ, indirectly.
The quarkbased hadrons — mesons and baryons — are likewise virtualpair components of the 𝛾 and/or ℋ.
When you "see" (measure) a real OV e^{}, it acts like a hole — a void —in ST awaiting to be filled with a positive charge ST unit. The e^{} represents so much integral *"downness" (↓) as opposed to the *"upness" (↑) of the + ST unit.
(*Relative to their Singularity and PEH. Diagrammatic convention places ↑ at the top, ↓ at the bottom.)
As a virtual pair, both ↓ and ↑ are always together — only one is expressed in OV, the other to balance it in the AV.
)
The exact same superposition, synchronous entanglement and waveparticle doubleslit expression found in the photon is present in all the children ST units.
So where has all the missing antimatter gone?
Well, it never left. It's where it has always been: as part of the inflationdriven virtualpair — matter:antimatter — that ALL ST units express!
The key point is that Nature embodies the Universe with the harmony of the one by her Conservation Laws — Laws that by combining simple mathematical rules of counting (a.k.a. the ISL), her balance sheet ALWAYS — in the end — reflects symmetry.
The unfolding apparent asymmetry:
 Big Bang
 Matter
 Antimatter
 Inflation
 Dark Matter
 Dark Energy
 DSE
 QE
is given a resultant symmetry by the entangled generation of SpaceTime(ST). A ST that is conserved.
Thus:
 Big BangInflation
sequentially, over ST Inflation, separates the entangled
 MatterAntimatter
into:
EAO, alternating entangled virtualpairs whose internal frameworks of asymmetrical (contributed by the PTs), slows down and concentrates parts of the ST field into coalescing, entangled particles that pull part of the expansion back home to the common Ⓢ that balances the accounting ledger sheet, giving
 Dark Matter1/Dark Energy
 DSEQE.
We just happen to only "see" one side of the OV/AV coin!
DSEQEC: One last look at just the photon.
The time component will always ensure that the EAO spin is accounted for!
(Note: the "Inflation" expansion of the Big Bang — separating Matter  Antimatter — generates an EAO chiral spin of sequential, interlocking, doublehelical spiral. This large scale structure is itself but a fractal reiteration of the ST units.)
AFPOP Introanimation:A Fresh Piece of Pi(e) ... and the √2, too ... Fractal  Fractal  Fractal from Reginald Brooks on Vimeo.
see: A Fresh Piece of Pi
SUMMARY
*Have you ever wondered why that area between the sun's surface photosphere (~5000° C) and its corona (~110 M° C) is not hotter, indeed it is actually cooler than the surface?
Think of the corona as "Soft Hair," a term coined in Stephen Hawking's, et al, last published paper:* Black Hole Entropy and Soft Hair, as an accumulation of photons that are gradually released (along with their ST information) away from the outer surface of a black hole event horizon. Only here there is no black hole as the mass of the sun is too small. Yet there is this huge accumulation of photons at some distance from the sun's surface. Much less so in the chromosphere area between. How so?
(*Black holes and soft hair: why Stephen Hawking's final work is important and Stephen Hawking's final scientific paper released)
The broad spectrum photons radiating out from the thermonuclear core (15 M° C) of the sun generate a harmonic wave band that forms the corona. Here the massive battle of constructive vs. destructive wave interference favors the former. A dense band of constructively interfering photons forms this 1st harmonic. Many of the photons will have matching coherent spin1 vectors, forming gravitons in the process.
The gravitons — representing local high curvature ST, i.e., Dark Matter — will gravitationally attract and curve and effectively "slow down" other photons within this "temporary" photon "soft hair" accretion disk.
Some of the interfering photons — and some of the ensuing interfering graviton waves — will destructively interact — opposite spin 1 vectors — to generate Higgs Bosons. The Higgs Boson field (spin 0) inherently favors ST expansion: witness the singularity of the Big Bang and the multiple singularities of the individual pp ST units.
One may visualize a number of harmonic interference bands radiating out from a "point source"— accumulated mass — giving rise to "structural nodes" along the way. The corona is one. The formation and distribution of the planets, et al, within the Solar System is yet another.
The accretion and consolidation of smaller bits of massenergy into preferential orbits about a central "point source" would be a natural consequence of these photongravitonHiggs resonance bands.
After firmly establishing a deep connection between Pythagorean Triples (and PRIMES) with the BIMdepicted Inverse Square Law, we have ventured back out to the world of quantum physics — or should we say quantum relativity physics. What have we put forward that might meaningfully inform our world?
In LightspeedST, we showed that nothing can physically travel faster than the speed of light precisely because the photon — the basic ST unit that pp into and out of existence forming a single quantum of ST — is itself composed of and generates 1 unit of lightspeed ST with each pulse.
Decoupling the ideal Higgs Boson field ST unit gives the photon, and from further decoupling, all the other ST units downstream.
Recoupling two coherent, likeminded, similar spin 1 photons generates the graviton (spin 2). This ellusive ST unit — that must be absolutely inundating the Universe — is really just the temporary constructive interference of two coherent superimposed photon ST unit waves at the moving point of their crossing.
As in all wave interferences, opposite spins interacting will neutralize the gravitational contracting action in favor of the ST expansion of the newly formed Higgs.
This +, , 0 wave interference of photon ST units has many, many consequences, and dare we say, that — along with the accompanying BIM — gives insights, if not downright clarifications of a number of fascinating, yet heretofore, inexplicable physical phenomena.
To be complete, we must reintroduce (Dark Matter = Dark Energy
(the inverse of); The History of the Universe in Scalar Graphics; The History of the Universe  update  The Big Void; and, The Conservation of SpaceTime) the Our View (OV), Alternative View (AV), and the role of how symmetry, matterantimatter, inflation, singularity, Planck Event Horizon, etc., fit into the pp of the ST field and units.
We thus have:
 BIM — the information
 PP — the structure
 Symmetry — the mechanism.
The BIM — along with the PTs and PRIMES — provides the information on how ALL ST will be informed: the rate of and relationship to all expansion of ST. Key structural nodes — representing quantum number states — are ubiquitously embedded within the expanding BIM.
The underlying quantum is the photon ST unit, created and always expressing the constant, ultimate and maximum velocity of light expression (gauge bosons) that every subsequently derived ST unit with mass will necessarily be moving at less than light speed (i.e., fermions and other bosons).
The +, , 0 wave interference effects are precisely the guiding dictates — along with the dictates of 1. and 2. — informing the physical manifestations of both the seen and unseen (Dark Matter, Dark Energy, as well as the phenomena underlying the waveparticle duality of the DoubleSlit Experiment and Quantum Entanglement, DSEQEC) Universe.
The symmetry — that which informs the Conservation of Energy, the Conservation of Momentum and Angular Momentum, the Conservation of Charge, the Conservation of the Velocity of Light, and ultimately, all in service to the Conservation of ST — divides the Universe — or should we say Cosmos — into polemic parts whose equal part ratios ultimately demand a separation — a far from equilibrium separation — precisely such that the dynamics may unfold at these edges (OVAV) in the drive to reestablish the equilibrium and a return to the ultimate neutral, perfect symmetry state of the ideal Higgs Boson ST field/particle.
This model addresses several outstanding issues:
Big Bang : The Singularity, or 0 or 1 (depending on your intial state reference preference) = ideal Higgs Boson ST field/particle of spin 0, mass 0, charge 0. Because ST expansion of the Singularity — Higgs — is inherently built into this perfect symmetry, we must conclude and invoke that such a spin 0, mass 0, charge 0 state is, however temporary, expansive. The same holds true for the individual Singularities of the individual pp ST units.
Quantum : The Big Bang is the 1st quantum. It quickly decomposes into its fractal quantum units, each pp into and out of existence at lightspeed from the Higgs Boson field. c=𝛾ⱱ
ST Units : The ST units — all derived from the photon (itself derived from the Higgs) — all pp into and out of existence as aggregates of quarks and leptons. A photonic ST field — like a ripple in a pond — is generated with each pulse and continues to expansively pulsepropagate outward until it is neutralized by interference effects along the way.
The builtin symmetry of the massless gauge boson ST units allows them to travel 1 full quantum of spatial extension with each pulse frequency expression (lightspeed), while the other lesssymmetrical ST units spend part of their ST pp within their own spatial expression with each pulse (sublightspeed).
The ubiquitous information of the BIM/PTs/PRIMES is present and guides every pp from its birth at the Singularity (Higgs) out past its Planck Event Horizon and on out to every possible ST extension throughout its possible expansion paths throughout the Universe. This information is encoded within the ST pp and it never needs to be communicated or propagated — as per any carrier, particle or pilotwave — to any such expansion as the information is "always known" — from the beginning to the end of each and every ST unit pp. This becomes critical to our understanding of everything!
Inflation : Both the 1st quantumBig Bang and all subsequent, subset quantum ST units pp, expand outward at c=𝛾ⱱ. This builtin fractal expansion relationship defines the very existence of our Universe. Without so much spatial extension over so much temporal periodicity we have no ST and thus no Cosmos.
Inflation separates, i.e., pushes the ST units into farfrom equilibrium status such that a reorganization into seenunseen, OVAV, matterantimatter domains both prevents the senseless, chaotic annihilation of opposites — even if they are to repeat their expansion — and allows a very measured, controlled, ordered and fully accountable return to their neutral, perfect symmetry SingularityHiggs state while maintaining the balance sheets of equal and opposites the Conservation Laws require. Inflation allows and drives the Creation and Conservation of SpaceTime (CaCoST) to exist.
As noted above, the inherent ST expansion force of the SingularityHiggs underlies this inflation.
Entanglement : Without a doubt, in some sense, the entire Universe — every ST unit and field — is entangled. The BIM ensures that. Every ST unit — whether generated in OV as two real, Equal And Opposite (EAO) entangled ST units or as part of the virtualpair — one member in OV, the other in AV — that every ST unit has upon its pp, is entangled in several or more ways. This has to be! How else would Nature be able to account for and maintain the symmetry of the Conservation Laws?
Interference : (waveparticle duality, DoubleSlit Experiment, Dark Matter, Dark Energy, gravitons, quantum gravity,...)
waveparticle duality : Every ST unit pp as a ST wave and its ST unit particle expression is simply a perceptual (measurement) of it within a local interpretation of its field. This is consistent with the "Copenhagen Interpretation" of Quantum Mechanics.
DoubleSlit Experiment (DSE) : Getting in the way — i.e., interfering with the information of the wavestate — the BIM information —of a pp ST wave collapses its DSE interference precisely because the perception/measuring process blocks the BIM information. The information that informs ALL possible slit paths — thereby collapsing any possibility that even a single photon could interfere with itself. ALL ST units carry ALL possible path information within the embedded BIM information.
Dark Matter — Dark Energy : Together, these two "mysteries" are really just part — albeit opposite ends — of the same interference of ST unit wavefield effects (+/ curved ST via gravitons).
Two or more pp ST units will have some expression of each others presence as to energy density. The overlapping of the ST unit wave fronts will generate interference effects. A massive body of high energy ST unit photons will output dense bands of constructive interference. The constructive overlap of two similar, coherent photons will generate a node we call — when thinking or perceiving it as a particle ST unit — a graviton (spin 2).
It acts to *curve the ST locally.
Like any interfering wavecrest, it expands and moves outward from its "point source"— following the dictates of the ISL (BIM). It is "quantum" in the sense that its component parts are quantum pp photon ST units whose c=𝛾ⱱ wavelength frequency properties match enough (coherence) to form a "temporary" curvature of ST by their constructive interference (spin 1 + spin 1 = spin 2), giving rise to our mysterious graviton. Again, the graviton ST unit as a particle is really only our designation of this moving interference node. It does NOT actually exist as a "particle" except for our convenience in defining it superposition wave state. Both the photon and graviton node at ST unit wave packets travel at the velocity of light.
The graviton as a vectorgauge exchange boson — passing the curvature of space information between bodies — needs to be looked at as the interfering field dynamic that it is rather than a mechanical "particle" exchange picture commonly held. Every corner of the Universe is flooded with these gravitons — i.e., flooded with +, , 0 interference effects of ALL the other pp ST units.
(*The spin 1 photon ST unit may itself be "sticky"— i.e., it contributes to the gravitational curvature of ST because it — as the creator of ST (via the Higgs) — inherently wants its spin 1 vector to act as an attractor, a ST curver while its ST pp is pushing to expand outward with each pulse. The graviton node is simply double the spin vector without doubling that expansive push.)
In Dark Matter, the concentration — and importantly — the resonance banding of waves of constructively interfering coherent photons generates the graviton ST curvature we know as gravity. When densely concentrated, it gives rise to the global effects of Dark Matter. Resonance plays a key role in that harmonics of interference wave/frequency expressions play out in the expanding bands of gravitonST curvature.
Sun's Corona is a simple expression of this phenomena. The corona's temperature is a reflection of its energy and its high energy is a reflection of the resonant concentration — the Dark Matter accumulation — of gravitons — constructively interfering photon ST unit pulses of the enormously dense photonenergy at its core.
Black Hole Event Horizon, a case of increasing the energy density of the core will increase the Dark Matter energy density of the surrounding "corona" up to a point whereby its density is so great that the photons can no longer and/or barely escape, effectively forming an Event Horizon — and the now "unseen" core inside as a Black Hole. (Hawking, et al, "Soft Hair" referenced above.)
Dark Energy, not to be, by any means, outdone, is simply the reflection of the opposite. The Conservation Laws require the total ST expression to be constant. *ST=energy. Concentration of ST in one part must necessarily generate dilution of ST in another part. The dilution of ST reflecta a curvature of ST property that is expansive — opposite to the contractive ST curvature property of the graviton. So if the same expanding photon + photon = graviton constructive interference node gets away from its contractive curvature, how does it later become expansive? Remember, that in any interfering wave phenomenal, one has +, , and 0 expressions. The + and  of the interfering photon ST units represent the superposition addition — spin 1_{R} + spin 1_{R} and spin 1_{L} + spin 1_{L} — of two similarspin photons giving rise to the graviton (spin 2). Combining two EAO photons — spin 1_{R} + spin 1_{L} — gives rise through superposition subtraction to our expansionloving Higgs Boson ST unit field (spin 0). Dilute ST unit density equals the formation of the expansive Higgs field = Dark Energy. Dark Matter = Dark Energy (the inverse of).
A ST field can be divided between:
positive (+) curvature (contraction via constructive interference)
negative () curvature (expansion via destructive interference)
(* c = 𝛾ⱱ is the first step in ST creation: one dimension of space/one dimension of time;
c2 = (𝛾ⱱ)2 = potential difference=2 spatial/2 temporal dimensions;
c3 = (𝛾ⱱ)3 = currentamperage=3 spatial/3 temporal dimensions;
c4 = (𝛾ⱱ)4 = force = 4 spatial/4 temporal dimensions;
c5 = (𝛾ⱱ)5 = energy/time = 5 spatial/5 temporal dimensions; see The LUFE Matrix:The distillation of System International (SI) units into more fundamentally base units of SpaceTime (ST) dimensions)
The alternating bands of contraction and expansion may be visualized as:
Everywhere within the Universe, one may look out and "see" expansion and contraction — i.e., Dark Energy and Dark Matter.
Just as the Higgsexpansion drives the Big Bang — as well as each and every individual ST unit pp — within its expansion are the seeds of its contraction — via the superposition of coherent photons.
Matter  Antimatter : In quantum #1, the Big Bang, inflation — a natural consequence of the HIggs Boson ST field expansion — obviated annihilation by the separation of matter and antimatter ST into EAO domains. The former as Our View (OV) that take our privileged view as "reality," and the Alternate View (AV), were all the EAO antimatter hangs out in the "unseen" view.
Between the two, and the source and balancing factor of both, is the SingularityHiggs field. Upon collapse it is the "one," the only one, the "zero" from which all ST expression — all energy — all fields and particles — arises. Each individual pp ST unit is but a local fractal component of the larger global hologram.
The individual, fractal ST unit itself is composed ot two domains as well — OV and AV. EAO. The Lshaped and Ishaped quarks are all consistently modeled giving rise to their net quantum state expressions of spin, mass and charge. Every ST unit ("particle") is accounted for. (Please see LightspeedST.)
Symmetry : A=B. A is not B. A=EAO B.
Break and recombine symmetries informs it all.
The tension — the asymmetrical imbalance — of far from equilibrium (pure symmetry) that gives rise to the dynamics of ST expression only exists to resolve to the apparently neutral, equilibrium threshold of perfect symmetry. The real mystery of the Cosmos is whether pure, perfect symmetry is anything other than an ideal, a portal, a perfect interference — and a temporary one at that!
EP = EPR :
Open letter addressing the "ER=EPR" work of Leonard Susskind, Juan Maldacena, and Mark Van Raamsdonk, …
The ER = EPR conjecture is a very interesting insight into the joining of quantum mechanics with quantum gravity through the geometry of spacetime.
What is needed is the other, complementary side that seals the deal: any entanglement of spacetime (ST) and ST units/particles, i.e. photons, gravitons, Higgs Bosons,… must reckon with:
 ST itself is formed in such a way that it obeys the Inverse Square Law (ISL), is always consistent with the Conservation of Energy, and its very formation must be one and the same with the formation of its progeny ST units and the builtin velocity of light, c;
 ST itself, and thus its ST progeny, must be conserved;
 ST itself, and thus it ST progeny, must be fully accountable for all intricacies of the DoubleSlit Experiment.
One can satisfy all three constraints by allowing ST to be pulsepropagated from its singularity (S) into and back out of our ordinary view, forming the photon with its builtin velocity, c, with each pulse. A graviton is nothing more than the positive interference of two similarspin photons, while the Higgs Boson, in the pure state, is two photons with oppositespins interfering. The Higgs, of course, gains mass as it decouples into ST units with 1/2 spin, be it quarks or electronsneutrinos, or their composites. Note that one could start with the Higgs and decouple, first to the photon and thereafter to its progeny. (LightspeedST)
Key to the ordered formation of ST is the ISL....
Experimental verification : Despite our knowing how incredibly weak the gravitational force is, we nevertheless should be able to devise experimental verification of the CaCoST model.
Knowing that photons can transfer momentum while at the same time testing for the gravitational effects of constructively and destructively interfering photons, might one devise a setup up to null out the former while testing for the latter.
• A series of high density, high energy coherent, polarized (spin 1 vectors all pointing in the same direction) converged into a superposition state — a focus point —upon which lies a "target." The object is to show flicking the laser lights on locally curves the ST immediately surrounding the "target," causing it to deflect towards the light source (spin 1_{R} + spin 1_{R} or spin 1_{L} + spin 1_{L} — of two similarspin photons giving rise to the graviton (spin 2)).
• A matched set of the same laser setup — only wtih 1/2 the photons polarized one way, the other 1/2 the other (Combining two EAO photons — spin 1_{R} + spin 1_{L} — gives rise through superposition subtraction to our expansionloving Higgs Boson ST unit field (spin 0)). The "target" should be deflected away from the light sources.
• A hybrid of the two: move the later Higgsgenerating laser setup to the opposite side of the "target." Repeat experiment with one, then the other, then both laser setups turned on. The "target" does not deflect. Some fine tuning will undoubtedly be required.
~~ ~~ ~~BACK: > Part I of II CaCoSTDSEQEC ~~ ~~ ~~
References (specific to ER=EPR and some of the surrounding issues):
ER=EPR
https://en.wikipedia.org/wiki/ER%3DEPR EPR
https://en.wikipedia.org/wiki/Quantum_entanglement Quantum Entanglement
https://en.wikipedia.org/wiki/Wormhole = EinsteinRosen Bridge (EP)
https://en.wikipedia.org/wiki/Black_hole Black Hole
https://en.wikipedia.org/wiki/Event_horizon Event Horizon
https://en.wikipedia.org/wiki/EPR_paradox EPR Paradox
https://en.wikipedia.org/wiki/Einstein%27s_thought_experiments Einstein's Thought Experiments
https://en.wikipedia.org/wiki/Firewall_(physics) AMPS firewall
Mark Van Raamsdonk
https://en.wikipedia.org/wiki/Mark_Van_Raamsdonk
https://arxiv.org/pdf/1005.3035.pdf Building up spacetime with quantum entanglement
https://sitp.stanford.edu/topic/quantumgravityandblackholes Gravity and Entanglement
https://www.nature.com/news/thequantumsourceofspacetime1.18797 The quantum source of spacetime
https://www.sciencenews.org/blog/context/neweinsteinequationwormholesquantumgravity A new ‘Einstein’ equation suggests wormholes hold key to quantum gravity
Leonard Susskind
https://en.wikipedia.org/wiki/Leonard_Susskind Leonard Susskind
http://theoreticalminimum.com/home
https://sitp.stanford.edu/people/leonardsusskind
https://sitp.stanford.edu/topic/quantumgravityandblackholes ER = EPR" or "What's Behind the Horizons of Black Holes?" (Lecture 1 & 2)
https://arxiv.org/pdf/hepth/9306069v2.pdf The Stretched Horizon and Black Hole Complementarity
Leonard Susskind, La ́rus Thorlacius, and John Uglum, Department of Physics
Stanford University, Stanford, CA 943054060
https://arxiv.org/pdf/1606.08444.pdf Space from Hilbert Space: Recovering Geometry from Bulk Entanglement
ChunJun Cao,1, ∗ Sean M. Carroll,1, † and Spyridon Michalakis1, 2, ‡
https://arxiv.org/pdf/1604.02589.pdf Copenhagen vs Everett, Teleportation, and ER=EPR
https://arxiv.org/pdf/1306.0533.pdf Cool horizons for entangled black holes Juan Maldacena and Leonard Susskind
Juan Martîn Maldacena
https://en.wikipedia.org/wiki/Juan_Mart%C3%ADn_Maldacena Juan Maldacena
https://www.ias.edu/scholars/maldacena
https://en.wikipedia.org/wiki/AdS/CFT_correspondence Antide Sitter/Conformal Field Theory correspondence, sometimes called Maldacena duality or gauge/gravity duality
https://www.ias.edu/ideas/2013/maldacenaentanglement Entanglement and the Geometry of Spacetime
https://www.ias.edu/ideas/2015/generalrelativityat100conference General Relativity at 100
https://www.ias.edu/ideas/2011/maldacenablackholesstringtheory Black Holes and the Information Paradox in String Theory
Patrick Hayden
https://sitp.stanford.edu/topic/quantuminformation Decoding Spacetime
https://quantumfrontiers.com/tag/patrickhayden/ Here’s one way to get out of a black hole!
http://iopscience.iop.org/article/10.1088/11266708/2007/09/120/meta;jsessionid=3A92BE27C55F2B4A3D8CFC70E14618CB.c3.iopscience.cld.iop.org Black holes as mirrors: quantum information in random subsystems
~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~
https://www.nature.com/articles/s4158601802005 Deterministic delivery of remote entanglement on a quantum network
Peter C. Humphreys, Norbert Kalb, Jaco P. J. Morits, Raymond N. Schouten, Raymond F. L. Vermeulen, Daniel J. Twitchen, Matthew Markham & Ronald Hanson
Nature, volume 558, pages 268–273 (2018)
http://www.theory.caltech.edu/∼preskill/ph229
http://www.theory.caltech.edu/people/preskill/ph229/notes/chap4.pdf Chapter 4 Quantum Entanglement
https://plato.stanford.edu/entries/qtentangle/ Quantum Entanglement and Information
https://en.wikipedia.org/wiki/Paul_Dirac Paul Dirac
https://arxiv.org/pdf/1503.06237.pdf Holographic quantum errorcorrecting codes: Toy models for the bulk/boundary correspondence
Fernando Pastawski,a Beni Yoshidaa Daniel Harlow,b John Preskill,a
https://journals.aps.org/pr/abstract/10.1103/PhysRev.48.73 The Particle Problem in the General Theory of Relativity
A. Einstein and N. Rosen, Phys. Rev. 48, 73 – Published 1 July 1935
https://journals.aps.org/pr/abstract/10.1103/PhysRev.47.777 Can QuantumMechanical Description of Physical Reality Be Considered Complete?
A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 – Published 15 May 1935
https://www.sciencenews.org/article/entanglementgravityslongdistanceconnection
Entanglement: Gravity's longdistance connectionhttps://arxiv.org/pdf/0709.0390.pdf* Entanglement, EPRcorrelations, Bellnonlocality, and Steering
S. J. Jones, H. M. Wiseman, and A. C. Doherty, (Dated: May 28, 2018)
https://www.quantamagazine.org/entanglementmadesimple20160428/ Entanglement Made Simple Frank Wilczek, April 28, 2016
http://science.sciencemag.org/content/360/6387/40* Spatial entanglement patterns and EinsteinPodolskyRosen steering in BoseEinstein condensates
Matteo Fadel, Tilman Zibold, Boris Décamps, Philipp Treutlein, Science 27 Apr 2018:
Vol. 360, Issue 6387, pp. 409413 DOI: 10.1126/science.aao1850
https://phys.org/news/201804einsteinpodolskyrosenparadoxmanyparticle.html as above
https://www.quantamagazine.org/closedloopholeconfirmstheunrealityofthequantumworld20180725/
https://phys.org/news/201803gravityquantummechanics.html
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.240401 Spin Entanglement Witness for Quantum Gravity
Sougato Bose, Anupam Mazumdar, Gavin W. Morley, Hendrik Ulbricht, Marko Toroš, Mauro Paternostro, Andrew A. Geraci, Peter F. Barker, M. S. Kim, and Gerard Milburn, Phys. Rev. Lett. 119, 240401 – Published 13 December 2017
Read more at: https://phys.org/news/201803gravityquantummechanics.html#jCp
https://phys.org/news/201802twowayquantumparticle.html?utm_source=nwletter&utm_medium=email&utm_campaign=weeklynwletter 'Twoway signaling' possible with a single quantum particle
Lisa Zyga, Phys.org, February 26, 2018
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.120.060503
TwoWay Communication with a Single Quantum Particle
Flavio Del Santo and Borivoje Dakić
Phys. Rev. Lett. 120, 060503 – Published 8 February 2018
Read more at: https://phys.org/news/201802twowayquantumparticle.html#jCp
~
~more youtube ER=EPR video referenceshttps://en.wikipedia.org/wiki/ER%3DEPR
https://www.youtube.com/watch?v=OBPpRqxY8Uw
https://www.youtube.com/watch?v=0x9AgZASQ4k
https://www.youtube.com/watch?v=pMNSLsjjdo
https://www.youtube.com/user/mrtp
https://www.youtube.com/watch?v=4uaDMU8iiPA
https://www.youtube.com/watch?v=LndrOIXG3i8
https://www.youtube.com/watch?v=KR3Msi1YeXQ
https://www.youtube.com/watch?v=VvOZd_tbZw
https://www.youtube.com/watch?v=X9mEhcPbGsM
~~ ~~ ~~
Additional References:
ER=EPR_quantumEntanglement DoubleSlit DELAYED
https://www.physicsoftheuniverse.com/scientists_wheeler.html
https://en.wikipedia.org/wiki/Doubleslit_experiment
https://en.wikipedia.org/wiki/Wheeler%27s_delayed_choice_experiment
https://www.physicsoftheuniverse.com/scientists_feynman.html
Tangled Up in Entanglement
https://www.newyorker.com/tech/elements/tangledupinentanglementquantummechanics
Strange Numbers Found in Particle Collisions  Quanta Magazine
https://www.quantamagazine.org/strangenumbersfoundinparticlecollisions20161115/
Closed Loophole Confirms the Unreality of the Quantum World  Quanta Magazine
https://www.quantamagazine.org/closedloopholeconfirmstheunrealityofthequantumworld20180725/
Hyperuniformity Found in Birds, Math and Physics  Quanta Magazine
https://www.quantamagazine.org/hyperuniformityfoundinbirdsmathandphysics20160712/
Physicists Hunt for the Big Bang’s Triangles  Quanta Magazine
https://www.quantamagazine.org/physicistshuntforthebigbangstriangles20160419/
Matter  Antimatter
https://en.wikipedia.org/wiki/Antimatter
What is antimatter?  Scientific American
https://www.scientificamerican.com/article/whatisantimatter20020124/
Antimatter  symmetry magazine
https://www.symmetrymagazine.org/article/octobernovember2004/explainitin60seconds
more on antimatter
10/19/17
Scientists make rare achievement in study of antimatter
Scientists on the BASE experiment vastly improved their measurement of a property of protons and antiprotons.
04/11/17
What’s left to learn about antimatter?
Experiments at CERN investigate antiparticles.
02/28/17
Our universe should be a formless fog of energy. Why isn’t it?
01/30/17
Sign of a longsought asymmetry
A result from the LHCb experiment shows what could be the first evidence of matter and antimatter baryons behaving differently.
01/19/17
Matterantimatter mystery remains unsolved
Measuring with high precision, physicists at CERN found a property of antiprotons perfectly mirrored that of protons.
11/24/15
Matter and antimatter behave differently. Scientists hope that investigating how might someday explain why we exist.
11/20/15
Physicists get a supercomputing boost
Scientists have made the firstever calculation of a prediction involving the decay of certain matter and antimatter particles.
04/23/15
Extreme cold and shipwreck lead
Scientists have proven the concept of the CUORE experiment, which will study neutrinos with the world’s coldest detector and ancient lead.
Where did all the antimatter go?  The United States at the LHC
http://unitedstates.cern/physics/antimatter
The five greatest mysteries of antimatter  New Scientist
Antimatter mysteries 1: Where is all the antimatter?  New Scientist
https://www.newscientist.com/roundup/antimattermysteries/
Q: Where is all the antimatter?  Ask a Mathematician / Ask a Physicist
http://www.askamathematician.com/2018/01/qwhereisalltheantimatter/
Where did all the antimatter go?  Quora
https://www.quora.com/Wheredidalltheantimattergo
Explainer: What is antimatter?
https://phys.org/news/201601antimatter.html
The Higgs Boson Simplified Through Animation  YouTube
https://www.youtube.com/watch?v=L6AN6UwTTjU Higgs Boson
Is the Higgs boson really the Higgs boson?  YouTube
https://www.youtube.com/watch?v=Jvy0V_AFYfQ Higgs Boson
Brian Greene explains some math behind the Higgs Boson  YouTube
https://www.youtube.com/watch?v=KWj00MCqSxs Brian Greene explains some of the math behind the Higgs Boson
The Higgs Mechanism Explained  Space Time  PBS Digital Studios
https://www.youtube.com/channel/UC7_gcs09iThXybpVgjHZ_7g * Space Time : PBS Studios
https://www.youtube.com/watch?v=kixAljyfdqU Higgs Boson
_{~}
Particle Data Group
Wikipedia Particle Physic, Standard Model, Quantum Field Theory
https://en.wikipedia.org/wiki/Quantum_field_theory
https://en.wikipedia.org/wiki/Standard_Model
https://en.wikipedia.org/wiki/Standard_Model#Fermions
https://en.wikipedia.org/wiki/Standard_Model#Gauge_bosons
https://en.wikipedia.org/wiki/Standard_Model#Higgs_boson
https://en.wikipedia.org/wiki/Lepton
https://en.wikipedia.org/wiki/Quark
https://en.wikipedia.org/wiki/Baryon
https://en.wikipedia.org/wiki/Hadron
https://en.wikipedia.org/wiki/Meson
https://en.wikipedia.org/wiki/Gluon
https://en.wikipedia.org/wiki/W_and_Z_bosons
https://en.wikipedia.org/wiki/Graviton
https://en.wikipedia.org/wiki/Photon
https://en.wikipedia.org/wiki/List_of_particles
https://en.wikipedia.org/wiki/Wave–particle_duality
https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics
https://en.wikipedia.org/wiki/Quantum_gravity
Cosmology
http://abyss.uoregon.edu/~js/cosmo/index.html
http://abyss.uoregon.edu/~js/cosmo/lectures/lec20.html
Javascript Wave Superposition Model
Black Hole Information Paradox
http://www.pbs.org/wgbh/nova/blogs/physics/2018/05/blackholeinformationparadox/
https://m.youtube.com/watch?v=vzQT74nNGME
http://www.physics.ohiostate.edu/~mathur/sissa.html
https://en.m.wikipedia.org/wiki/Black_hole_information_paradox
https://en.m.wikipedia.org/wiki/Holographic_principle
https://www.sciencedaily.com/terms/holographic_principle.htm
https://www.livescience.com/62028hawkingdeathparadoxquestionscience.html
https://www.sciencealert.com/stephenhawkingpublishedsolutionblackholeinformationparadoxdied
https://www.theguardian.com/science/2018/oct/10/stephenhawkingsfinalscientificpaperreleased
https://www.quantamagazine.org/blackholefirewallsconfoundtheoreticalphysicists20121221/
https://www.quantamagazine.org/newfoundwormholeallowsinformationtoescapeblackholes20171023/.
https://www.physicsforums.com/threads/theinformationparadoxandnonlocalityerepr.875792/
https://en.m.wikipedia.org/wiki/ER%3DEPR
Unsolved/ Unanswered Problems/Questions in Physics
List of unsolved problems in physics
https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics
The 18 Biggest Unsolved Mysteries in Physics
https://www.livescience.com/34052unsolvedmysteriesphysics.html
Crunch time for physics: What's next?
https://www.newscientist.com/roundup/physicscrunch/
The 11 Greatest Unanswered Questions of Physics
http://discovermagazine.com/2002/feb/cover/
The Greatest Unsolved Problem in Theoretical Physics
Five Great Problems in Theoretical Physics
https://www.thoughtco.com/fivegreatproblemsintheoreticalphysics2699065
Physics: What We Do and Don’t Know
https://www.nybooks.com/articles/2013/11/07/physicswhatwedoanddontknow/
The 10 Biggest Unsolved Problems in Physics
http://files.aiscience.org/journal/article/html/70310003.html
Open Problems In Mathematics And Physics
FUNDAMENTAL UNSOLVED PROBLEMS IN PHYSICS AND ASTROPHYSICS
http://www.calphysics.org/problems.pdf
5 Unanswered Questions that Will Keep Physicists Awake at Night
The Secret History of Gravitational Waves
The Secret History of Gravitational Waves ***an important read
https://www.americanscientist.org/article/thesecrethistoryofgravitationalwavesNew Quantum Paradox 2018!
New Quantum Paradox Clarifies Where Our Views of Reality Go Wrong ***another important read
https://www.quantamagazine.org/frauchigerrennerparadoxclarifieswhereourviewsofrealitygowrong20181203/Quantum theory cannot consistently describe the use of itself (Daniela Frauchiger & Renato Renner , 2018)
https://www.nature.com/articles/s41467018057398x
My References:
Page 55:
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And last, but not least, is the connection between the BIM+PTs and:
 the pentagonal geometry of the pentagon, decagon (doublepentagon), icosahedron and dodecahedron
 the Fibonacci Numbers series
 the 360° composite, axial view of the DNA doublehelix
 the icosahedronbased capsid form of the Zika, EpsteinBarr, and the majority of human viruses
that will be covered in the APPENDIX.
See Appendices C_D_E: BIM PPTS, Pentagons, Decagons, Phi (ϕ), Fibonacci, Pentagon Connections to the PTs, and, BIM + PT + DNA + Zika, EpsteinBarr and other IcosahedralStructured Human Viruses for ALL the Figures and Tables relating to these topics.
VII. Summary
Brief Synopsis:
 Invention/discovery that the Inverse Square Law (ISL) could be both visualized and mathematically consistent within a simple matrix grid: Brooks Base SquareInverse Square Law Matrix —>BBSISL Matrix —>BIM; (MathspeedST) more…
 Within the BIM, it was found that the “children” of the 345 Primitive Pythagorean Triple (PPT) triangles were ubiquitous in their distribution; (MathspeedST & TPISC I: Basics, The PythagoreanInverse Square Connection) more…
 Thereafter, ALL PTs — primitive “parents” (PPT) and nonprimitive “children” (nPPT) —are found on select Rows of the BIM by simply following the squared numbers on the the Prime Diagonal down to intersecting Rows; ( TPISC I: Basics & TPISC II: Advanced) more…
 The mathematical basis relating the PPTs was found, as was the consistent FractalTemplate Profile (FTP) that every PT follow; (TPISC II: Advanced & TPISC III: Clarity) more…
 The FTP allowed all the PPTs to be sorted out and organized into a definitive Tree of Primitive Pythagorean Triples (ToPPT) that coextends infinitely throughout the infinitely expanding BIM —>BIMtree or BIMToPPT; (TPISC III: Clarity) more… and more…
 Marking (YELLOW) all BIM cells evenly ➗ by 24 generates a striking diamondgrid, crisscrossing pattern with 4 additional YELLOW marked cells in the center of each diamond; (TPISC IV: Details) See 24 Summary.
 Every PPT is found to exist on —and ONLY on — those Rows whose 1st Column grid cell are ➗ by 24 (YELLOW), though not every such YELLOW marked Row contains a PPT. nPPT are only present on such Rows if accompanied by a PPT; (TPISC IV: Details) See 24 Summary.
 The “stepsister” of any given PT Row is found rsteps down the Axis from that Row and it, too, always and only exists on a YELLOW marked Row (“r” is part of the FTP originally derived from the Dickson Method for algebraically calculating all PTs); (TPISC IV: Details)
 The PPT Row (and nPPT Row) always contains the 4A (A=area) value of that PT and the “stepsister” PPT Row always contains the 8A value, both landing exclusively on YELLOW marked grid cells, giving a striking visualization of ALL PPTs and their rbased “stepsisters;” (TPISC III: Clarity & TPISC IV: Details)
 The significance of the “stepsister” is that it becomes the mathematical link to the “NEXT” PPT within the ToPPT — like the RussianDoll model; (TPISC III: Clarity)
 The significance of the expanding and increasingly interconnected PTs, as the BIM itself expands, is one in which the perfectsymmetry geometry of regular shapes and solids — equal triangles, squares, circles,… of the BIM allows — at certain articulation nodes (i.e., Rows) — the introduction of the slightly lessperfectsymmetry geometry (i.e., bilateral symmetry) of the full rectangle and oval that the nonisosceles right triangle PTs represent, into the unfolding structural framework, working from the ground up, if you will. The roots of fractalsbased self organization are first to grow here! (TPISC III: Clarity & TPISC IV: Details) more…
ALL self organization of any sort — be it force field or particulate matter — must have an organizing mathematical layer below driving it!
24 SUMMARY
SUMMARY:
The BIM is Symmetrical down the Diagonal.
ALL Natural Whole Integer Numbers (WIN) on the Axis.
ALL WINs ➗ by 24 in YELLOW (24Y).
ALL PPTs and PRIMES (>2) are ODD #s.
ALL PPTs and PRIMES (>2) are NEVER ➗3.
ALL PTs & ALL PRIMES on 24Y Rows/Columns in PURPLE (= “ACTIVE” Rows).
Some PURPLE 24Y Rows/Columns have neither PT/PRIME.
ODD #s ➗3 are NEVER PPTs or PRIMES, thus NEVER on ACTIVE (PURPLE) Rows.
ODD #s ➗3 ALWAYS follow two ACTIVE Rows forming a repetitive set.
Addition of 24 to ANY ACTIVE Row ODD # = another ACTIVE Row ODD #, while addition to an nonACTIVE ODD # = another nonACTIVE ODD # ➗3.
ALL Squared #s that are PPTs, remain PPTs. ANY PPT #(x) times itself, times its square (x^{2}) and/or times it serial products = NEW PPT
Example1: 5x5=25, 5x25=125, 5x125=625, 5x625=3125, 5x3125=15625=125^{2}, 5x15625=78125, 5x78125=390625=625^{2},… products are ALL PPTs.
Example 2: 97x97=9409, 97x9409=912,673, 97x912,673=88,529,281=9409^{2}=ALL PPTs.
ALL Squared #s that are NOT PPTs, remain NOT PPTs when x^{2} or √x as above.
DETAILS:
 Take BIM and divide all numbers evenly divisible by 24.
 This gives you a crisscross pattern based on 12, i.e. 12,24,36,48,… from Axis.
 Halfway between, are the EVEN # rows based on 6.
 On either side of this 6based and 12based frequency, the rows just before and just after, are ACTIVE Rows. These are ALWAYS ODD # Rows. They form an Active Row Set (ARS).
 Their Axis #s are NEVER ➗3. They ALWAYS have their 1st Col value ➗ by 24.
 Adding 24 to ANY of the ODD # NOT ➗3 ACTIVE Row Axis values ALWAYS sums to a value NOT ➗3 and thus to another ACTIVE Row Axis value (as adding 2 + 4 = 6, ➗3 added to a value NOT ➗3 = NOT ➗3 sum*).
 Another ODD Axis # Row lies before and after each pair of ACTIVE Rows, i.e. between EVERY set of two ACTIVE Rows, is an ODD nonACTIVE Row and their Col 1 value is NOT ➗by 24.
 Adding 24 to ANY of these ODD # ➗3 Axis values ALWAYS sums to a value also ➗3 (as 2 + 4 = 6, ➗3 added to a value already ➗3 = ➗3 sum *).
 While not an exclusive condition, it is a necessary condition, that ALL PPTs and ALL Primes have Col 1 evenly ➗ by 24.
 Together, two ACTIVEs + one nonACTIVE form a repetitive pattern down the Axis, i.e. ARS + nonActive Row.
 *While 24 seems to define this relationship, any EVEN # ➗3 will pick out much if not all of this pattern, e.i., 6, 12, 18,…
 It follows that:
ALL PTs (gray with small black dot) fall on an ACTIVE Row.
ALL PRIMES (red with faint RED circle) fall on an ACTIVE Row.
The difference, ∆, in the SQUARED Axis #s on any two ACTIVE Rows is ALWAYS divisible by 24.
The difference, ∆, in the SQUARED Axis #s on an nonACTIVE ODD Row and an ACTIVE Row is NEVER divisible by 24.
The difference, ∆, in the SQUARED Axis #s on any nonACTIVE ODD Row and another nonACTIVE ODD Row is ALWAYS divisible by 24.
Going sequentially down the Axis, every ODD number in the series follows this pattern:
nA—AA—nA—AA—nA—AA—
#3—57—9—1113—15—1719—21… Every 3rd ODD # (starting with 3) is ➗by 3 = nA .
#57—9—1113—15—1719 Every 1st & 2nd, 4th & 5th, 7th & 8th,… ODD # is NOT ➗ by 3 = A. In other words, the two consecutive ODD #s, between the the nA ODD #s, are A ODD #s and are NOT ➗ by 3.
3,4,5 restated: let A = ACTIVE Row Axis #, nA = nonACTIVE Row Axis
A_{2}^{2}A_{1}^{2}= ➗ 24 and A ≠ ➗by 3
nA^{2}A^{2}≠ ➗ 24
nA_{2}^{2}nA_{1}^{2}= ➗ 24 and nA = ➗by 3
 6,7,8 restated:
ODD Axis #s ➗by 3 (every 3rd ODD #) are NEVER ACTIVE Row members — thus never PT/PRIME
ODD Axis #s NOT ➗by 3 (every 1,2 — 4,5 — 7,8….ODD #s ) are ALWAYS ACTIVE Row members and candidates for being PT and/or PRIME.
In brief:
An ACTIVE Row ODD Axis # squared + a multiple of 24 (as 24x) = Another ACTIVE Row ODD Axis # squared , and the Square Root = a PT and/or a PRIME # candidate :
A_{1}^{2}+ 24x = A_{2}^{2} and √A_{2}^{2} = A_{2} = a PT and/or PRIME candidate;
ODD_{1}^{2}+ 24x = ODD_{2}^{2} and √ODD_{2}^{2} = ODD_{2} = a PT and/or PRIME candidate, if and only if, its 1st Col. value is ➗ by 24.
The difference in the squared values of any two PTs/PRIME #s (>3) is ALWAYS a multiple of 24!
On the Prime Diagonal, the ODD #s follow the same pattern as on the Axis (see No.7)
BIM➗PPTs and PRIMES: (Latest: as this work was being prepared, a NEW relationship was found.) See below under Why?
(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)
SubMatrix 2:
Once again:
These (colored inset boxes) SubMatrix 2 values:
 ALL PPTs have Col 1 ➗4
 NO PPTs have Col 1 NOT ➗4
 For any given Active Rows Set, only 1 Row is a ➗4 Row
 SOME Col 1 ➗4 Rows are NOT PPTs ( starred )
 The NOT PPTs ( starred ) Axis #s are ➗Prime Factors*.
Foolproof Steps to Find ALL PPTs
Axis# must be ODD, NOT ➗3 = Active Row Set (ARS) member
Only 1 of the 2 ARS can be a PPT
SubMatrix Col 1 # MUST be ➗4
SOME may NOT be PPT if ➗Prime Factor (>5)
Remaining Axis # is a PPT. Exceptions:
Squared #s that are PPTs, remain PPTs when x2 or √x,
 ALL Squared #s that are PPTs, remain PPTs. ANY PPT #(x) times itself, times its square (x2) and/or times it serial products = NEW PPT
 Example1: 5x5=25, 5x25=125, 5x125=625, 5x625=3125, 5x3125=15625=1252, 5x15625=78125, 5x78125=390625=6252,… products are ALL PPTs.
 Example 2: 97x97=9409, 97x9409=912,673, 97x912,673=88,529,281=94092=ALL PPTs.
Squared #s that are NOT, remain NOT when x2 or √x, as above.
As to answers to the open questions called above:
what exactly is the relationship between PPTs and PRIMES?
 a loosely threaded connection is quite apparent;
why do they BOTH land on ARS?
 they both must be ODD #s, not ➗3, whose (x^{2}1)/24 is true;
why do some ARS have both, neither, or one or the other?
 some clouds, some clarity, at least for the PPTs;
can the PRIMES be used to predict the PPTs?
 yes, in the sense that if the PPT candidate is ➗Prime Factor (>5), it will not be a PPT;
can the PPTs be used to predict the PRIMES?
 currently, NO, yet the threaded connections are so great that the pattern will eventually emerge!
The difference in the squared values of any two PTs/PRIME #s (>3) is ALWAYS a multiple of 24!
On the Prime Diagonal, the ODD #s follow the same pattern as on the Axis (see No.7)
BIM÷24: SubMatrix Sidebar: What is the role of 24 in the underlying structure?
SEE: Tables: 33a, 33b and 33c. towards the end of Appendix B for some very NEW INFO on the BIM÷24.
The underlying geometry of the BIM÷24 PRESELECTS the Axis Rows into TWO Groups: ARs and NONARs. The PPTs and PRIMES are EXCLUSIVELY — as a sufficient, but not necessary condition — found on the ARs and NEVER on the NONARs. While both Groups follow (PD^{2}  PD^{2})÷24, they do so ONLY within their own respective Groups. They do NOT crossover. This Grouping divide occurs naturally within the BIM as shown in these images below.
The ISL as presented in the BIM is deeply, intimately structured around the number 24 — and its factors: 4,6, 3,8 2,12, and 1,24.
The interplay between these small sets of Numbers generates an incredible amount of richness and complexity with seemingly simplistic BIM itself. This has led to TPISP: The PythagoreanInverse Square Connection, and the PRIMES.
Open in separate browser tab/window to see all.
BIM➗PPTs and PRIMES
BIM➗PPTs and PRIMES: (Latest: as this work was being prepared, a NEW relationship was found. SEE: Tables 3132 in Appendix B for a great deal more info and proofs!)
A dovetailing of PPTs and PRIMES on the BIM
The discovery of the Active Row Sets (ARS) — the direct result of the BIM ➗24 — in which it has been found that ALL PPTs and ALL PRIMES are exclusively found on, was in and of itself, a slow an arduous journey.
Once found, it has added a great deal of visual graphic clarity! In simple terms, it simply marks out the obvious. Both the PPTs and the PRIMES can not be on Axis Row #s that are EVEN, nor ➗3. This leaves ONLY Rows that are ODD #s and not ➗3.
The BIM ➗24 marked those Active Row Sets indirectly, by being on either side — i.e., +/ 1 — of the Axis Row # intercepted by the ➗24. Directly, the ARS was shown to be picked out by Submatrix 1 and 2 values of the 1st cell column of those Rows.
So we have the PPTs and the PRIMES occupying the same footprint rows, the ARS Rows. Both as a necessary, but not sufficient for primality requirement, i.e., some ARS Rows do NOT have a PPT or PRIME, or both. ALL PPTs and ALL PRIMES are ALWAYS found on an ARS Row, NEVER on a nonARS Row. Some ARS Rows may have none, either a PPT or a PRIME, or both.
Nevertheless, on this vast matrix array of ISL whole integer numbers, that the PPTs and PRIMES exclusively occupy the same ➗24based footprint points to an underlying connection!
The 1st connection was found and written about in the three white papers of 20056 on PRIMES:
 Page 2b: Butterfly Primes...let the beauty seep in.
 Page 2c: Butterfly Prime Directive...metamorphosis.
 Page 2d: Butterfly Prime Determinant Number Array (DNA) ~conspicuous abstinence~.
The 2nd connection, as referenced below, has been the latest discovery that Euler’s 6n+1 and 6n1 pick out, as a necessary — but not sufficient for primality — condition ALL the PRIMES.
When you look at the BIM➗24, you can readily see how this theorem simply picks out the very same ARS Rows! (For ARS 5 and above.)
The BIM➗24 becomes a DIRECT GRAPHIC VISUALIZATION of EULER’s PRIMES = 6n+1 and 6n1, where n=1,2,3,..
The 3rd connection is that for those ARS Rows that do NOT contain PRIMES — e.i., 25, 35, 49, … and have been shown to negate the possibility of the # being prime because it is itself prime factorable — divisible by another set of primes — is ALSO DIRECTLY VISUALIZABLE ON THE BIM➗24 AS THE INTERSECTING PRIME COLUMNS!!!
The 4th connection reveals that the BIM Prime Diagonal (PD) — the simple squares of the Axis #s — defines:
 The ISL itself, as every BIM Inner Grid cell value is simply the difference between its horizontal and vertical PD values;
 The Pythagorean Triples, as every PD cell value points to a PT when one drops down from it to its ARS intercept;
 The PRIMES, as the difference in the squares of any two PRIMES (≧5) — every PD value of a PRIME Axis # — is evenly ➗24.
The 5th connection is that for those ARS Rows— that may or may not contain PPTs and/or PRIMES — their 1,5,7,11,13,17,19,23,25,.. ODD intersecting Columns NOT ➗3, ARE ALL➗24, and, this is ALSO DIRECTLY VISUALIZABLE ON THE BIM➗24 AS THE INTERSECTING COLUMNS (usually depicted graphically in YELLOWORANGE boxes/cells on the BIM as part of the diamond with centers pattern) !!!
￼
￼ ￼ If P = X, evenly ➗24, then P = a PRIME Candidate where, X=evenly➗24, P=Prime, P^{2}_{2}=larger Prime Squared, P^{2}_{1}=smaller PRIME Squared, n=1,2,3,…TPISC, The PythagoreanInverse Square Connection, has evolved into a connection with the PRIMES (TPISCP).
How Visualization of the PRIMES on the BIM ÷24 Simplifies Much of the Mysteries of the PRIME Conjectures and What This Means!
Much is made of the PRIME Conjectures: https://primes.utm.edu/notes/conjectures/ & site:primes.utm.edu Prime Gap conjecture
 Goldbach (Strong) Conjecture (every even # is made of two primes)
 Goldbach Tertiary (Weak) Conjecture (every odd # is made of three primes)
 Twin Prime Conjecture (there are an infinite # of primes separated by 2)
 Twin Primes Conjecture (there are an infinite # of primes separated by a fixed # gap)
 Prime Gap (size between consecutive primes)
 Prime Triple Conjecture (there are an infinite # of 3 consecutive primes with ∆ of 6, first and last)
 Prime Quadruple Conjecture (there are an infinite # of 4 consecutive primes with ∆ of 8, first and last)
 Prime ktuplet Conjecture (there are an infinite # of prime ktuplets for each* k*)
 Dickson’s Conjecture (there are a lot of primes, Twin, Sophie Germain, ktuplet,…)
 Pythagorean Primes Conjecture(Pythagorean triples with prime # hypotenuse)
A look at the BIM÷24 grid will quickly give one a visualization of the limits — and expanses — of these conjectures!
(1 above.) This was examined and a proofsolution was offered in 2010 with the Periodic Table of Primes (PTOP), hidden within the BIM.
(2 above.) Work remains.
(3 above.) ALL Primes fall on Active Rows (ARs) within an Active Row Set (ARS) of three Axis #s: two ODDs with an EVEN inbetween. The difference (∆) between the ODDs = 2. Thus ANY and ALL Twin Primes — separated by 2 — are seen right here, and only here, directly, on the ARS.
(48 above.) Because the ARSs follow a strict Number Pattern Sequence (NPS) in that the EVEN #s are ÷12 — being the endpoint of BIM÷24 — and the bookcased ODD #s are NEVER ÷3, there is a builtin ∆, a natural gap, between the ARS and ODD #s ÷ 3. This means the ARs — that provide a necessary, but not sufficient requirement for ANY PRIME — must necessarily have gaps that are:
taken from the lower ARS #:
 (Twins of) 2, 6, 8, 12, 14, 18, 20,… i.e., up 2, 4, 2, 4, 2, 4, 2,….
taken from the higher ARS #:
 (Twins of) 4, 6, 10, 12, 16, 18, 22,… i.e., up 4, 2, 4, 2, 4, 2, 4,….
(9 above.) Dickson’s Conjecture leads to many of the individual conjectures above. One, the Sophie Germain Prime Conjecture states that if a PRIME #, p, has another PRIME # generated at 2p + 1, it is a Sophie Germain PRIME and that there are an infinite number of these. On the BIM, it is easy to see that any Sophie PRIME is simply taking the lower AR # and ADDING a multiple of 6 — 6x — to get to the next PRIME. Try it: (≧5)5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, ... OEIS: A005384. (https://en.wikipedia.org/wiki/Sophie_Germain_prime)
*the lower AR + 2 = higher AR — neither is ÷ by 3, while the lower AR  2 = ODD# that is ÷ by 3 and NOT an AR.
This works for p≧5:
5+6=11, 11+(2)6=23, 23+6=29, 29+(2)6=41, 53+(5)6=83, 83+6=89, 89+(4)6=113, 113+(3)6=131, 131+(7)6=173, 173+(1)6=179, 179+(2)6=191, 191+(7)6=233, 233+(1)6=239, 239+(2)6=251, 251+(5)6=281, 281+(2)6=293, 293+(11)6=359, 359+(10)6=419, 419+(2)6=431, 431+(2)6=443, 443+(8)6=491, 491+(3)6=509, 509+(14)6=593, 593+(8)6=641, 641+(2)6=653, 653+(1)6=659, 659+(4)6=683, 683+(6)6=719, 719+(4)6=743, 743+(3)6=761, 761+(8)6=809, 809+(17)6=811, 911+(7)6=953, …
It appears that this conjecture really is simply restating Euler’s 6n+1 and 6n1 — where n=1,2,3,… method for determining ALL PRIMES.
(10 above.) Ironically, the Pythagorean Triples were first to be found on the ARS, and only after reviewing earlier work — the Butterfly Primes — that the connection of the PRIMES to the BIM÷24 was made. Both the Primitive Pythagorean Triples (PPTs) and the PRIMES have the same — again, necessary, but not sufficient requirement — that they occupy EXCLUSIVELY the ARs potentially of any ARS. When the PPTs overlap the PRIMES, we get the Pythagorean Primes. An interesting note is that unlike the PRIMES that can occupy BOTH ARs of a given ARS, the PPTs will NEVER do this, occupying either one or the other, but NEVER BOTH, ARs within a ARS.
Fermat's (FermatEuler) 4n + 1 = Sum of Two Squares Theorem, where 4n + 3 ≠ Sum of Two Squares. The 4n + 1 = Sum of Two Squares = Pythagorean Primes (PTs where the hypotenuse, c = Prime #).
Remember, not only does the BIM÷24 reveal the ARs within the ARS, the∆ BETWEEN THE SQUARES OF ANY PAIR OF AR Axis #s (≧5) IS A MULTIPLE OF 24. The latter becomes a necessary, but not sufficient test of both primality, and, PPT validity.
Note that this same ∆ occurs with pairs of two nonAR ODD #s, i.e. 9 and 15. Both are ÷3. 152  92 = 144, and 144/24 = 6. A hybrid of an AR and a nonAR ODD pair set will NOT show a ∆ of 24x.
This points to underlying NPS within the BIM (see the pattern below the links):
For the ODD #s^{2}:
∆÷24 vs NOT ∆÷24
∆÷24 in ARS vs ∆÷24 in nonARS, NEVER a mix of the two
For the EVEN #s^{2}:
∆÷24 vs NOT ∆÷24
∆÷24 includes those in progressive NPS series ∆2, ∆4, ∆6, ∆8, ∆10,… (see Table in Appendix ).
The PPTs and PRIMES are STRICTLY following this NPS within the BIM!
Perhaps the biggest finding here is: The PRIMES — Inverse Square Law Connection! Just like the PPTs! TPISC stands for both The PythagoreanInverse Square Connection, AND, The PRIMESInverse Square Connection. TPPISC, TP/PISC, TP/PISC, TPPISC, or TPISPC,...
The connection between that Universal Law, the ISL, — the underlying law of ALL of spacetime —and the PRIMES — the fundamental "quarks" of the number quantity system, AND, the Pythagorean Triples — the fundamental righttriangle/rectangle form of that same geometry — is without out a doubt the most intriguing, beguiling, and misunderstood relationship we are only just NOW getting a real glimpse at. The future looks very promising!
Additional useful links:
See Appendix B: Tables 3133 for more info.
https://en.wikipedia.org/wiki/Goldbach%27s_conjecture
https://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture
https://en.wikipedia.org/wiki/Pythagorean_triple
https://en.wikipedia.org/wiki/Twin_prime
https://en.wikipedia.org/wiki/Prime_gap
https://en.wikipedia.org/wiki/Prime_triplet
https://en.wikipedia.org/wiki/Prime_quadruplet
https://en.wikipedia.org/wiki/Prime_quadruplet#Prime_quintuplets
https://en.wikipedia.org/wiki/Prime_ktuple
https://en.wikipedia.org/wiki/Dickson%27s_conjecture
https://en.wikipedia.org/wiki/Sophie_Germain_prime
https://en.wikipedia.org/w/index.php?search=Pythagorean+Primes&title=Special%3ASearch&go=Go
https://en.wikipedia.org/wiki/Prime_ktuple
https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares
https://en.wikipedia.org/wiki/Pythagorean_prime
https://en.wikipedia.org/wiki/Fermat%27s_little_theorem
https://en.wikipedia.org/wiki/Euler%27s_theorem
https://en.wikipedia.org/wiki/Euclid–Euler_theorem
http://mathworld.wolfram.com/Eulers6nPlus1Theorem.html
What is the underlying NPS of the BIM ÷24?
Earlier we found:
For the ODD #s^{2}
 ∆÷24 vs NOT ∆÷24
 ∆÷24 in ARS vs ∆÷24 in nonARS, NEVER a mix of the two
For the EVEN #s^{2}
 ∆÷24 vs NOT ∆÷24
 ∆÷24 includes those in progressive NPS series ∆2, ∆4, ∆6, ∆8, ∆10,… (see Table in Appendix ).
The PPTs and PRIMES are STRICTLY following this NPS within the BIM!
But what about just the BIM ÷24. How does this pattern out?
Let’s dispense with the EVENs first.
If one takes the EVEN #s on either side of the ARS + the EVEN # in the MIDDLE of the ARS, we have a new set. Let’s call it the EVENs Set (ES).
Well, it turns out there are two versions of the ESs and they alternate down the PD. Let’s call them ES_{1} and ES_{2} and they go: ES_{1}ES_{2}ES_{1}ES_{2},…
If we plot the ESs along the PD it will look like this (remember these are the squares of the EVEN Axis #s):
16 — 36 — 64 100 —144 — 196
256 —324— 400 484 — 576 — 676
784 —900—1024 1156—1296—1444
1600—1764—1936 2116—2304—2500
The ES_{1} are on the Left, the ES_{2} are on the Right.
The ÷24 NPS of the ES_{1} andES_{2} play out as:
Both group sets bookend the ARS with an ODD #÷3 (NONARS ODD) between.
The OUTER ends of ES_{1} are ∆/24 and the OUTER ends of ES_{2} are ∆/24.
The MIDDLE of ES_{1} are ∆/24 and the MIDDLE of ES_{2} are ∆/24.
There is NO MIXING between sets and NO MIXING OUTER with MIDDLE #s.
 The MIDDLE ES_{1} is ÷ by the 1st MIDDLE of the whole ES_{1} set.
 The MIDDLE ES_{2} is ÷ by the 1st MIDDLE of the whole ES_{2} set.
ES_{1} ÷ 16 ÷ 8 ÷ 4 ÷ 2 OUTER; MIDDLE ÷ 36, NEVER ÷ 24. (NOTE: these are the PD#s and not the ∆ in PD#s.)
ES_{2} ————÷ 4 ÷ 2 OUTER; MIDDLE ÷ 36, ALWAYS ÷ 24. (NOTE: these are the PD#s and not the ∆ in PD#s.)
A look at the BIM ÷24 shows that ALL MIDDLE PDs of their ESs are ÷12 and graphically:Table 31d. Snapshot of the PDF showing the EVENS role in the BIM ÷24 Distribution.
ES_{1} fall along the INBETWEEN YELLOW DIAMOND PATTERN formed from the BIM ÷24.
 NEVER ÷ 24.
ES_{2} fall at the Axial POINTS of the YELLOW DIAMOND PATTERN.
 ALWAYS ÷ 24.
Table31d PDF sequence. The EVENS. Open the PDFs in a separate tab/window to see all pages.
Table 33c. The EVENS
So now that we have established that the EVENS, too, reveal a ÷24 expression within the BIM, let's move on the ODDS.
What about the ODDs?
We know that the ODDs ÷3 are the separators of the EVEN's ES_{1} and ES_{2} . They are also a separate player in the whole ODDs set in that though their PDs are ÷24, they are NOT part of the ARS. And as NONARs they do NOT MIX with the AR ODDs.
We will continue to label the two groups as ARs and NONARs.
Earlier we found:
For the ODD #s^{2}
 ∆÷24 vs NOT ∆÷24
 ∆÷24 in ARS vs ∆÷24 in nonARS, NEVER a mix of the two.
More precisely:
ALL ARs^{2} are ∆/24, and, NEVER ÷3, and after subtracting (1) are ALWAYS ÷24, and, ALL now ÷3.
ALL NONARs^{2} are ∆/24, and, ALWAYS ÷3, and after subtracting (1) NEVER ÷24, but ALL ÷9 ÷3.
There is NO MIXING between groups.
Additionally, the ODD AR group may be further divided into:
ODD Set 1 (OS_{1}): (Like the EVENS, ES_{1}) for INBETWEEN YELLOW DIAMOND Rows 6,18,30,42,...
ODD Set 2 (OS_{2}): (Like the EVENS, ES_{2}) for the POINTS of the YELLOW DIAMOND Rows 12,24,36,...
However, here, BOTH groups ARE MIXABLE, i.e. ∆/24 across OS_{1} and OS_{2}.
What really becomes interesting about the ODDs on the BIM ÷24 is that, all by themselves without regard to the PPTs or PRIMES, they form a deep, underlying NPS ALL BASED ON THEIR ∆s/24!
In a nutshell, if you take each ODD #, square it (that's the PD#) and now sequentially substract the ODD PDs below it, and then ÷24, the results will generate a tightly connected NPS across the ODDS on the BIM.
The ARs^{2} 12549121169289361529625,… (1^{2}5^{2}7^{2}11^{2}13^{2}17^{2}19^{2}23^{2}25^{2},) subtraction=ALL ARs.
The NONARs^{2} 981225441729,… (3^{2}9^{2}15^{2}21^{2}27^{2},…) subtraction=ALL NONARs.
Here is a quick look at the AR and NONAR Distribution. Open the PDFs below in a separate tab/window.
Here is a quick look at the AR and NONAR Distribution. Open the PDFs below in a separate tab/window.
Table 33a. ODDs AR and NONAR Distribution PDF. Open the PDFs in a separate tab/window to see all pages.
Table 33b. ODDs sequenced PDF. Open the PDFs in a separate tab/window to see all pages.
Table 33c ODDs_more detail PDF. Open the PDFs in a separate tab/window to see all pages.
Table 33d21
Table 33d22
Table 33d2
~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~
NEWLY ADDED (after TPISC IV published):BACK: > PRIME GAPS on a separate White Paper BACK: > PeriodicTableOfPrimes(PTOP)_GoldbachConjecture on a separate White Paper BACK: > Make the PTOP with Fractals on a separate White Paper BACK: > TPISC IV: Details White Paper
Back to Part I of the BIMGoldbach_Conjecture.
Back to Part II of the BIMGoldbach_Conjecture.
Back to Part III of the BIMGoldbach_Conjecture.
BACK: > TPISC IV: Details: BIM + PTs + PRIMES on a separate White Paper BACK: > TPISC_IV: Details:_PRIMES_vs_NOPRIMES on a separate White Paper
Artwork referencing TPISC.TPISC_IV:_Details ~PRIMES vs NOPRIMES~
Copyright©2019, Reginald Brooks. All rights reserved.
(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24. Note: open the PDFs into a separate tab/page to see the full content. The png image files give a quicklook only.)
PTs: 40+ FEDM derived profiling focus point parameters:
While the actual Area (A) and Perimeter (P) of the PT in question can be shown directly on the Matrix as grid cell areas — as can the PT proof — it is the 4x the A, and, the difference (∆) in the length of the sides, squared, that present the true Pythagorean  Inverse Square Connection.
4A + (ba)^{2} = c^{2} = a^{2} + b^{2}
where, (ba)^{2} = (ts)^{2} = ƒ^{2}, giving 4A + ƒ^{2} = c^{2} = a^{2} + b^{2}
Finding the A and P on the Matrix necessarily introduced some additional key focus point values — giving rise to the FEDM.
The 40+ key focus points are found consistently in each and every rset Template — defining the PTs.
Every PT Row contains at least 8 key values directly;
 c — x_{f} — o — b^{2} — 4A — d — a^{2} — c^{2}
And another 8 values indirectly by counting STEPS;
 r/4 — r/2 — r — a —t —p — s — b
Giving another 8 values, by calculation/STEPS or a mixture of both;
 A — P — U — e — ƒ — ƒ^{2} — F — r^{2}
And additionally;
 8A — @ — Ƀ — d/p — g —h — i — J — √J — k — l — m — n — p^{2} — q — v — V — W — √W — ¥.
If you plot these out along their rset STEPS (rsteps) spacings, the ∆ between values and their PD for any given Column follows 1r^{2} — 4r^{2} — 9r^{2} — 16r^{2} — 25r^{2} —…;
c^{2}o=1r^{2} 169153=16=1r^{2}, c^{2}d=4r^{2} 169105=64=4r^{2}, c^{2}g=9r^{2} 16925=144=9r^{2}
Naturally, the PD sequence, up from the Row values, follows the same 1—4—9—16—25—...
The "downward" Diagonal, perpendicular from the PD, back to the Row Axis gives the Ƀ — V — ¥ and cƒ points, and, 8A is always rsteps down the grid from the 4A location on that PT Row.
PTs: 10 ways to approach the BIM :
The PT Axis Row: ALL PTs have c — X_{f} — o — b^{2} — 4A — d — a^{2} — c^{2}
 c is on the Left SIDE Axis;
 X_{ƒ} is r/2 steps in from Axis;
 o is rsteps in from Axis;
 b^{2} is asteps in from Axis;
 4A = cƒ steps coming back from the PD;
 d is 2rsteps in from the Axis;
 a^{2} is bsteps in from the Axis;
 c^{2} is on the PD at the intersect of the PT Row.
The Main Diagonals (Template) for ALL PTs:
PD — O (origin)—>c^{2} as o—ƒ^{2}—r^{2}—a^{2}—b^{2}—c^{2};
⊥PD — c^{2}—>2c (Axis) follows 4c—8c—12c—16c sequence.
The Secondary Diagonals:
 p—U—d this important diagonal defines the Common Diagonal of the Golden Diamond ToPPTs;
 ⊥d—e—d/p (on Axis).
The Tertiary Diagonals:
 √W—o;
 ⊥o—√J (on Axis).
The Horizontal Axis (TOP) of Columns:
 0—ALL of the above—from c —>c^{2};
 r/4—r/2—r—2r—3r—4r—.. and ba=ts.
The Vertical Axis (Left SIDE) of Rows:
 0—ALL of the above—from c —>2c;
 s—p—√W—c—√J—d/p—t_{n}—2c—r—2r—3r—4r—....
The 4A—8A—c^{2}—ƒ^{2} Rectangle Connection:
The Proof a^{2} + b^{2} = c^{2} = 4A + ƒ^{2}:
The Complementary Pair (Square Pairs) Sets of ANY PT:
ALL PTs have matching 4A values on both the PD Row itself and rsteps down that PD Column:
BBSISL Matrix (BIM): Basic, fundamental Rules of the Symmetrical Matrix grid:
10 Basic, fundamental rules of the symmetrical BBSISL Matrix
Basic BBSISL Rule 1: All numbers (#s) related by the 149...PD sequence
Basic BBSISL Rule 2: Every # in the PD sequence is the square of an Axial #.
Basic BBSISL Rule 3: The OddNumber Summation sequence forms the PD sequence.
Basic BBSISL Rule 4: Every EVEN Inner Grid (IG) # is divisible by 4 & all are present.
Basic BBSISL Rule 5: Every IG# is:
A: The difference (∆) between its two PDsequence #s. (Note: A=B=C=D, and, E.)
 Ex: PD 25  PD9 = 16
B: The sum (∑) of the ∆s of each of its PD#s between its two PDsequence #s (as above).
 Ex: (PD 25  PD16) + (PD16  PD9) = 16
C: The ∆ between the squares of the two Axial #s forming that IG# (as above).
 Ex: 5^2  3^2 = 16
D: The product of the Addition & Subtraction of the two Axial #s forming that IG# (as above).
 Ex: (5 + 3) x (5  3) = 16
E: Also, the product of its 2 Axial #s intersected by that IG#'s 90° diagonals.
 Ex: 2 x 8 = 16
Basic BBSISL Rule 6: Every ODD IG# is NOT PRIME & all are present.
 Corollary: NO EVEN NOT divisible by 4 #s are present on the IG.
Basic BBSISL Rule 7: The ODDNumber sequence, and the 149...PD sequence, forms the sequential ∆ between ALL IG#s.
Basic BBSISL Rule 8: The ∆ between #s within the Parallel Diagonals is a constant 2 x its Axial #.
Basic BBSISL Rule 9: The ∆ between #s in the Perpendicular Diagonals follow:
A: From EVEN PD#s, √PD x 4 starts the sequence & follows x1x2x3x4....
B: From ODD PD#s, √PD x 4 starts the sequence & follows x1x2x3x4....
C: From ODD Perpendicular Diagonals between the EVENODD diagonals (above), the sequence starts with the same value as the Axis number ending the diagonal, the sequence following x1x3x5x7..
Basic BBSISL Rule 10: Every #, especially the #s in the ONEs Column, informs both smaller and larger Subset symmetries (much larger grids required to demonstrate).
BBSISL Matrix Inner Grid Golden Rules (IGGR)
5 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner Grid
IGGR 1: The IG is formed of two equal & symmetrical 90°right, isosceles triangles that are bilaterally symmetrical about the PD — and, infinitely expandable.
IGGR 2: The 90°righttriangle — inherent to ALL squares and rectangles by definition — both forms the alternating EVENODD square grid cells within the Matrix, and, is responsible for all major patterns and sequences, thereupon.
IGGR 3: SubtractionAddition: Every IG# is simply the ∆ between its two PD#s (subtraction), and, the sum (∑) of any IG# + its PD# above = the PD# on the end of that Row (or, Column).
IGGR 4: MultiplicationDivision: Every IG# is simply the product of the two AXIAL #s intersected by the two diagonals — of that said IG# — pointing back to the Axis at a 90° angle (multiplication), and, the dividend of the Axial divisor and quotient (division).
IGGR 5: The actual # of gridcell steps — i.e., the actual # of STEPS from a given IG# to another by a strictly horizontal, vertical, or 45° diagonal path — forms a simple, yet often fundamental descriptor to the patternsequence templates that inform the more advanced patterns, e.i., Exponentials and especially the Pythagorean Triples (PTs). STEPS are particularly important in the geometric visualizations within the BBSISL Matrix (as alluded to in IGGR 2, above).
Pythagorean Triples and BBSISL Fundamentals (TPISC: The PythagoreanInverse Square Connection)
3 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner Grid that encompass the PTs.
 TPISCBBSISL Rule 1: Every IG EVEN Squared # is part of a Square Paired Set (SPS) that:
 A: Has reciprocal SPS members on the PD vertically above.
 B: Both SPS members reside on the SAME Row.
 C: They represent the a^{2} and b^{2} values of a PT, whose c^{2} value is on the PD intersection
TPISCBBSISL Rule 2: Every PT is found on the BBSISL Matrix and can be located by this intersection of EVERY PD (9>) and a Row with SPSs.
TPISCBBSISL Rule 3: Every PT — including its sides, perimeter, area and proof — can also be found and fully profiled (and, predicted) as rset, s,tset members of the Dickson Method (DM), Expanded Dickson Method (EDM), and the Fully Expanded Dickson Method (FEDM), shown herein. The role of the rvalue for any given PPT (or nPT) can not be overstated: the rvalue is as equally important in defining any PT as the a, b and csides.
Witness:
r = a + b  c
and,
(a + b + c + r)/2 = c + r = a + b.
And as
a = r + c  b = (c + r ƒ)/2
solving for ƒ gives:
ƒ= ba
and,
ƒ^{2}= (ba)^{2}
giving the The Proof
a^{2} + b^{2} = c^{2} = 4A + ƒ^{2}.
With
p = c  2r
we can subsitute in
c = a + b  r
to give:
p = a + b  3r
giving us all the basics to profile any PT, knowing that:
4A = rc + r^{2} = 2ab
Matrix Flow:
 ALL WINs;
 Axis (vertical, horizontal) +IG + PD = BIM;
 BIM  (Axes & PD) = Inner Grid (IG);
 IG  1^{st} Parallel Diagonal (P∥D)= Strict Inner Grid (SIG);
 Axes = ALL WINs (X) sequentially on both vertical (Left SIDE) and horizontal (TOP) Axis;
 PD = Axis^{2} = X^{2} = a^{2} — b^{2} — c^{2} of ALL Pythagorean Triples (PTs);
 Axis Rows & Columns define the PD, IG, ALL PTs and ALL Exponentials X^{n};
 Parallel Diagonals (to the PD) define ALL Exponentials X^{n} (Exp X^{n});
 ALL PTs and ALL Exp X^{n} are on the BIM;
 ALL Squared #s, by definition, are on the PD;
 ALL Squared #s are ALSO on the IG;
 ALL Squared Sides (a^{2}, b^{2}, c^{2}) of ALL PTs (across Rows) & ALL Exp X^{>2} (down Parallel Diagonals) are also on the IG;
 Therefore, ANY Squared # on the IG is PART OF A PT & some, but NOT ALL, are also Exp X^{>2};
 ALL Exp X^{n} are found on the IG, specifically along their Parallel Diagonals (P∥Ds) ;
 ALL Exp X^{n} values, like ALL IG #s, are simply the ∆ between their two PD Area values.
Exponentials Summary (see TPISC V: Exponentials):
For any given # X, located on the Axis, its respective x^{2} is, of course, located on the PD, while X^{3}, X^{4}, X^{5},.. (X^{n}), are ALL found on that X Diagonal Parallel ( P∥D) to the PD;
The distance — # of steps diagonally — between successive Exponentials X^{1,2,3..}. for a given X, follows a Number Sequence Pattern (NPS) equal to is X^{n} sequence value:
The Sum (∑) of the Axis Column & Row # values x that Diagonal Axis # X, equals the X^{n} value:
 ∑(Axis_{Col} + Axis_{Row}) x X = X^{n} as does the Axis_{Col} x Axis_{Row} product;
 ∑(6+10) x 4 = 64 = 4^{3}, where X=4, and, Ax_{Col} x Ax_{Row} = 4 x 16 =64 = 4^{3}.
The Sums (∑s) of the ∆s between the PD #s of a given X^{3}_{5}=X^{3} and is simply an expression of the IGGR:
 2^{3} = 149 with ∆s of 3 & 5, where 3 + 5 =8 = 2^{3}.
The Area (# of grid cells) of a given X^{x} = Area X^{x}~_{uo}~  Area X^{x}~_{LD}~, and flollows the same NPS progression sequence as X^{3}, X^{4}, X^{5},… in Area and in # of PD steps.
 1^{st} P∥D = # ∆ 2, ODD #s (Prime & NotPrime [NP]);
 2^{nd} P∥D = # ∆ 4, EVEN #s ➗ 4 = ALL Exp 2^{n};
 3^{rd} P∥D = # ∆ 6, ODD #s NP= ALL Exp 3^{n};
 4^{th} P∥D = # ∆ 8, EVEN #s ➗ 4 = ALL Exp 4^{n};
 5^{th} P∥D = # ∆ 10, ODD #s NP= ALL Exp 5^{n};
 6^{th} P∥D = # ∆ 12, EVEN #s ➗ 4 = ALL Exp 6^{n};
 7^{th} P∥D = # ∆ 14, ODD #s NP= ALL Exp 7^{n};
 8^{th} P∥D = # ∆ 16, EVEN #s ➗ 4 = ALL Exp 8^{n}.
Summation:
Here’s the thing. Amongst a myriad of other connections, there exist an intimate connection between three number systems:
 The ISL as laid out in the BIM;
 PTs — and most especially PPTs — as laid out on the BIM;
 The PRIME numbers — PRIMES — as laid out on the BIM.
The BIM is the FIXED GRID numerical array of the ISL.
Amongst its vast array of interconnecting Number Pattern Sequences (NPS) — i.e., number systems — two such systems stick out and do so in such an overtly visual — as well as mathematical — way that their connection to each other is more than implied.
You see, both the PPTs and PRIMES strictly align themselves on the SAME paths within the BIM.
Now, their footprints upon these paths are not identical, yet their paths chosen are. If you divide the number array — i.e., the Inner Grid numbers — of the BIM by 24, a SubMatrix 1 grid is formed.
Upon that SubMatrix 1, pathways are formed on every ODD Axis number NOT ➗3.** EVERY PPT and PRIME lies on these paths!**
Yes, while in the details we show how:
 The 1st Col. on the BIM — that which is the square of the Axis number (i.e., the Prime Diagonal number)  1, when then ➗ by 24, equals a Whole Integer Number (WIN);
 This defines the path — the “Active Row” upon the BIM;
 While every Active Row path may or may not contain a PPT and/or PRIME, every PPT and/or PRIME is ALWAYS located on one of these paths;
 The difference (∆) between the squares of any two PRIMES (>5) is also ➗24;
 The serial — and exponential — products of ANY and ALL PPTs remain PPTs, whilst those NOT remain NOT.
Furthermore, by distilling the BIM to SubMatrix 2 — i.e., every number across a Row is progressively divided by a growing sequence series starting with the Col. 1 as the Axis number  1 — every such serialexponential PPT is clearly predicted by its neighbors within the sequence. One more example of the extremely intimate relationship between The Pythagorean  Inverse Square Connection (TPISC).
One simply can not ignore how the PRIMES, the PTs and ultimately the ISL define the architecture of SpaceTime!
In the original MathspeedST, an artificial division was made, separating the content into:
 TAOST, The Architecture Of SpaceTime,
 TCAOP, The Complete Absence Of Primes.
Now we have come full circle.
ALL PRIMES and ALL PPTs follow — although individually with their own respective footprints — the SAME, HIGHLY PATTERNED NPS path on the BIM.
This is no coincidence. The ➗24 Active Row Pattern that defines this path on the BIM does so in a highly ordered pattern. The energy density that expresses itself as curved ST does so precisely by the numerical architect of the builtin ISL.
We now have some very strong evidence that the numbers that define the PPTs and the numbers that define the PRIMES are ubiquitously linked throughout the entire number pattern array that defines the ISL. This is revealed on the BIM!
Surely their interplay provides some very significant contributions to the overall Architecture of SpaceTime!
feel the beat: +  +    +  +    +  +    +  +    ...
VIMEO "BIM_PRIMES_24"1! )
BIM_PT_PRIMES_24 from Reginald Brooks on Vimeo.
UPDATE: Finally, the NEW relationship between the BIM÷24 and the PRIMES has been found! See the bottom of APPENDIX B for the tables that document this work! From the APPENDIX B:
Back to the Table 31 sequences. Table 31a... series, along with the previous Table33 series, uncovers how the PRIMES were found within the BIM both by algebra and geometry. This has been reduced to a focus white paper: "PRIMES vs NOPRIMES that is a condensed summary of this work. One may benefit from looking over this paper prior to the study of these more elaborate tables. A summary from the paper is presented here:
VIII. ConclusionSUMMARY*
(~from "*Simple Path from the BIM to the PRIMES" that presents the algebraicgeometry method.)
How do we go from the simple grid of the BIM (BBSISL Matrix) to identifying the PRIMES?
The BIM is a symmetrical grid — divided equally down its diagonal center with the Prime Diagonal (PD) — that illuminates the Number Pattern Sequence (NPS) of the Inverse Square Law (ISL) via simple, natural Whole Integer Numbers (WIN).
The BIM Axis numbers are 1,2,3,.. with 0 at the origin.
The Inner Grid (IG) contains EVEN and ODD WIN, but except for the 1st diagonal next to the PD — a diagonal that contains ALL the ODD WIN — there are NO PRIMES (NOPRIMES, NP) on the SIG (Strict Inner Grid).
The PD WIN are simple the square of the Axis WIN.
ALL the IG WIN result from subtracting the horizontal from the vertical intersection of the PD.
Dropping down a given PD Squared WIN (>4) until it intersects with another squared WIN on a Row below will ALWAYS reveal that Row to be a Primitive Pythagorean Triple (PPT) Row, whose hypotenuse, c, lies on the intersecting PD. ALL PPTs may be identified this way.
Dividing the BIM cell values by 24 — BIM÷24 — forms a crisscrossing DIAMOND NPS that divides the overall BIM into two distinct and alternating Row (and Column) bands or sets:
 ODD WIN that are ÷3 and referred to as NONARs;
 ODD WIN that are NOT ÷3 and referred to as ARs, or Active Rows;
 The ARs ALWAYS come in pairs — with an EVEN WIN between — as the UPPER and LOWER AR of the ARS (Active Row Set);
ALL PPTs and ALL PRIMES ALWAYS are found exclusively on the ARs — no exceptions.
By applying:
(1) *
let y = odd number (ODD) 3, 5, 7,… and x = 1, 2, 3,... one generates a NP table containing ALL the NP;
*True if ÷3 ODDs are first eliminated, otherwise ADD exponentials of 3 to the NP pool;
Eliminating the NP — and the NP contain a NPS — from ALL the ODD WIN, reveals the PRIMES (P).
A necessary, but not sufficient confirmation — but not proof — of primality is found by finding the even division of 24 into the difference of the square of ANY two PRIMES as:
(2) (P_{2})^{2}  (P_{1})^{2} =
let
Be aware that this also holds true for ALL the AR NP. The P and NP are NOT ÷3, and are both part of the ARS and therefore any combination of the two squared differences will be ÷24:
(3) (NP_{2})^{2}  (NP_{1})^{2} =
(4) (NP_{2})^{2}  (P_{1})^{2} =
(5) (P_{2})^{2}  (NP_{1})^{2} =
The ÷3 NONAR set is separately ÷24, but can NOT be mixed with members of the AR set (ARS) as:
(6) (NONARNP_{2})^{2}  (NONARNP_{1})^{2} = n24
(7) (NONARNP_{2})^{2}  (NP_{1})^{2} ⧣
(8) (NP_{2})^{2}  (NONARNP_{1})^{2} ⧣
(9) (P_{2})^{2}  (NONARNP_{1})^{2} ⧣
(10) (NONARNP_{2})^{2}  (P_{1})^{2} ⧣
The division into AR and NONAR sets has a NPS that ultimately define the elusive pattern of the P.
Furthermore, may be rearranged to:
(11)
(12)
asking whether any given ODD (>3) is a P or NP, it is exclusively a NP if, and only if, y reduces to the same value after applying x. As y is effectively an ODD of either a PRIME or composite of PRIMES factor*, one only needs to satisfy a single instance to validate NONPrimality.
One can also obtain ALL the P by eliminating the BIM SIGO and O^{2} from the 1^{st} Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O^{2} = ODDs^{2}, and the 1^{st} Diagonal = the 1^{st} Diagonal Parallel to the PD.
This itself is further simplified by switching out the ODD AXIS values with the O^{2} — the O^{2} being the PD values — such that we now have:
(13) SIGO(A^{2}) = Strict Inner Grid ODDS & ODD AXIS^{2} giving a distinct visualization advantage;(14) NP = SIGO(A^{2})
(15) 1st Diagonal  SIGO(A^{2}) = P.
This second method — the algebraic geometry method — as presented here.
Introduction
Two methods — one pure algebraic and the other a more visual algebraic geometry presented here — have been found that capture ALL the NOPRIMES (NP). While they process slightly different, they dovetail nicely into a very visual Number Pattern Sequence (NPS) here on the BIM. They both give identical NP results.
So what is the significance of capturing ALL the NP?
The NP are the highly NPS that define the elusive pattern of the P. P + NP = ALL ODD WINs (≥3).
In any group of WIN, if you know the NP, you also know the P. Here is the highly visualizable geometric method for capturing ALL NP. In fact, it is just as simply stated in 12. of the SUMMARY.
Algebraic Geometry Method
One can obtain ALL the P by eliminating the BIM SIGO and O^{2} from the 1^{st} Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O^{2} = ODDs^{2}, and the 1^{st} Diagonal = the 1^{st} Diagonal Parallel to the PD.
This itself is further simplified by switching out the ODD AXIS values with the O^{2} — the O^{2} being the PD values — such that we now have:
One can also obtain ALL the P by eliminating the BIM SIGO and O^{2} from the 1^{st} Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O^{2} = ODDs^{2}, and the 1^{st} Diagonal = the 1^{st} Diagonal Parallel to the PD.
This itself is further simplified by switching out the ODD AXIS values with the O^{2} — the O^{2} being the PD values — such that we now have:
(13) SIGO(A^{2}) = Strict Inner Grid ODDS & ODD AXIS^{2}(14) NP = SIGO(A^{2})
(15) 1st Diagonal  SIGO(A^{2}) = P.
The PDFs will follow these animated gifs. Videos and other supporting graphics thereafter.
Animated Gifs:
PRIMES vs NOPRIMES2: algebraic method.
PRIMES vs NOPRIMES1: algebraicgeometry method.
PRIMES vs NOPRIMES3: algebraic and algebraic geometry method.
PRIMES vs NOPRIMES4: algebraic method in detail.
PDFs +
PRIMES vs NOPRIMES: snapshot1 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below.
PRIMES vs NOPRIMES PDF: snapshot1 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The NPS of the NP define the elusive pattern of the P.
PRIMES vs NOPRIMES: snapshot2 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. Here the x=1,2,3,... base value sets  Lower y an Upper +yas doublewide, Lshaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3or divisible by 9 Axis squaredpaths. The larger PDF is below.
PRIMES vs NOPRIMES PDF: snapshot2 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the x=1,2,3,... base value sets  Lower y an Upper +yas doublewide, Lshaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3or divisible by 9 Axis squaredpaths.
PRIMES vs NOPRIMES: snapshot3 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. Here the x=1,2,3,... base value sets  Lower y an Upper +yas doublewide, Lshaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3or divisible by 9 Axis squaredpaths. The larger PDF is below.
PRIMES vs NOPRIMES PDF: snapshot3 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the x=1,2,3,... base value sets  Lower y an Upper +yas doublewide, Lshaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3or divisible by 9 Axis squaredpaths.
PRIMES vs NOPRIMES PDF: snapshot4 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the x=1,2,3,... base value sets  Lower y an Upper +yas doublewide, Lshaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3or divisible by 9 Axis squaredpaths.
Table31a4: PRIMES vs NOPRIMES PDF: Open in a separate tab/window to see all 11 pages. Here the x=1,2,3,... base value sets  Lower y an Upper +yare shown individually and collectively as sets. The full Upper and Lower tables conclude.
Table31a6_2: PRIMES vs NOPRIMES PDF: If you look at the ODD Axis ÷3 NOPRIMES (NP) that lie in the paths between the Lshaped Doublewide xbase sets (x=1,2,3,)…), one finds a distinct Number Pattern Sequence (NPS) between successive NP values. Those shown in BLUE are NOT included in the crisscrossing Lshaped Doublewide paths (seen in the snapshots above), while those in GRAY are. The NPS seen here, based ultimately on the 1,3,5,7,… ODD number summation series that defines the whole BIM distribution (including the 1st Diagonal, the PD, and the successive differences in sequential Inner Grid cell values) reiterates that of both the Lshaped Doublewide paths as well as the individual x paths. Both give a NPS of the NP that reveal the elusive pattern of the PRIMES. There remains little doubt that the PRIMES , as well as the Primitive Pythagorean Triples (PPTs), are intimately related to the INVERSE SQUARE LAW (ISL)!
Videos:
PRIMES vs NOPRIMES1long from Reginald Brooks on Vimeo.
PRIMES vs NOPRIMES2long from Reginald Brooks on Vimeo.
PRIMES vs NO_PRIMES 3 from Reginald Brooks on Vimeo.
PRIMES_vs_NOPRIMES4 from Reginald Brooks on Vimeo.
...more supporting graphics and tables:
Table31a45_+RunDiff_EQUATIONsDEMO(divide3Filter)+10x10+. A simple table to demo the NP process.
Table31a4_+RunDiff_EQUATIONPATTERNSLOWER+UPPER+Annotated. NP Table with layout notes.
Table31a4_+RunDiff_EQUATIONs(SqrdAxisseqPD)+50x500+.pdf. From the BIM to the NP/P Tables.
BIM35Table31a45_NOPRIMES_FactoreswithARSYELLOWnumbAnnotated. The NP pattern on the BIM.
Table33a,c,d_ODDsqrddivide24_sheets. The NP pattern – an amazing NPS – defines the P.
BIMrows11000+Primes_sheets+SubMatrix2. A BIM reference showing ÷24 ARS, PPTs, NP an P.
For the full documentation of all the graphics and pdfs — including history, development, expanded tables, and more — see the link references immediately below. Appendix B has all the documentation.
REFERENCES
While there are no specific references for this work other than referring back to my own original work, there are many references involved in the study and research on the PRIMES in general. These have been well documented in the TPISC_IV: Details_BIM+PPT+PRIMES focus white paper. This paper also contains the background graphics and tables leading up to this current work. The focus page is part of the much larger TPISC_IV: Details ebook project that is freely available as an HTML webpage. The quick reference outline can be found here.
From SQUARES to RECTANGLES — CIRCLES to OVALS — ISOSCELES to nonISOSCELES TRIANGLES: That's the story of TPISC: The Pythagorean  Inverse Square Connection.
The Areas found on the PD become the very SAME Areas on the IG in the BIM. These IG Square Areas are the Squared Sides of the PTs, and thus are part of the "PROOF" of the Pythagorean Theorem, and, they are each present on the Matrix as both a grid cell value and literally as actual square grid cell units giving us a beautiful visual proof of the algebraic geometry they express!
In forming the proofs, each PT — because of its presence 4x — becomes engaged in an even larger set of correlations to itself, individually, as well as to other PTs. The interconnections are beyond extensive, they are so deeply ingrained into the PTs as a whole — as with the Tree of Pythagorean Triples (ToPPT) — that structurally they become a fractaldriven substructure framework within the larger confines of the BIM.
The simple beauty of the 4A and 8A Areas both proving the Theorem and informing another PT generation, belies what also is going on here: as we have pounded home repeatedly, strict symmetry is given focus nodes for the introduction of some slight asymmetry — provided by PTs whose Templatedriven fractal expression gives both harmony and balance to the evolving SpaceTime structure that the BIM is forming — and this very process, so intimately informed by the BIM, that pure ST is given the possibilities of unique expression. Expression of ST energy in the form of waveparticle ST units whose categorical identities are dependent on just such variations in the ST configuration patterns — mass, spin, charge,… — and yet are always related back to the very ISL driven ST from which they arose.
The implication for these PTdriven ST expressions become an intriguing challenge to the interpretation of the DoubleSlit Experiment and the very notion of Quantum Entanglement. The DSEQEC (DoubleSlit Experiment Quantum Entanglement Conjecture) suggests that these two phenomena are essentially just twosides of the same coin. The waveparticle ST unit supposition — that a ST unit may simultaneously present itself as the sum total of all its possible quantum states and yet definitely collapse its "wave" portion to be expressed as a "particle" ST unit is — the conjecture maintains — a built in property of the very BIM and ToPPT forming any and all ST units pulsepropagating from their singularities to an expression within our Universe.
The quantum universe begs the question: what is the quantum?
Perhaps we can say the basic pulsepropagation of ST from its singularity — a singularity connected to the larger pool of all singularities by the Conservation of Energy — out into full spacetime extension, and back, is the fundamental quantum. The ST so formed has articulation points, nodes of structural formation made by the embedded Pythagorean Triples. These nodes are also quantized as they both are dependent upon the unfolding Inverse Square Lawbased ST, and they have distinct vector parameters inherently build into their asymmetric forms. They have both magnitude and direction — and as vectors their net expression can be the result of the total added or subtract interfering ST units. The fact that every primitive PT comes in both its parent PPT form as well as its child nPPT form provides a built in fractallike structure, that, when combined with the fact that each and every PT — PPT and nPPT — has 4 iterations of expression within the unfolding ST unit pulsepropagation, and we we now have a fractalbased, holographic quantum universe(s)! The Creation and Conservation of SpaceTime (CaCoST) model brings it all together in an attempt to answer some of the most outstanding unanswered questions of physics today.
Final Thoughts
Even mathematically, Nature can be expressed in more than one way. While the view may change, Nature seems to prefer efficiency.
In order to satisfy the Conservation Laws — from the initial Conservation of SpaceTime to its outter flanks: Conservation of Momentum, Angular Momentum, Charge and, of course, Energy — that means an accounting system that is primarily "payasyougo." Rather than balancing the entries across the expanse of the Universe at some distant point in time, it's much more expedient to balance the parts as you go.
When ST — and that includs all ST unit entities — is formed, its balance sheet requries both a matter and antimatter accounting. The Equal And Opposite (EAO) quantum state parameters — mass, charge and spin — that are parts of its energy expression, are accounted for right from the get go. A pairing — a virtual pairing is one way of looking at it — that is part and parcel of the very creation of ST.
The BIM itself informs that ST, and, if, as proposed, the 4 iterations of each PT within the BIM represent the quantum state numbers of any ST unit — and their information is ubiquitously known for every ST unit pp — we now have great insights into the Creation and Conservation of SpaceTime (CaCoST).
Like the Big Bang itself, every little bang, pp ST unit is but a fractal mimicking the process and accounting for the quantum as:
 Singularity
 PEH
 Inflationexpansion
 Matter  Antimatter
 Dark Matter  Dark Energy
 DSE
 QE
 (EP = EPR)
 CaCoST
This is the CaCoST model!
Lastly, the hugely important finding that ALL PTs and ALL PRIMES ALWAYS fall on ONLY Active Rows — either side of BIM ÷24 Rows, and, the very same Rows algebraically described by Euler's Prime=6n =/1 Theorem — is not to be dismissed as it points to yet another very deep relationship between The Pythagorean  Inverse Square Connection (TPISC), the "Details" of which we have laid out in this work!
The difference in the Squares of EVERY Active Row number — including ALL PTs and ALL PRIMES — is ALWAYS an even multiple of 24! This is built into the Prime Diagonal (PD) of the BIM itself!
CaCoSTDSEQEC with Hands representing "spin."
CaCoST_HANDS_DSEQEC from Reginald Brooks on Vimeo.
Every pp unfolds a BIM grid where ALL info is known throughout its expansion. Letting the Pythagorean Triples (PTs) represent "spin" and their vector "direction" orientation (Left, L or Right, R), one finds 4 iterations of spin (2L and 2R) automatically built into the ISL matrix. Those 4 iterations are graphically represented as the opposing Arrow Circles. We have two simple possibilities: 1. The Equal And Opposite (EAO) "spin" values are simply those in the AV Antimatter; 2. Every ST unit actually reveals as a virtualpair with both EAO "spin" values in BOTH the OV Matter AND the AV Antimatter. In #1, the single photon interferes (and entangles) with its EAO Antimatter partner. In #2, interference (and entanglement) occur in BOTH Matter and Antimatter domains. In Entanglement, we have 2 ST units — e.i. photons — created & sharing the same Quantum State numbers/parameters, i.e., "spin." Regardless of distance of separation, they ALWAYS maintain EAO "spin" values. Within the BIM, each PT has 4 iterations of its triangle simultaneously present. One can designate the short side "a" to be the "spin" vector direction. That means, there are 4 "spin" vector direction informationdirections ALWAYS known. In either scenario #1 or #2, measuring the "spin" direction on one will always yield the EAO "spin" direction vector on the other photon. Scenario #1 — being simpler — gives an easier visualization of the process. Scenario #2 — where both the Matter & Antimatter photon are virtually present as pairs of EAO entities — gives a more complex, but perhaps closer to "reality" picture as here the "KNOWN" information of "spin" direction allows the photon to be in eitheror/both "spin" directions at once, and coinciding with known results, the manner, method and actual act of measurement "selects" — at that moment — for a particular "spin" vector. If you orientate your test to select the "spin" state, it will automatically be mirrored to the information — and thus state — of the EAO entangled photon. There is no information transported, it is ALWAYS known! DSEQEC Every ST unit forms from the instruction set of the Inverse Square Law (BIM). The PTs provide the basis of the Quantum State Numbers/Parameters. Each PT has its 4iterations of its triangle ubiquitously KNOWN for all ST. The formation of any & all ST units generates EAO MatterAntimatter expressions. Both the Matter & Antimatter expressions are guided by the ISL of the BIM. Thus, any given ST unit —e.i., photon —is Entangled from the start, acting like a virtualpair. The "spin" information —encoded in the BIM —is EAO in the Entangled pairs. The DoubleSlit Experiment (DSE) exemplifies how a single photon can interfere with itself. Quantum Entanglement (QE) exemplifies how 2 entangled photons —born of and sharing the same quantum state — reveal how ALL ST units are formed as 2 Entangled, EAO MatterAntimatter virtualpairs, the INFORMATION of their QUANTUM STATE ALWAYS built right into their ST formation via the EAO iterations of the PTs within the expanding BIM. There is NO HIDDEN or TRANSFERRED INFORMATION. ALL INFORMATION IS ALWAYS KNOWN.
IX. References
TPISC_IV:_Details ~A MathspeedST Supplement~
Copyright©201819, Reginald Brooks. All rights reserved.Preface*TPISC ( pisque—silent “T”): The Pythagorean — Inverse Square Connection*3Steps to Nirvana:Something Very NEW! ➗24: PPTs and PRIMES(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)DSEQEC: DoubleSlit Experiment — Quantum Entanglement Conjecture, and, CaCoST: Creation and Conservation of SpaceTimeIntroList of 40 characters/focus points:I. Who?PTs — Pythagorean Triples — and the squares and rectangles they represent.II. What?PTs are nonisosceles, 90°righttriangles composed of Whole Integer Numbers (WINS).III. Where?PTs — and their proofs — are ubiquitously located, in five (5) easy steps, throughout the infinitely expandable BIM. (see Chapter V: How for details)IV. When?PTs are uniquely located on any Row (or Column) containing Paired Sets — 2 Squared Areas — that represent a^{2} and b^{2}, with c^{2} at the endpoint on the PD.V. How?PTs have been eloquently and succinctly defined by algebraic geometry.PTs: 40+ FEDM derived profiling focus point parameters:PTs: 10 ways to approach the BIM (return to this again after completing the sections that follow):BBSISL Matrix (BIM): Basic, fundmental Rules of the Symmetrical Matrix grid: 10 Basic, fundamental rules of the symmetrical BBSISL MatrixBBSISL Matrix Inner Grid Golden Rules (IGGR)5 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner GridPythagorean Triples and BBSISL Fundamentals (TPISC: The PythagoreanInverse Square Connection)3 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner Grid that encompass the PTs.Matrix Flow:Exponentials Summary (see TPISC V: Exponentials):EQUATIONS:VI. Why?A. The PTs are so intimately and ubiquitously intertwined, interconnected and resonantly echoed throughout the BIM that one could almost rewrite TPISC: The Pythagorean  Inverse Connection to Pythagorean = Inverse Square Law, i.e. the Pythagorean Theorem is really just a subset of the Inverse Square Law (ISL)!B. Areas, Perimeters, Proofs.C. How — but NOT why — 24 connects the PTs and PRIMES to the BIM.SubMatrixTogether SubMatrix 1 and SubMatrix 2 will provide the visualgeometric and algebraicgeometric location of ALL PPTs and PRIMES.First SubMatrix 1:SubMatrix 2:SubMatrix 2 Sidebar: Exponentials of the PPTs(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)From TPISC III: Clarity Conclusion:Open letter to Leonard Susskind, Juan Maldacena, and Mark Van Raamsdonk, …DSEQEC: DoubleSlit Experiment — Quantum Entanglement Conjecture, and, CaCoST: Creation and Conservation of SpaceTimeReferences (specific to ER=EPR and some of the surrounding issues):Closing thoughtsSee Appendices C_D_E: BIM PPTS, Pentagons, Decagons, Phi (ϕ), Fibonacci, Pentagon Connections to the PTs, and, BIM + PT + DNA + Zika, EpsteinBarr and other IcosahedralStructured Human Viruses for ALL the Figures and Tables relating to these topics.VII. Summary24 SUMMARYSUMMARY:DETAILS:In brief:(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)SubMatrix 2:(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)PTs: 40+ FEDM derived profiling focus point parameters:PTs: 10 ways to approach the BIM (return to this again after completing the sections that follow):BBSISL Matrix (BIM): Basic, fundamental Rules of the Symmetrical Matrix grid: 10 Basic, fundamental rules of the symmetrical BBSISL MatrixBBSISL Matrix Inner Grid Golden Rules (IGGR)5 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner GridPythagorean Triples and BBSISL Fundamentals (TPISC: The PythagoreanInverse Square Connection)3 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner Grid that encompass the PTs.Matrix Flow:Exponentials Summary (see TPISC V: Exponentials):Summation:feel the beat: +  +    +  +    +  +    +  +    ...VIII. ConclusionFrom SQUARES to RECTANGLES — CIRCLES to OVALS — ISOSCELES to nonISOSCELES TRIANGLES: That's the story of TPISC: The Pythagorean  Inverse Square Connection.IX. ReferencesX. AppendixAppendix A: BIM PPTs: Details of their Portrait ProfilesAppendix B: BIM ➗24 PPTs and PRIMESTable VI b fully expanded 11010 PTs and Primes with ALL ACTIVE Rows colorcoded in VioletSubMatrix 2 Sidebar: Exponentials of the PPTsAppendix C: BIM_PT_Pentagons BIM + PT + Decagon (double Pentagons) BIM + PT + Decagon (double Pentagons of the DNA doublehelix)Appendix D: Phi (ϕ), Fibonacci, Pentagon Connections to the PTsComments:TABLES I  IV: A deeper look:SUMMARYLINKS:Golden Ratio, phi (ϕ)Platonic SolidsPhi (ϕ) , Fibonacci Numbers and PentagonsImagesPythagorean Triples and Fibonacci NumbersKepler's Triangle and phi (ϕ) and the Pythagorean TheoremUseful links for educators:Appendix E: BIM + PT + DNA + Zika, EpsteinBarr and other IcosahedralStructured Human Viruses BIM + PT + DNA + Zika Virus BIM + PT + DNA + EpsteinBarr Virus BIM + PT + DNA + EpsteinBarr virus Appendix F: BIM + misc: BIM: How to Make
X. Appendix
Appendix A: BIM PPTs: Details of their Portrait Profiles
(Table number reference is for Appendix A.)
Note: Often both quick view png/jpg and full view pdf versions presented.
NOTES: Sections 110
Section 1: 2018 (29pgs)
Section 2: 2018 (20pg)
Section 3: 2018 (19pgs)
Section 4A: 2017 (23pgs)
Section 4B: 2017 (19pgs)
Section 4C: 2017 (68pgs)
Section 5: 201617 (32pgs)
Section 6: 201617 (7pgs)
Section 7: 201617 (16pgs)
Section 8: 201617 (22pgs)
Section 9: 2014 (17pgs)
Section 10: 201618 (15+pgs)
TPISC_IV:_Details ~A MathspeedST Supplement~
Copyright©201819, Reginald Brooks. All rights reserved.Preface*TPISC ( pisque—silent “T”): The Pythagorean — Inverse Square Connection*3Steps to Nirvana:Something Very NEW! ➗24: PPTs and PRIMES(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)DSEQEC: DoubleSlit Experiment — Quantum Entanglement Conjecture, and, CaCoST: Creation and Conservation of SpaceTimeIntroList of 40 characters/focus points:I. Who?PTs — Pythagorean Triples — and the squares and rectangles they represent.II. What?PTs are nonisosceles, 90°righttriangles composed of Whole Integer Numbers (WINS).III. Where?PTs — and their proofs — are ubiquitously located, in five (5) easy steps, throughout the infinitely expandable BIM. (see Chapter V: How for details)IV. When?PTs are uniquely located on any Row (or Column) containing Paired Sets — 2 Squared Areas — that represent a^{2} and b^{2}, with c^{2} at the endpoint on the PD.V. How?PTs have been eloquently and succinctly defined by algebraic geometry.PTs: 40+ FEDM derived profiling focus point parameters:PTs: 10 ways to approach the BIM (return to this again after completing the sections that follow):BBSISL Matrix (BIM): Basic, fundmental Rules of the Symmetrical Matrix grid: 10 Basic, fundamental rules of the symmetrical BBSISL MatrixBBSISL Matrix Inner Grid Golden Rules (IGGR)5 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner GridPythagorean Triples and BBSISL Fundamentals (TPISC: The PythagoreanInverse Square Connection)3 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner Grid that encompass the PTs.Matrix Flow:Exponentials Summary (see TPISC V: Exponentials):EQUATIONS:VI. Why?A. The PTs are so intimately and ubiquitously intertwined, interconnected and resonantly echoed throughout the BIM that one could almost rewrite TPISC: The Pythagorean  Inverse Connection to Pythagorean = Inverse Square Law, i.e. the Pythagorean Theorem is really just a subset of the Inverse Square Law (ISL)!B. Areas, Perimeters, Proofs.C. How — but NOT why — 24 connects the PTs and PRIMES to the BIM.SubMatrixTogether SubMatrix 1 and SubMatrix 2 will provide the visualgeometric and algebraicgeometric location of ALL PPTs and PRIMES.First SubMatrix 1:SubMatrix 2:SubMatrix 2 Sidebar: Exponentials of the PPTs(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)From TPISC III: Clarity Conclusion:Open letter to Leonard Susskind, Juan Maldacena, and Mark Van Raamsdonk, …DSEQEC: DoubleSlit Experiment — Quantum Entanglement Conjecture, and, CaCoST: Creation and Conservation of SpaceTimeReferences (specific to ER=EPR and some of the surrounding issues):Closing thoughtsSee Appendices C_D_E: BIM PPTS, Pentagons, Decagons, Phi (ϕ), Fibonacci, Pentagon Connections to the PTs, and, BIM + PT + DNA + Zika, EpsteinBarr and other IcosahedralStructured Human Viruses for ALL the Figures and Tables relating to these topics.VII. Summary24 SUMMARYSUMMARY:DETAILS:In brief:(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)SubMatrix 2:(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)PTs: 40+ FEDM derived profiling focus point parameters:PTs: 10 ways to approach the BIM (return to this again after completing the sections that follow):BBSISL Matrix (BIM): Basic, fundamental Rules of the Symmetrical Matrix grid: 10 Basic, fundamental rules of the symmetrical BBSISL MatrixBBSISL Matrix Inner Grid Golden Rules (IGGR)5 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner GridPythagorean Triples and BBSISL Fundamentals (TPISC: The PythagoreanInverse Square Connection)3 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner Grid that encompass the PTs.Matrix Flow:Exponentials Summary (see TPISC V: Exponentials):Summation:feel the beat: +  +    +  +    +  +    +  +    ...VIII. ConclusionFrom SQUARES to RECTANGLES — CIRCLES to OVALS — ISOSCELES to nonISOSCELES TRIANGLES: That's the story of TPISC: The Pythagorean  Inverse Square Connection.IX. ReferencesX. AppendixAppendix A: BIM PPTs: Details of their Portrait ProfilesAppendix B: BIM ➗24 PPTs and PRIMESTable VI b fully expanded 11010 PTs and Primes with ALL ACTIVE Rows colorcoded in VioletSubMatrix 2 Sidebar: Exponentials of the PPTsAppendix C: BIM_PT_Pentagons BIM + PT + Decagon (double Pentagons) BIM + PT + Decagon (double Pentagons of the DNA doublehelix)Appendix D: Phi (ϕ), Fibonacci, Pentagon Connections to the PTsComments:TABLES I  IV: A deeper look:SUMMARYLINKS:Golden Ratio, phi (ϕ)Platonic SolidsPhi (ϕ) , Fibonacci Numbers and PentagonsImagesPythagorean Triples and Fibonacci NumbersKepler's Triangle and phi (ϕ) and the Pythagorean TheoremUseful links for educators:Appendix E: BIM + PT + DNA + Zika, EpsteinBarr and other IcosahedralStructured Human Viruses BIM + PT + DNA + Zika Virus BIM + PT + DNA + EpsteinBarr Virus BIM + PT + DNA + EpsteinBarr virus Appendix F: BIM + misc: BIM: How to Make
Appendix B: BIM ➗24 PPTs and PRIMES
(Table number reference is for Appendix B and TPISC IV.)
TABLES 19 (rsets, Z➗24,Axis squared, differences, groundwork tables)
Table VI b fully expanded 11010 PTs and Primes with ALL ACTIVE Rows colorcoded in Violet
(Once renamed, ALL Tables — as pdf/pngs—will be listed here with links to and fro main article)
TABLES 1019 (PRIMEsgaps and random differences)
TABLES 2023 (PRIMEPPT or not with rvalues)
TABLES 2428 (SubMatrix 2 )
TABLES 2930 (SubMatrix 2 Sidebar: Exponentials of the PPTs)
SubMatrix 2 Sidebar: Exponentials of the PPTs
Table 29 Exponentials of the first 10 PPTs cvalues to be used in Tables 30ag.
Tables 30ag The SubMatrix 2, when ➗4, and the difference (∆) between this and the next exponential PPT treated this way, is subsequently ➗ by its SubMatrix 2 variable, the PREVIOUS exponential within the series is revealed. Restated as an example: When one subtracts 1 from the exponential values of c (the cvalue of the PPT) you get the SubMatrix 2 value. Divide that by 4 and take the difference (∆) between it and the next. Divide that by 3 to give the PREVIOUS PPT cvalue in the series.
The SubMatrix 2 variable divisor = 3 = SubMatrix 2 value/4 = 12/4. These variables run: 1,3,4,6,7,9,10,...
Tables: 31a, a+, a++, ++MP, a+++ and 31b
Tables: 32a, 32b and 32c
Tables: 33a, 33b and 33c.
The underlying geometry of the BIM÷24 PRESELECTS the Axis Rows into TWO Groups: ARs and NONARs. The PPTs and PRIMES are EXCLUSIVELY — as a sufficient, but not necessary condition — found on the ARs and NEVER on the NONARs. While both Groups follow (PD^{2}  PD^{2})÷24, they do so ONLY within their own respective Groups. They do NOT crossover. This Grouping divide occurs naturally within the BIM as shown in these images below.
The ISL as presented in the BIM is deeply, intimately structured around the number 24 — and its factors: 4,6, 3,8 2,12, and 1,24.
The interplay between these small sets of Numbers generates an incredible amount of richness and complexity with seemingly simplistic BIM itself. This has led to TPISP: The PythagoreanInverse Square Connection, and the PRIMES.
(NOTE: OPEN in separate browser tab/window to see all.)Supporting Graphics forTables: 33a, 33b, 33c and 33d. Open in separate browser tab/window to see all.
Back to the Table 31 sequences. Table 31a... series, along with the previous Table33 series, uncovers how the PRIMES were found within the BIM both by algebra and geometry. This has been reduced to a focus white paper: "PRIMES vs NOPRIMES that is a condensed summary of this work. One may benefit from looking over this paper prior to the study of these more elaborate tables. A summary from the paper is presented here:
xxxxxxxxSUMMARY*
(~from "*Simple Path from the BIM to the PRIMES" that presents the algebraicgeometry method.)
How do we go from the simple grid of the BIM (BBSISL Matrix) to identifying the PRIMES?
The BIM is a symmetrical grid — divided equally down its diagonal center with the Prime Diagonal (PD) — that illuminates the Number Pattern Sequence (NPS) of the Inverse Square Law (ISL) via simple, natural Whole Integer Numbers (WIN).
The BIM Axis numbers are 1,2,3,.. with 0 at the origin.
The Inner Grid (IG) contains EVEN and ODD WIN, but except for the 1st diagonal next to the PD — a diagonal that contains ALL the ODD WIN — there are NO PRIMES (NOPRIMES, NP) on the SIG (Strict Inner Grid).
The PD WIN are simple the square of the Axis WIN.
ALL the IG WIN result from subtracting the horizontal from the vertical intersection of the PD.
Dropping down a given PD Squared WIN (>4) until it intersects with another squared WIN on a Row below will ALWAYS reveal that Row to be a Primitive Pythagorean Triple (PPT) Row, whose hypotenuse, c, lies on the intersecting PD. ALL PPTs may be identified this way.
Dividing the BIM cell values by 24 — BIM÷24 — forms a crisscrossing DIAMOND NPS that divides the overall BIM into two distinct and alternating Row (and Column) bands or sets:
 ODD WIN that are ÷3 and referred to as NONARs;
 ODD WIN that are NOT ÷3 and referred to as ARs, or Active Rows;
 The ARs ALWAYS come in pairs — with an EVEN WIN between — as the UPPER and LOWER AR of the ARS (Active Row Set);
ALL PPTs and ALL PRIMES ALWAYS are found exclusively on the ARs — no exceptions.
By applying:
(1) *
let y = odd number (ODD) 3, 5, 7,… and x = 1, 2, 3,... one generates a NP table containing ALL the NP;
*True if ÷3 ODDs are first eliminated, otherwise ADD exponentials of 3 to the NP pool;
Eliminating the NP — and the NP contain a NPS — from ALL the ODD WIN, reveals the PRIMES (P).
A necessary, but not sufficient confirmation — but not proof — of primality is found by finding the even division of 24 into the difference of the square of ANY two PRIMES as:
(2) (P_{2})^{2}  (P_{1})^{2} =
let
Be aware that this also holds true for ALL the AR NP. The P and NP are NOT ÷3, and are both part of the ARS and therefore any combination of the two squared differences will be ÷24:
(3) (NP_{2})^{2}  (NP_{1})^{2} =
(4) (NP_{2})^{2}  (P_{1})^{2} =
(5) (P_{2})^{2}  (NP_{1})^{2} =
The ÷3 NONAR set is separately ÷24, but can NOT be mixed with members of the AR set (ARS) as:
(6) (NONARNP_{2})^{2}  (NONARNP_{1})^{2} = n24
(7) (NONARNP_{2})^{2}  (NP_{1})^{2} ⧣
(8) (NP_{2})^{2}  (NONARNP_{1})^{2} ⧣
(9) (P_{2})^{2}  (NONARNP_{1})^{2} ⧣
(10) (NONARNP_{2})^{2}  (P_{1})^{2} ⧣
The division into AR and NONAR sets has a NPS that ultimately define the elusive pattern of the P.
Furthermore, may be rearranged to:
(11)
(12)
asking whether any given ODD (>3) is a P or NP, it is exclusively a NP if, and only if, y reduces to the same value after applying x. As y is effectively an ODD of either a PRIME or composite of PRIMES factor*, one only needs to satisfy a single instance to validate NONPrimality.
One can also obtain ALL the P by eliminating the BIM SIGO and O^{2} from the 1^{st} Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O^{2} = ODDs^{2}, and the 1^{st} Diagonal = the 1^{st} Diagonal Parallel to the PD.
This itself is further simplified by switching out the ODD AXIS values with the O^{2} — the O^{2} being the PD values — such that we now have:
(13) SIGO(A^{2}) = Strict Inner Grid ODDS & ODD AXIS^{2} giving a distinct visualization advantage;(14) NP = SIGO(A^{2})
(15) 1st Diagonal  SIGO(A^{2}) = P.
This second method — the algebraic geometry method — as presented here.
Introduction
Two methods — one pure algebraic and the other a more visual algebraic geometry presented here — have been found that capture ALL the NOPRIMES (NP). While they process slightly different, they dovetail nicely into a very visual Number Pattern Sequence (NPS) here on the BIM. They both give identical NP results.
So what is the significance of capturing ALL the NP?
The NP are the highly NPS that define the elusive pattern of the P. P + NP = ALL ODD WINs (≥3).
In any group of WIN, if you know the NP, you also know the P. Here is the highly visualizable geometric method for capturing ALL NP. In fact, it is just as simply stated in 12. of the SUMMARY.
Algebraic Geometry Method
One can obtain ALL the P by eliminating the BIM SIGO and O^{2} from the 1^{st} Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O^{2} = ODDs^{2}, and the 1^{st} Diagonal = the 1^{st} Diagonal Parallel to the PD.
This itself is further simplified by switching out the ODD AXIS values with the O^{2} — the O^{2} being the PD values — such that we now have:
One can also obtain ALL the P by eliminating the BIM SIGO and O^{2} from the 1^{st} Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O^{2} = ODDs^{2}, and the 1^{st} Diagonal = the 1^{st} Diagonal Parallel to the PD.
This itself is further simplified by switching out the ODD AXIS values with the O^{2} — the O^{2} being the PD values — such that we now have:
(13) SIGO(A^{2}) = Strict Inner Grid ODDS & ODD AXIS^{2}(14) NP = SIGO(A^{2})
(15) 1st Diagonal  SIGO(A^{2}) = P.
The PDFs will follow these animated gifs. Videos and other supporting graphics thereafter.
Animated Gifs:
PRIMES vs NOPRIMES2: algebraic method.
PRIMES vs NOPRIMES1: algebraicgeometry method.
PRIMES vs NOPRIMES3: algebraic and algebraic geometry method.
PRIMES vs NOPRIMES4: algebraic method in detail.
PDFs +
PRIMES vs NOPRIMES: snapshot1 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below.
PRIMES vs NOPRIMES PDF: snapshot1 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The NPS of the NP define the elusive pattern of the P.
PRIMES vs NOPRIMES: snapshot2 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. Here the x=1,2,3,... base value sets  Lower y an Upper +yas doublewide, Lshaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3or divisible by 9 Axis squaredpaths. The larger PDF is below.
PRIMES vs NOPRIMES PDF: snapshot2 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the x=1,2,3,... base value sets  Lower y an Upper +yas doublewide, Lshaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3or divisible by 9 Axis squaredpaths.
PRIMES vs NOPRIMES: snapshot3 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. Here the x=1,2,3,... base value sets  Lower y an Upper +yas doublewide, Lshaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3or divisible by 9 Axis squaredpaths. The larger PDF is below.
PRIMES vs NOPRIMES PDF: snapshot3 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the x=1,2,3,... base value sets  Lower y an Upper +yas doublewide, Lshaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3or divisible by 9 Axis squaredpaths.
PRIMES vs NOPRIMES PDF: snapshot4 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the x=1,2,3,... base value sets  Lower y an Upper +yas doublewide, Lshaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3or divisible by 9 Axis squaredpaths.
Table31a4: PRIMES vs NOPRIMES PDF: Open in a separate tab/window to see all 11 pages. Here the x=1,2,3,... base value sets  Lower y an Upper +yare shown individually and collectively as sets. The full Upper and Lower tables conclude.
Table31a6_2: PRIMES vs NOPRIMES PDF: If you look at the ODD Axis ÷3 NOPRIMES (NP) that lie in the paths between the Lshaped Doublewide xbase sets (x=1,2,3,)…), one finds a distinct Number Pattern Sequence (NPS) between successive NP values. Those shown in BLUE are NOT included in the crisscrossing Lshaped Doublewide paths (seen in the snapshots above), while those in GRAY are. The NPS seen here, based ultimately on the 1,3,5,7,… ODD number summation series that defines the whole BIM distribution (including the 1st Diagonal, the PD, and the successive differences in sequential Inner Grid cell values) reiterates that of both the Lshaped Doublewide paths as well as the individual x paths. Both give a NPS of the NP that reveal the elusive pattern of the PRIMES. There remains little doubt that the PRIMES , as well as the Primitive Pythagorean Triples (PPTs), are intimately related to the INVERSE SQUARE LAW (ISL)!
Videos:
PRIMES vs NOPRIMES1long from Reginald Brooks on Vimeo.
PRIMES vs NOPRIMES2long from Reginald Brooks on Vimeo.
PRIMES vs NO_PRIMES 3 from Reginald Brooks on Vimeo.
PRIMES_vs_NOPRIMES4 from Reginald Brooks on Vimeo.
...more supporting graphics and tables:
Table31a45_+RunDiff_EQUATIONsDEMO(divide3Filter)+10x10+. A simple table to demo the NP process.
Table31a4_+RunDiff_EQUATIONPATTERNSLOWER+UPPER+Annotated. NP Table with layout notes.
Table31a4_+RunDiff_EQUATIONs(SqrdAxisseqPD)+50x500+.pdf. From the BIM to the NP/P Tables.
BIM35Table31a45_NOPRIMES_FactoreswithARSYELLOWnumbAnnotated. The NP pattern on the BIM.
Table33a,c,d_ODDsqrd÷24_sheets. The NP pattern – an amazing NPS – defines the P.
BIMrows11000+Primes_sheets+SubMatrix2. A BIM reference showing ÷24 ARS, PPTs, NP an P.
For the full documentation of all the graphics and pdfs — including history, development, expanded tables, and more — see the link references immediately below. Appendix B has all the documentation.
REFERENCES
While there are no specific references for this work other than referring back to my own original work, there are many references involved in the study and research on the PRIMES in general. These have been well documented in the TPISC_IV: Details_BIM+PPT+PRIMES focus white paper. This paper also contains the background graphics and tables leading up to this current work. The focus page is part of the much larger TPISC_IV: Details ebook project that is freely available as an HTML webpage. The quick reference outline can be found here.
Various supporting graphics
xxxxxxxxxx
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TPISC_IV:_Details ~A MathspeedST Supplement~
Copyright©201819, Reginald Brooks. All rights reserved.Preface*TPISC ( pisque—silent “T”): The Pythagorean — Inverse Square Connection*3Steps to Nirvana:Something Very NEW! ➗24: PPTs and PRIMES(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)DSEQEC: DoubleSlit Experiment — Quantum Entanglement Conjecture, and, CaCoST: Creation and Conservation of SpaceTimeIntroList of 40 characters/focus points:I. Who?PTs — Pythagorean Triples — and the squares and rectangles they represent.II. What?PTs are nonisosceles, 90°righttriangles composed of Whole Integer Numbers (WINS).III. Where?PTs — and their proofs — are ubiquitously located, in five (5) easy steps, throughout the infinitely expandable BIM. (see Chapter V: How for details)IV. When?PTs are uniquely located on any Row (or Column) containing Paired Sets — 2 Squared Areas — that represent a^{2} and b^{2}, with c^{2} at the endpoint on the PD.V. How?PTs have been eloquently and succinctly defined by algebraic geometry.PTs: 40+ FEDM derived profiling focus point parameters:PTs: 10 ways to approach the BIM (return to this again after completing the sections that follow):BBSISL Matrix (BIM): Basic, fundmental Rules of the Symmetrical Matrix grid: 10 Basic, fundamental rules of the symmetrical BBSISL MatrixBBSISL Matrix Inner Grid Golden Rules (IGGR)5 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner GridPythagorean Triples and BBSISL Fundamentals (TPISC: The PythagoreanInverse Square Connection)3 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner Grid that encompass the PTs.Matrix Flow:Exponentials Summary (see TPISC V: Exponentials):EQUATIONS:VI. Why?A. The PTs are so intimately and ubiquitously intertwined, interconnected and resonantly echoed throughout the BIM that one could almost rewrite TPISC: The Pythagorean  Inverse Connection to Pythagorean = Inverse Square Law, i.e. the Pythagorean Theorem is really just a subset of the Inverse Square Law (ISL)!B. Areas, Perimeters, Proofs.C. How — but NOT why — 24 connects the PTs and PRIMES to the BIM.SubMatrixTogether SubMatrix 1 and SubMatrix 2 will provide the visualgeometric and algebraicgeometric location of ALL PPTs and PRIMES.First SubMatrix 1:SubMatrix 2:SubMatrix 2 Sidebar: Exponentials of the PPTs(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)From TPISC III: Clarity Conclusion:Open letter to Leonard Susskind, Juan Maldacena, and Mark Van Raamsdonk, …DSEQEC: DoubleSlit Experiment — Quantum Entanglement Conjecture, and, CaCoST: Creation and Conservation of SpaceTimeReferences (specific to ER=EPR and some of the surrounding issues):Closing thoughtsSee Appendices C_D_E: BIM PPTS, Pentagons, Decagons, Phi (ϕ), Fibonacci, Pentagon Connections to the PTs, and, BIM + PT + DNA + Zika, EpsteinBarr and other IcosahedralStructured Human Viruses for ALL the Figures and Tables relating to these topics.VII. Summary24 SUMMARYSUMMARY:DETAILS:In brief:(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)SubMatrix 2:(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)PTs: 40+ FEDM derived profiling focus point parameters:PTs: 10 ways to approach the BIM (return to this again after completing the sections that follow):BBSISL Matrix (BIM): Basic, fundamental Rules of the Symmetrical Matrix grid: 10 Basic, fundamental rules of the symmetrical BBSISL MatrixBBSISL Matrix Inner Grid Golden Rules (IGGR)5 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner GridPythagorean Triples and BBSISL Fundamentals (TPISC: The PythagoreanInverse Square Connection)3 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner Grid that encompass the PTs.Matrix Flow:Exponentials Summary (see TPISC V: Exponentials):Summation:feel the beat: +  +    +  +    +  +    +  +    ...VIII. ConclusionFrom SQUARES to RECTANGLES — CIRCLES to OVALS — ISOSCELES to nonISOSCELES TRIANGLES: That's the story of TPISC: The Pythagorean  Inverse Square Connection.IX. ReferencesX. AppendixAppendix A: BIM PPTs: Details of their Portrait ProfilesAppendix B: BIM ➗24 PPTs and PRIMESTable VI b fully expanded 11010 PTs and Primes with ALL ACTIVE Rows colorcoded in VioletSubMatrix 2 Sidebar: Exponentials of the PPTsAppendix C: BIM_PT_Pentagons BIM + PT + Decagon (double Pentagons) BIM + PT + Decagon (double Pentagons of the DNA doublehelix)Appendix D: Phi (ϕ), Fibonacci, Pentagon Connections to the PTsComments:TABLES I  IV: A deeper look:SUMMARYLINKS:Golden Ratio, phi (ϕ)Platonic SolidsPhi (ϕ) , Fibonacci Numbers and PentagonsImagesPythagorean Triples and Fibonacci NumbersKepler's Triangle and phi (ϕ) and the Pythagorean TheoremUseful links for educators:Appendix E: BIM + PT + DNA + Zika, EpsteinBarr and other IcosahedralStructured Human Viruses BIM + PT + DNA + Zika Virus BIM + PT + DNA + EpsteinBarr Virus BIM + PT + DNA + EpsteinBarr virus Appendix F: BIM + misc: BIM: How to Make
Appendix C: BIM_PT_Pentagons
BIM + PT + Decagon (double Pentagons)
NEW! Connections between:
 BIM (BBSISL Matrix) + PTs (Pythagorean Triples)
\2. PT + Pentagon & Decagon (double pentagons of DNA)
 BIM + PT + DNA
 BIM + PT + DNA + Zika Virus (cryoem imagery)
It is suggested in the previous work that the pentagonal geometry of the virus allows it to insert itself within the decagonal geometry of our DNA, perhaps spiraling in along the DNA doublehelical axis, looking for a simple match opening!
In this work, a closer look at how the base 345 PT relates to the pentagon, doublepentagon (decagon) and the 3 concentric decagonal geometry found in the axial view of the DNA doublehelix molecule will be visually examined.
BIM_PT_Pentagons from Reginald Brooks on Vimeo.
BIM + PT + Decagon (double Pentagons of the DNA doublehelix)
See a full slideshow of the BIM + PT + Decagon : here
See the video: here on Vimeo at https://vimeo.com/263223746
See white papers: here
See more on TPISC: The Pythagorean  Inverse Square Connection: here
Comment: Here we continue with TPISC: The Pythagorean  Inverse Square Connection. This time looking for a connection between the 345 Pythagorean Triple and the concentric, decagons (double pentagons) that form the geometric structure of the DNA doublehelix molecule as view down its axis composited from 1 full, 360 degree spiral.
While emphatically NOT AN EXACT MATCH, the 345 PT is ridiculously close to being an exact match both from the the angles of a given pentagon to that of the decagon and most especially connecting key angles and vertices between the concentric decagons.
In 2001, I wrote up “GoDNA: the Geometry of DNA (axial view),” and, “SCoDNA: Structure & Chemistry of DNA” about the pure doublepentagon (decagon) geometry of the doublehelix molecule. It is the basis for the concentric decagonal geometry — acting a template.
The question remains: is the "ridiculously close to being an exact match" good enough — sufficient enough — to define an actual connection between the BIM + PT, and, the decagon? Does this slight wiggle room actually allow for the connection to be made within the organic presentation and interaction of Nature. "Soft geometry" versus the crisp, hardedged, no room for anything less than perfect fits "hard geometry" of pure mathematics? Are the strict mathematics that informs all of spacetime operative in a slightly looser, more generous manner in the actual physical manifestations of the organic world of Nature? Are not the Fibonacci sequence numbers of the Golden Mean "approximations" to the actual value of the irrational number, phi (Φ)? (for more on 5, Φ and pentagons see "The Golden Number.")
here
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Appendix E: BIM + PT + DNA + Zika, EpsteinBarr and other IcosahedralStructured Human Viruses
BIM + PT + DNA + Zika Virus
NEW! Connections between:
 BIM (BBSISL Matrix) + PTs (Pythagorean Triples)
 PT + Pentagon & Decagon (double pentagons of DNA)
 BIM + PT + DNA
 BIM + PT + DNA + Zika Virus (cryoem imagery)
It is suggested that the pentagonal geometry of the virus allows it to insert itself within the decagonal geometry of our DNA, perhaps spiraling in along the DNA doublehelical axis, looking for a simple match opening!
See the video: here on Vimeo at https://vimeo.com/262123322
See white papers: here
http://www.brooksdesignps.net/Reginald_Brooks/Code/Html/arthry5.htm
See more on TPISC: The Pythagorean  Inverse Square Connection: here
Comment: In pushing for a further connection between pure, elemental natural whole integer numbers and the simple geometry that they inform, I was looking for a straightforward connection between the BIM, the PTs, and the pentagons as the BIM heavily favors the 5 resonance (base 10).
Waiting in the dentist’s office, I saw the magnificent imagery of the Zika virus as generated from the Nobel Winning cryoelectron micrograph imaging technique right there in the pages of the latest National Geographic.
The pentagonal structure of the Zika Virus leaped out and rang up all sorts of bells.
In 2001, I wrote up “GoDNA: the Geometry of DNA (axial view),” and, “SCoDNA: Structure & Chemistry of DNA” about the pure doublepentagon (decagon) geometry of the doublehelix molecule. Surely, there is a connection between these two geometries: both based on the pentagon!
We know viruses insert themselves into the chromosome and redirect it to make copies of itself (and more).
That the 345 fundamental PT — parent to all subsequent PTs (both primitive and nonprimitive) — nicely (though NOT PERFECTLY) matches the angles and slopes of the pentagon and decagon, gives one wonder if the simple geometries of the Inverse Square Law (BIM) and those ever so simple Pythagorean Triples might be in the perfect position to inform the formation of the pentagonal geometry that itself informs so much of the sublime beauty and structure we see manifested throughout the cosmos!
Would not the highly geometric viral population be of the same or similar geometry of the host population it exploited to its own ends?
References:
https://www.wired.com/story/cryoelectronmicroscopywinsthenobelprizeinchemistry
https://gizmodo.com/herearesomeincredibleimagesmadepossiblebythisy1819137180
https://www.nobelprize.org/nobel_prizes/chemistry/laureates/2017/press.html?source=techstories.org
http://www.cell.com/trends/biochemicalsciences/abstract/S09680004(17)(17)300488
https://cdn.rcsb.org/pdb101/learn/resources/zika/zikapapermodel.pdf
https://www.nih.gov/newsevents/nihresearchmatters/zikavirusstructurerevealed
https://www.sciencedaily.com/releases/2016/04/160419144741.htm
http://science.sciencemag.org/content/early/2016/03/30/science.aaf5316/tabfiguresdata
https://www.chemistryworld.com/news/explainerwhatiscryoelectronmicroscopy/3008091.article
http://www.virology.ws/2016/04/05/structureofzikavirus/
https://en.wikipedia.org/wiki/Zika_virus
https://reliawire.com/structureimmaturezika/
https://www.ncbi.nlm.nih.gov/Structure/pdb/5IRE
Cryoem Structure of Zika Virus
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BIM + PT + DNA + EpsteinBarr Virus
NEW! Connections between:
 BIM (BBSISL Matrix) + PTs (Pythagorean Triples)
\2. PT + Pentagon & Decagon (double pentagons of DNA)
 BIM + PT + DNA
 BIM + PT + DNA + Zika Virus + EpsteinBarr Virus + many more(cryoem imagery)
Number 4. should read: BIM + PT + DNA + the Icosahedral structure of the majority of human viruses
It is suggested that the pentagonal geometry of the virus allows it to insert itself within the decagonal geometry of our DNA, perhaps spiraling in along the DNA doublehelical axis, looking for a simple match opening!
BIM+PT+DNA+EB Virus from Reginald Brooks on Vimeo.
BIM + PT + DNA + EpsteinBarr virus
See a full slideshow of the BIM + PT + DNA + EpsteinBarr : here
See the video: here on Vimeo at https://vimeo.com/266783704
See white papers: here
See more on TPISC: The Pythagorean  Inverse Square Connection: here
Comment: In pushing for a further connection between pure, elemental natural whole integer numbers and the simple geometry that they inform, I was looking for a straightforward connection between the BIM, the PTs, and the pentagons as the BIM heavily favors the 5 resonance (base 10).
Previously, I wrote on the connection between the BIM + PT + DNA and specifically the Zika Virus.
A recent paper on the EpsteinBarr Virus lead to this brief follow up that ultimately relates to the majority of viruses that infect humans, from the common cold to HIV.
The pentagonal structure — at the core of the icosahedral structure that the majority of human viruses reveal — is at the heart of the connection between geometry and biology.
In 2001, I wrote up “GoDNA: the Geometry of DNA (axial view),” and, “SCoDNA: Structure & Chemistry of DNA” about the pure doublepentagon (decagon) geometry of the doublehelix molecule. Surely, there is a connection between these two geometries: both based on the pentagon!
We know viruses insert themselves into the chromosome and redirect it to make copies of itself (and more).
That the 345 fundamental PT — parent to all subsequent PTs (both primitive and nonprimitive) — nicely (though NOT PERFECTLY) matches the angles and slopes of the pentagon and decagon, gives one wonder if the simple geometries of the Inverse Square Law (BIM) and those ever so simple Pythagorean Triples might be in the perfect position to inform the formation of the pentagonal geometry that itself informs so much of the sublime beauty and structure we see manifested throughout the cosmos!
Would not the highly geometric viral population be of the same or similar geometry of the host population it exploited to its own ends?
more...(345 PT + Decagon connections)
References:
https://pdbj.org/emnavi/emnavi_detail.php?lgc=1&id=2092
https://pdbj.org/emnavi/emnavi_movie.php?id=2092
https://pdbj.org/emnavi/quick.php?id=2092
https://pdbj.org/emnavi/doc.php?id=developer
https://pdbj.org/emnavi/doc.php?tag=
https://blog.tagesanzeiger.ch/outdoor/wpcontent/uploads/sites/12/2014/05/EpsteinBarrVirus3.jpg
http://epsteinbarrvirus.purzuit.com
https://3dprint.nih.gov/discover/3dpx/001280/x3d
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4128116/
https://www.sciencedaily.com/news/plants_animals/viruses/
http://virology.net/big_virology/BVretro.html
http://virology.net/big_virology/Special/Nermut/Retro.html
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3075868/
https://www.liebertpub.com/doi/abs/10.1089/aid.1993.9.929
What is the basic structure of a virus?
“Structure of eukaryotic viruses The viral genome is surrounded by a protein shell known as capsid. Capsid encloses the genetic material of the virus and consists of protein subunits known as capsomeres. The nucleic acid genome plus the protective protein coat is called the nucleocapsid which may have icosahedral, helical or complex symmetry. The majority of viruses have capsids with either helical or icosahedral structure. The icosahedral shape, which has 20 equilateral triangular faces, approximates a sphere. All faces of an icosahedron are identical. In the icosahedral structure, the individual polypeptide molecules form a geometrical structure that surrounds the nucleic acid.”
Icosahedral Viruses include: Adenovirus Papovaviruses Herpes viruses Picornaviruses Reoviruses Retroviruse
https://en.wikipedia.org/wiki/Capsid
https://en.wikibooks.org/wiki/Structural_Biochemistry/Proteins/CryoElectron_Microscopy
https://en.wikibooks.org/wiki/Structural_Biochemistry/Carbohydrates/Virus
https://www.britannica.com/science/virus/Theproteincapsid#ref256445
The3Dimagestopcoldscientistsvirtuallookinsidevirus.html
https://www.huffingtonpost.com/2014/01/13/deadlyvirusesbeautifulphotos_n_4545309.html
Appendix D: Phi (ϕ), Fibonacci, Pentagon Connections to the PTs
Phi (ϕ), Fibonacci, Pentagon Connections to the PTs
While the LINKS section below the Comments references the historically established connections, the Comments section will address some specific findings — some of them new!
Comments:
The phi (ϕ), Fibonacci and pentagon connections — along with the Kepler Triangle — have some interesting properties that deserve a closer look. Even more so in that they relate back to the DNA doublehelix molecule and the geometry of many viruses that transform it.
For reference (23 decimals here):
(✓ϕ)^{2} = 1.27
ϕ = 𝞪^{2}/𝛾 = 𝞪𝛾 = 1.618
ϕ^{2} = 𝞪^{3} = 2.618
𝞪 = ϕ/𝛾 = 1.378 (*finestructure constantlike approximation, see NOTE)
𝞪^{2} = 𝛾 ϕ = 1.90 ( as 1.8988)
𝞪^{3} = ϕ^{2} = 2.618
𝞪 ϕ = 2.23
𝞪^{2}ϕ = 3.07
𝛾 = ^{3}✓ϕ = ✓𝞪 = ϕ/𝞪 = 𝞪^{2}/ϕ = 1.174
NOTE:
ϕ = 1.61803 ^{3}√ϕ = 1.174 = 𝛾 𝛾^{3} = 1.1743 = ϕ 𝞪 = 𝛾^{2} = 1.3782 𝛾= ϕ/𝞪 = 1.61803/1.3782 = 1.1739 These are the original phi (ϕ), 𝞪 and 𝛾 relationships found in the DNA Master Chart. PLEASE NOTE: In all references to the early DNA triangle work, 𝞪 should be correctly designated 𝞪^{1}. Back in 2001, the 𝞪 was used as 𝞪 = 1.3782, when in fact it is the reciprocal of 𝞪, as 𝞪 = 1/137, thus 𝞪^{1} is, from this current work on, correctly used as 𝞪 = 0.72…….. The differences in 𝞪like numbers are shown below this Kepler TriangleDNA Triangle section in the Tables sections. The correction starts there. This does not preclude the relationships described here, it is only one of proper symbol use. Here 𝞪 = 1.3782, down below in the Tables section, 𝞪 = 0.72…….
What happens when you square the Kepler Triangle? (square all sides) ;
you get
If we set the Kepler Triangle = to the DNA Triangle with hypotenuse=phi (ϕ), it works as we solve for 1.
𝞪^{2} + (1/𝛾)^{2} = ϕ^{2} = 1^{2} + (✓ϕ)^{2}
𝞪^{2} + (1/𝛾)^{2} = 1^{2} + (✓ϕ)^{2}
𝞪^{2} + (1/𝛾)^{2}  (✓ϕ)^{2} = 1^{2} = 1
1.90 + 0.73  1.618 = 1
NOTE: The DNA triangles are taken directly from earlier work, NOT scaled here to match the Kepler Triangle.
Another interesting relationship:
as, ϕ^{2} = 𝞪^{3} = 2.618
And:
as, ϕ^{2} = 𝞪^{3} = 2.618
And:
are equal to each other.
*My use of the close approximation to the numbers of alpha, not the actual value, was based on number theory work I was doing back then (2001 and before) on the axial view of the DNA doublehelix molecule. I used the 1.3784... number because it kept popping up with my work on phi and the decagons. Something is most definitely there, but my use of it as the fine structure constant per se would probably require a more rigorous proof to be fully accepted as alpha in the physics community. Its approximation to alpha is maintained by the 𝞪 symbol. Key relationships:
 ϕ = 𝞪^{2}/𝛾 = 𝞪𝛾 = 1.618
 ϕ^{2} = 𝞪^{3} = 2.618
 𝞪 = ϕ/𝛾 = 1.378
 𝞪^{2} = 𝛾 ϕ = 1.90 ( as 1.8988)
 𝞪^{3} = ϕ^{2} = 2.618
 𝞪 ϕ = 2.23
 𝞪^{2}ϕ = 3.07
 𝛾 = ^{3}✓ϕ = ✓𝞪 = ϕ/𝞪 = 𝞪^{2}/ϕ = 1.174
As a pure number — 1/1.378 = 0.7255.. — is a fascination that its not to be be ignored! SEE DOWN BELOW!!!
From wikipedia https://en.m.wikipedia.org/wiki/Finestructure_constant see Feynman quote:
"There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!
— Richard Feynman, Richard P. Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129. ISBN 0691083886."
The above connection of the concentric doublepentagons (decagons) of the axial view of the DNA doublehelix molecule to the Kepler Triangle provides yet another connection between phi (ϕ), right triangles and the Pythagorean Theorem, and by inference, the right triangles of the Pythagorean Triples, as they too, have a phi (ϕ) connection via their strong connection with the Fibonacci numbers. The original work, “GoDNA: the Geometry of DNA (axial view),” and, “SCoDNA: Structure & Chemistry of DNA,” generated 3 concentric decagons that were related to each other by phi (ϕ), alpha (𝞪) and the ratio of the two, gamma (𝛾 = ϕ/𝞪). The relationship generated the DNA Master Chart in which every segment of any one decagon relates mathematically to all other segments in the other two. From this a DNA Triangles image was formed:
For more, see the original paper…here.
*The connection between phi (ϕ), the Fibonacci numbers and the BIM (BBSISL Matrix) has been extensively covered in the original white paper and ebook "Brooks Base Square and The Inverse Square Law" (201011): the specific reference (Rules 161168) to phi (ϕ), the Fibonacci numbers and the IBM (BBSISL Matrix) connection: here
See more of the white papers: here
See more on TPISC: The Pythagorean  Inverse Square Connection: here
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If A = B and B = C, then A = C.
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APPENDIX D TABLES I  IV: A deeper look:
Table I. The Fibonacci Numbers are sequentially divided by phi (ϕ) in each Column. Each resolves down to a 𝛾 value in BLUE and a 𝞪 like value in ORANGE. The reciprocal of the 𝞪like value, 𝞪1, is shown on the last ROW.
Table II. While the ratios of random, nonFibonacci numbers added sequentially resolve to phi (ϕ), successive divisions by phi (ϕ) — as shown in each Column — do NOT regenerate themselves, nor do they resolve to 𝞪 or 𝛾like number values like the Fibonacci does.
Table III. Below, corresponding to three slightly different iterations of √α1 = 𝛾 as seen in Columns 1,2 — Columns 4,5 — Columns 6,7
Column 1 Reciprocal of 𝞪** 1= 1.3819 as
𝞪** = 0.7236292487 taken from working down Fibonacci Number 89 (Table I)— dividing each Fibonacci number sequentially by ϕ.Working from the 0.7236292487 on up — multiplying sequentially by ϕ — only works up to 10946 before becoming too large. Col 1,2
Column 4 Reciprocal of the published finestructure constant = 𝞪**−1** = 137.035999139(31) as α = 0.0072973525664(17).
Working from the 0.72973525664 on up — multiplying sequentially by ϕ — only works up to 55 before becoming too large. Col 4,5 Reference: https://en.wikipedia.org/wiki/Finestructure_constant
Column 6 Reciprocal of the natural fitting 0.7236067977 = 1/1.3819660113
Working from the 0.7236067977 on up — multiplying sequentially by ϕ — works perfectly at least up to 20,365,011,074. Col 6,7
1.3819660113 × 1.17082 = 1.6180334454 1.170822 = 1.3708194724 1.170823 = 1.6049828547
Here we see that this nearly “perfect” fit 𝞪 = 0.7236067977 = 1/1.38196601 completely regenerates the Fibonacci Numbers Sequence.
As, 𝞪** 1= 1/𝞪 and (1/𝞪)ϕ = ϕ, so 1.3819660113 × 1.17082 = 1.6180334454 and 𝛾2 = 𝞪 1 , so ** 1.170822 = 1.3708
Less perfect, than the original DNA Triangle values (see above), is 1.170823 = 1.6049828547 — NOT quite ϕ.
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Clearly, these numbers are close — very close — to defining a definitive connection between the Phi (ϕ), the Fibonacci Number Sequence, the pentagon, the Kepler Triangle, the Pythagorean Triples, the DNA doublehelix molecule, and perhaps even the finestructure constant.
Tweaking the values slightly one way satisfies some sets, but not all. Tweaking the values a little the other way satisfies those sets outside, but sacrifices some of the formerly included sets. It may also be simple a matter of scale, i.e. molecular versus atomic.
On the level of physics, and their constants, one would prefer a more definitive connection.
On the level of biology — even though physics informs the chemistry that informs the biology — we can appreciate how the template of the perfect match may enjoy the fruits of diversity by exactly that type of subtle variation from an exact fit!
Table IV. ϕ = 1.61803 3√ϕ = 1.174 = 𝛾 𝛾3 = 1.1743 = ϕ 𝞪 = 𝛾2 = 1.3782 𝛾 = ϕ/𝞪 = 1.61803/1.3782 = 1.1739 The Columns in ORANGE are from the Original DNA Triangle data (𝞪1 designated as simple 𝞪)
SUMMARY
We have three (3) sets of ϕ—𝞪—𝛾 relationships:
 From the original DNA Triangle (as shown in ORANGE in Table IV, Cols. 16 ), 𝛾 = 1.1739 or 1.174
 From the Fibonacci Number Sequence (Table I, Col. 89; Table III, Col. 1,6; and Table IV, Cols. 7,8 ), 𝛾 = 1.17082 or 1.17086
 From the finestructure constant numbers (Table III, Col. 4), 𝛾 = 1.1807
The latter, somewhat mimicking the actual finestructure constant, strays the furthest from the others.
No. 1 and 2 are quite close, yet not exactly interchangeable as one plays out the ratios.
The DNA Triangle set is derived from the measurements and calculations of the concentric, doublepentagonal geometry of the axial view (i.e. one complete 360° rotation) of the doublehelix spiral. Its tight correlations make for some direct DNA Triangle correlations and a selfconsistent DNA Master Chart relating the various geometric segments of the concentric decagons to each other.
The Fibonacci set is taken strictly from calculations: dividing the known Fibonacci Numbers by ϕ. The difference comes from where you start in the sequence and what direction do you go. We know the ratio more perfectly registers the ϕ value, the further along the sequence one goes.
The common factors here are ϕ, the pentagon, and the Fibonacci numbers…and, now perhaps, some common binding factor that has been referred to as gamma, 𝛾.
The Fibonnaci numbers can generate the Pythagorean Triples, and, because of their relationship to ϕ, they are related to pentagons — especially in the form of concentric, doublepentagons.
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Review of the Pythagorean Triples from Wikipedia: here
Return to the BIM + PT + Pentagons page: here
LINKS:
While there are many excellent reference links available online, the GoldenNumber.net is a great starting point for organization, clarity and thoroughness:
https://www.goldennumber.net/sitemap/
https://www.goldennumber.net/geometry/
https://www.goldennumber.net/triangles/
https://www.goldennumber.net/fivephi/
https://www.goldennumber.net/math/
https://www.goldennumber.net/fibonacciseries/
https://www.goldennumber.net/fibonacci24pattern/
https://www.goldennumber.net/phipigreatpyramidegypt/
Another great site covering Numbers (and much more) for education:
http://schoolbag.info/mathematics/numbers/index.html Numbers: Index
http://schoolbag.info/mathematics/numbers/76.html Pythagorean Triples
http://schoolbag.info/mathematics/numbers/77.html Fibonacci +Pythagorean Triples
http://schoolbag.info/mathematics/numbers/82.html Pythagorean Curiosities
http://schoolbag.info/mathematics/numbers/107.html Fibonacci list
http://schoolbag.info/mathematics/numbers/108.html Primes list
http://schoolbag.info/mathematics/numbers/109.html Mers.Primes list
http://schoolbag.info/mathematics/numbers/110.html Perfect Numbers
Golden Ratio, phi (ϕ)
https://en.wikipedia.org/wiki/Golden_ratio
http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/phi.html
Platonic Solids
http://www.3quarks.com/en/PlatonicSolids/ (images courtesy 3quarks.com)
http://www.treenshop.com/Treenshop/ArticlesPages/FiguresOfInterest_Article/Introduction.htm
https://en.wikipedia.org/wiki/Geometry
https://en.wikipedia.org/wiki/Platonic_solid
https://en.wikipedia.org/wiki/Regular_dodecahedron
https://en.wikipedia.org/wiki/Regular_icosahedron
https://en.wikipedia.org/wiki/Icosahedral_symmetry
http://mathworld.wolfram.com/PlatonicSolid.html
Phi (ϕ) , Fibonacci Numbers and Pentagons
https://en.wikipedia.org/wiki/Fibonacci_number
https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers
http://www4.ncsu.edu/~njrose/pdfFiles/GoldenMean.pdf
http://www.matematicasvisuales.com/english/html/geometry/goldenratio/pentagondiagonal.html
http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fib.html
http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fibmaths.html#section3
http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/phi2DGeomTrig.html
https://www.mathsisfun.com/numbers/fibonaccisequence.html
https://math.temple.edu/~reich/Fib/fibo.html
Images
https://www.pinterest.com/pin/10414642860397615/
https://www.pinterest.com/goodjolt/fibonacci/
https://www.pinterest.com/goodjolt/fibonacci/
Pythagorean Triples and Fibonacci Numbers
http://nextlevelmaths.com/resources/wow/pythag_fibonacci/
http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fibmaths.html#section3
https://summathfun.wordpress.com/2010/12/02/fibonaccimethodoffindingpythagoreantriples/
Kepler's Triangle and phi (ϕ) and the Pythagorean Theorem
https://en.wikipedia.org/wiki/Kepler_triangle
https://www.goldennumber.net/triangles/
Useful links for educators:
http://www.nextlevelmaths.com/resources/useful_links/
https://summathfun.wordpress.com/2010/12/02/fibonaccimethodoffindingpythagoreantriples/
http://www.studentguide.org/theultimateresourceonthefibonaccisequence/
https://www.mathsisfun.com/numberpatterns.html
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TPISC_IV:_Details ~A MathspeedST Supplement~
Copyright©201819, Reginald Brooks. All rights reserved.Preface*TPISC ( pisque—silent “T”): The Pythagorean — Inverse Square Connection*3Steps to Nirvana:Something Very NEW! ➗24: PPTs and PRIMES(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)DSEQEC: DoubleSlit Experiment — Quantum Entanglement Conjecture, and, CaCoST: Creation and Conservation of SpaceTimeIntroList of 40 characters/focus points:I. Who?PTs — Pythagorean Triples — and the squares and rectangles they represent.II. What?PTs are nonisosceles, 90°righttriangles composed of Whole Integer Numbers (WINS).III. Where?PTs — and their proofs — are ubiquitously located, in five (5) easy steps, throughout the infinitely expandable BIM. (see Chapter V: How for details)IV. When?PTs are uniquely located on any Row (or Column) containing Paired Sets — 2 Squared Areas — that represent a^{2} and b^{2}, with c^{2} at the endpoint on the PD.V. How?PTs have been eloquently and succinctly defined by algebraic geometry.PTs: 40+ FEDM derived profiling focus point parameters:PTs: 10 ways to approach the BIM (return to this again after completing the sections that follow):BBSISL Matrix (BIM): Basic, fundmental Rules of the Symmetrical Matrix grid: 10 Basic, fundamental rules of the symmetrical BBSISL MatrixBBSISL Matrix Inner Grid Golden Rules (IGGR)5 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner GridPythagorean Triples and BBSISL Fundamentals (TPISC: The PythagoreanInverse Square Connection)3 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner Grid that encompass the PTs.Matrix Flow:Exponentials Summary (see TPISC V: Exponentials):EQUATIONS:VI. Why?A. The PTs are so intimately and ubiquitously intertwined, interconnected and resonantly echoed throughout the BIM that one could almost rewrite TPISC: The Pythagorean  Inverse Connection to Pythagorean = Inverse Square Law, i.e. the Pythagorean Theorem is really just a subset of the Inverse Square Law (ISL)!B. Areas, Perimeters, Proofs.C. How — but NOT why — 24 connects the PTs and PRIMES to the BIM.SubMatrixTogether SubMatrix 1 and SubMatrix 2 will provide the visualgeometric and algebraicgeometric location of ALL PPTs and PRIMES.First SubMatrix 1:SubMatrix 2:SubMatrix 2 Sidebar: Exponentials of the PPTs(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)From TPISC III: Clarity Conclusion:Open letter to Leonard Susskind, Juan Maldacena, and Mark Van Raamsdonk, …DSEQEC: DoubleSlit Experiment — Quantum Entanglement Conjecture, and, CaCoST: Creation and Conservation of SpaceTimeReferences (specific to ER=EPR and some of the surrounding issues):Closing thoughtsSee Appendices C_D_E: BIM PPTS, Pentagons, Decagons, Phi (ϕ), Fibonacci, Pentagon Connections to the PTs, and, BIM + PT + DNA + Zika, EpsteinBarr and other IcosahedralStructured Human Viruses for ALL the Figures and Tables relating to these topics.VII. Summary24 SUMMARYSUMMARY:DETAILS:In brief:(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)SubMatrix 2:(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24.)PTs: 40+ FEDM derived profiling focus point parameters:PTs: 10 ways to approach the BIM (return to this again after completing the sections that follow):BBSISL Matrix (BIM): Basic, fundamental Rules of the Symmetrical Matrix grid: 10 Basic, fundamental rules of the symmetrical BBSISL MatrixBBSISL Matrix Inner Grid Golden Rules (IGGR)5 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner GridPythagorean Triples and BBSISL Fundamentals (TPISC: The PythagoreanInverse Square Connection)3 Basic, fundamental rules of the symmetrical BBSISL Matrix Inner Grid that encompass the PTs.Matrix Flow:Exponentials Summary (see TPISC V: Exponentials):Summation:feel the beat: +  +    +  +    +  +    +  +    ...VIII. ConclusionFrom SQUARES to RECTANGLES — CIRCLES to OVALS — ISOSCELES to nonISOSCELES TRIANGLES: That's the story of TPISC: The Pythagorean  Inverse Square Connection.IX. ReferencesX. AppendixAppendix A: BIM PPTs: Details of their Portrait ProfilesAppendix B: BIM ➗24 PPTs and PRIMESTable VI b fully expanded 11010 PTs and Primes with ALL ACTIVE Rows colorcoded in VioletSubMatrix 2 Sidebar: Exponentials of the PPTsAppendix C: BIM_PT_Pentagons BIM + PT + Decagon (double Pentagons) BIM + PT + Decagon (double Pentagons of the DNA doublehelix)Appendix D: Phi (ϕ), Fibonacci, Pentagon Connections to the PTsComments:TABLES I  IV: A deeper look:SUMMARYLINKS:Golden Ratio, phi (ϕ)Platonic SolidsPhi (ϕ) , Fibonacci Numbers and PentagonsImagesPythagorean Triples and Fibonacci NumbersKepler's Triangle and phi (ϕ) and the Pythagorean TheoremUseful links for educators:Appendix E: BIM + PT + DNA + Zika, EpsteinBarr and other IcosahedralStructured Human Viruses BIM + PT + DNA + Zika Virus BIM + PT + DNA + EpsteinBarr Virus BIM + PT + DNA + EpsteinBarr virus Appendix F: BIM + misc: BIM: How to Make
Appendix F: BIM misc...
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BIM (BBSISL Matrix): How to Make
10 Easy steps to making the symmetrical BIM
After you set up your table with 10 x 10 Columns and Rows:
(Example using the Numbers app.)
A B C E F G H I J line Axis # = x PD # = x^{2} x^{2} 1 x^{2} 4 x^{2} 9 x^{2} 16 x^{2} 25 36
 Column Headers. After 25, simply copy and paste in the PD sequence 364964… that you can copy from your own list: 1^{2}, 2^{2}, 3^{2}, 4^{2},... as found in Column 3 (C) below.
 Column 2 (B) is just like Column 1 (A): type in 0,1,2,3… and drag down.
 POWER('Axis # = x',2) is the formula for squaring the Axis #s giving the PD#=x^{2} in Column 3 (C).
 $'PD # = x^{2}'−1 is the formula for ALL the remaining Columns ADJUSTED by changing the “1” with the PD cell value found in the Header immediately above. The “$” symbol means “absolute” and all subsequent calcs will use the fixed value from C.
 Once selected, you can dragcopy the formula across the Table and then make the ADJUSTMENTS quickly.
 Autofill the Table by selecting the TOP Row (under the HEADER) and pull down the orangedot symbol to the bottom of the Table.
 Because the BIM is bilaterally symmetrical about the PD, and, because the same PD #s forming the horizontal ROWS are also forming the vertical Columns, the BIM upper triangle — above the PD — will show a negative () sign in front of the cell values and this sign is to be ignored.
 Below the PD, all the cell values will be (+) numbers, without any sign.
 The PD itself — because it reflects subtracting the same PD # from itself — will be “0” and this may be color coded to reflect that and/or the ACTUAL PD values, as found on the HEADER above, can be inserted.
 To expand or to fill in a particular Axis # — or range of #s — simply add the appropriate Columns and Rows and duplicate the above process for those values.
Color coding the Table, Columns, Rows and Cells can be done in the Numbers App Cell section. Text in the Text section, etc.
Why the 10th Column and Row? Fill that in to check your work here.
Larger examples of How to Make the BIM with a spreadsheet app (Numbers) found here in Table 31 series found in Appendix B of TPISC IV: Details.Of course for smaller BIM grids, one can simply enter the cell values by hand as all the Inner Grid (IG) values are simply the difference between their Prime Diagonal (PD) Column and Row intercepts, e.i. 16 = 25  9 PD values.
BIM_How_to_Make from Reginald Brooks on Vimeo.
UPDATE: Once you get the basics down, there is a cleaner, more direct method that is also easily expandable.
.
Open below image in a new tab to see all 5 example sheets.NOTE: for extending beyond allowed (Numbers, Excel is much higher) max 255 Columns:
 1. Duplicate sheet made with the NEW method, rename new sheet;
 2. On Row 2, replace with the NEW squared #s that you are extending;
 3. On Row 3 (first nonheader row), place the first formula and drag across:
 $'PD # = x2'−D$2
 4. Drag Row 3 down to autofill the cells below. Each Col should have SAME formula, but will give individual values appropriate for its location. Double check. SAVE.
 5. Color code the PD (Prime Diagonal “0”s), Row 3, and Col. C (blue) to separate out the Axis and PD from the Inner Grid cell values. Replace the PD #s with the Row 2 values, center all cells, and adjust format as needed.
BIM+PT (2018) Introduction from Reginald Brooks on Vimeo.
A central hub for all math works can be found on the MathspeedST Media Center page.
BIM 50x50 (open image in separate tab/window)
BIM 75x75 (open image in separate tab/window)
BIM 125x125 (open image in separate tab/window)
BIM 250x250 (open image in separate tab/window)
Appendix G: BIM the PTOP and the Goldbach Conjecture
~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~
Reference:
"A New Kind of Prime
The twin primes conjecture’s most famous prediction is that there are infinitely many prime pairs with a difference of 2. But the statement is more general than that. It predicts that there are infinitely many pairs of primes with a difference of 4 (such as 3 and 7) or 14 (293 and 307), or with any even gap that you might want."
Quote is from Quanta Mag 9/26/19 article: Big Question About Primes Proved in Small Number Systems
by Kevin Hartnett
~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~
 Twin Primes
 Primes separated by other EVEN numbers
 Euler's 6n+1 and 6n1
 FermatEuler's 4n + 1= Sum of Two Squares Theorem (Pythagorean Primes)
 Dickson's Conjectures: Sophie Germain Primes
 Goldbach Conjecture (Euler's "strong" form)
 Primes  BIM  Pythagorean Triples
~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~
A new and easy visualization!
PRIME GAPS
Game Board
There is a new game board in town. It has actually been around for awhile, but few know of it. It's basically a matrix grid of natural numbers that defines the Inverse Square Law (ISL). It's called the BIM, short for the BBSISL Matrix. Every grid cell is uniquely occupied by a given number that is simply the difference between the horizontal and vertical intercept values of the main "Prime Diagonal" (not Prime number diagonal) that mirrordivides the whole matrix.
If you show all the matrix values that are evenly divisible by 24, a crisscrossing pattern of diamonddiagonal lines will appear and this ends up giving us a unique visual on the distribuition of ALL PRIMES!
Basically, all ODDs (3 and greater for this presentation) fall into a repetitive pattern of:
ODD ÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3...
The matrix ÷24 described above, nicely picks out these sets of —Not÷3—Not÷3 that we call Active Row Sets (ARS). Each ODD member of the ARS is referred to as an Active Row (AR).
The ÷3 ODD Rows that lie between are called NonActive Rows, or NA.
It is a necessary, but not sufficient condition, that ALL PRIMES are strictly located on ARs. NO exceptions, except for some ARs have NO PRIMES.
Another striking visualization is that ALL Primitive Pythagorean Triples (PPTs) also lie only on the ARs, following the same necessary, but not sufficient, condition. NO exceptions here as well, except for some ARs also have NO PPTs. In addition, only one of the ARs within an ARS can have a PPT.
So any AR within an ARS can have 0/1 PRIMES and/or 0/1 PPTs in any combination, only with the one caveat: that only 1 PPT/ARS is allowed.
Now, not to belabor this BIM game board, let's talk PRIME GAPS!
Prime Gaps
Let's refer to the lower number value for the ARS as "Lower" and the higher as "Upper." All are ODDs. As will become apparent, you can always tell if your ODD AR is "Lower/Upper" simply by adding 2 and ÷3: if the result evenly ÷3 it is "Upper" and if not, "Lower."
The Even Gap between primes strictly follows this pattern: (all are necessary but not sufficient conditions for primality)
For any given set of twin primes, or any single prime, that is the smaller "Lower" of an AR set:
 Even Gap must be 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32, 36, 38, 42, 44, 48, 50,…
 Example: P=5: 7, 11, 13, 17, 19, 23, (25), 29, 31, (35), 37, 41, 43, 47, (49), 53, (55)
 P + 2, P + 2+4, P + 2+6, P + 2+10, P + 2+12, P + 2+16, P + 2+18, P + 2+22, P + 2+24,…
 P+2^{1}, P+2+2^{2}, P+2^{3}, P+2^{2}+2^{3}, P+2+2^{2}+2^{3}, P+2+2^{4}, P+2^{2}+2^{4}, P+2^{3}+2^{4}, P+2+2^{3}2^{4}
 The Even Gap difference pattern follows: 242424….
For any given set of twin primes, or any single prime, that is the larger "Upper" of an AR set:
 Even Gap must be 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 34, 36, 40, 42, 46, 48, 52, 54.…
 Example: P=7: 11, 13, 17, 19, 23, (25), 29, 31, (35), 37, 41, 43, 47, (49), 53, (55), 59, 61
 P + 4, P + 2+4, P + 2+8, P + 2+10, P + 2+14, P + 2+16, P + 2+20, P + 2+22, P + 2+26, P + 2+28,
 P+2^{2}, P+2+2^{2}, P+2+2^{3}, P+2^{2}+2^{3}, P+2^{4}, P+2+2^{4}, P+2+2^{2}+2^{4}, P+2^{3}+2^{4}, P+2^{2}+2^{3}+2^{4}, P+2+2^{2}+2^{3}+2^{4},
 The Even Gap difference pattern follows: 424242…. and is identical in the middle, not the start!
One can readily see that all this is simply the natural result of two ARs alternating with a NA row:
AR—AR—NA—AR—AR—NA—AR—AR—NA—AR—AR—NA—…
“Lower” AR number
AR—5
AR—7
NA—9
AR—11
AR—13
NA—15
AR—17
AR—19
NA—21
AR—23
AR—25
NA—27…
It is 6 steps, i.e. +6, from any NA to the next NA, from any “Lower” AR to the next “Lower” AR, from any “Upper” AR to the next “Upper” AR. ALL of the various EVEN GAPS may be shown to be a direct consequence of the natural number sequence and easily visualized on the BIM.
To any “Lower” Active Axis Row ODD:
 Add 2, 4, or 6,…
 If sum is ÷3, it is a NA
 If sum is NOT÷3, it is an AR number and a PRIME (or PPT) candidate.
“Upper” AR number
AR—7
NA—9
AR—11
AR—13
NA—15
AR—17
AR—19
NA—21
AR—23
AR—25
NA—27…
To any “Upper” Active Axis Row ODD:
 Add 2, 4, or 6,…
 If sum is ÷3, it is a NA
 If sum is NOT÷3, it is an AR number and a PRIME (or PPT) candidate.
Again, It is 6 steps, i.e. +6, from any NA to the next NA, from any “Lower” AR to the next “Lower” AR, from any “Upper” AR to the next “Upper” AR. ALL of the various EVEN GAPS may be shown to be a direct consequence of the natural number sequence and easily visualized on the BIM.
The ARS pattern on the BIM clearly shows the above patterns and may be extrapolated to infinity:
PRIME Conjectures
A number of PRIME conjectures have been shown to be easily visualized on the BIM:
Reference: TPISC IV: Details: BIM + PTs + PRIMES
The factors of 24 — 1,24–2,12–3,8–4,6 — when increased or decreased by 1, ultimately pick out ALL ARs. Euler's 6n +/1 is the most direct, Fermat's 4n + 1 gets the Sums of Two Squares = Pythagorean Primes(while 4n + 3 gets the rest).
Fermat's Little Theorem (as opposed to the more familiar "Fermat's Last Theorem") tests for primality.
But now there is a dead simple way to test for primality:
The difference in the squares between ANY 2 PRIMES (≧5) ALWAYS = n24.
For example, take any random ODD #  25 —> it must be ÷24 n times to be PRIME. (n=1,2,3,...)
741
741252/24 = n= 22877.3 NOT PRIME
189
189252/24 = n = 1487.3 NOT PRIME
289
289252/24 = n = 3479 PRIME
PRIMES vs NOPRIMES
I would be remiss if I did not mention that throughout this long journey that began with the Inverse Square Law and the Primtive Pythagorean Triples that the PRIMES kinda of just fell out. They just kept popping up. Not the least, but certainly not without effort, the BIM actually directly visualizes ALL of the NOPRIMES. In doing so, one is left with information that is simply the inverse of the PRIMES. Subtract the NOPRIMES information from the list of ODDS (disregarding 2) and what remains are ALL the PRIMES! A simple algebraic expression falls out from that:
NP = 6yx +/ y
letting x=1,2,3,… and y=ODDs 3,5,7,… with the only caveat is that if you don't first eliminate all the ÷3 ODDS, you must include exponentials of 3 (3^{x}) in the NP tally.
Goldbach Conjecture (Euler's "strong" form)
In 200910, a solution to Euler's "strong" form of the Goldbach Conjecture "that every even positive integer greater than or equal to 4 can be written as a sum of two primes" was presented as the BBSISL Matrix Rule 169 and 170. This work generated a Periodic Table of Primes (PTOP) in which Prime Pair Sets (PPsets) that sequentially formed the EVEN numbers were laid out. It is highly patterned table. A recently annotated version is included with the original below. (See above link for details.)
It turned out this PTOP was actually embedded — albeit hidden — within the BIM itself as shown in Rule 170 and here, too, a recently annotated version is included below. (See above link for details.)
LINKS:Rule 169: Periodic Table of Primes.
Rule 169: annotated
Rule 170: Periodic Table of Primes (PTOP): embedded within Brooks (Base) Square.
Rule 170: annotated
Table34:PTOP 100
Table35: PTOP 200+
MathspeedST: TPISC Media Center
NEWLY ADDED (after TPISC IV published):
Back to Part I of the BIMGoldbach_Conjecture.
Back to Part II of the BIMGoldbach_Conjecture.
Back to Part III of the BIMGoldbach_Conjecture.
BACK: > Simple Path BIM to PRIMES on a separate White Paper BACK: > PRIMES vs NOPRIMES on a separate White Paper BACK: > TPISC_IV: Details_BIM+PTs+PRIMES on a separate White Paper BACK: > PeriodicTableOfPrimes(PTOP)_GoldbachConjecture on a separate White Paper BACK: > Make the PTOP with Fractals on a separate White Paper
With that review of the ongoing work, the full presentation of this work starts below. It will cover everything above plus the NEW work leading to the findings and proof of the Goldbach Conjecture. This will be the basis for the ebook: PTOP: Periodic Table of PRIMES & the Goldbach Conjecture
PTOP: Periodic Table Of PRIMES & the Proof of the Goldbach Conjecture
PTOP Goldbach Conjecture from Reginald Brooks on Vimeo.
PTOP: Periodic Table Of PRIMES & the
PTOP: Periodic Table Of PRIMES & theGoldbach ConjectureINTRODUCTIONSTATEMENT: Layout & EssentialsThere are five sets of data to consider:FINDINGS & PROOFSUMMARY and CONCLUSIONDATA: Images and TablesDATA: Images and TablesREFERENCES
Goldbach Conjecture
INTRODUCTION
While the (strong) Goldbach Conjecture has been verified up to 4x10^{18}, it remains unproven.
A number of attempts have demonstrated substantial, provocative and often beautiful patterns and graphics, none have proven the conjecture.
Proof of the conjecture must not rely solely on the notion that extension of a pattern to infinity will automatically remain valid.
No, instead, a proof must, in its very nature, reveal something new about the distribution and behavior of PRIMES that it is absolutely inevitable that such pattern extension will automatically remain valid. The proof is in the pudding!
Proof offered herein is just such a proof. It offers very new insights, graphical tables and algebraic geometry visualizations into the distribution and behavior of PRIMES.
In doing so, the Proof of the Euler Strong form of the Goldbach Conjecture becomes a natural outcome of revealing the stealthy hidden Number Pattern Sequence (NPS) of the PRIMES.
STATEMENT: Layout & Essentials
Proof of the Goldbach Conjecture (strong form, ≥6)
Natural (n), Whole Integer Numbers (WIN) — 0,1,2,3,…infinity — form horizontal and vertical Axis of a simple matrix grid.
The squares of such WINs — n^{2}=1^{2}=1, 2^{2}=4, 3^{2}=9,…infinity — forms the central Diagonal of said grid — dividing it into two bilaterally symmetric triangular halves.
Every Inner Grid (IG) cell within is simply the difference (∆) between its horizontal and vertical Diagonal intercept values. They extend to infinity. The Diagonal WINs form the base of a 90° Rangled isosceles triangle with said IG cell value at the apex.
Every IG cell within is also the product of two Axis WINs (Either horizontal or vertical, not both), that form the base of a 90° Rangled isosceles triangle with said IG cell value at the apex.
The complete matrix grid extends to infinity and is referred to as the BIM (BBSISL Matrix).The BIM forms — and informs — a ubiquitous map (algebraic geometry) to:
The Inverse Square Law (ISL);
 The Pythagorean Triples (PT);
 The PRIMES (stealthily hidden, but revealed by NPS.
The 1^{st} Diagonal that runs parallel to either side of the main Prime Diagonal (PD, not of PRIME numbers, but primary), is composed of the ODD WINs: 1,3,5,…infinity.
If we add +1 to each value, that 1^{st} Diagonal now becomes a sequence of ALL the EVEN WINs (≥4): 4, 6, 8,…infinity. NOTE: this is why the PTOP is hidden, in the normal, base BIM these remain ODDs.
Select ANY EVEN WIN and plot a line straight back to its Axis WIN — that Axis WIN = EVEN/2 = core Axis #.
Upon that same Axis, PRIME Pair sets (PPsets) — whose sum (∑) equals the EVEN WIN (on the 1^{st} Diagonal) — will be found that form the base of 90° Rangled isosceles triangle(s) whose apex lie(s) on that straight line between the EVEN and its 1/2 Axis WINs. PPsets with identical PRIMES = 1/2 Axis value.
The proof that every EVEN WIN has ≥1 PPsets can be seen in the Periodic Table Of PRIMES (PTOP) that stealthily informs the BIM of how each and every EVEN WIN is geometrically related to one or more PPsets.
These PPsets are NOT randomly contributing their ∑s to equal the EVEN WINs, rather they come about as the consequence of a strict NPS: the sequential — combining, linking, concatenation — addition of the PRIMES Sequence (PS) — 3,5,7,11,13,17,19,…—to a base PS — 3,5,7,11,13,17,19,….
The NPS of this addition forms the PTOP: for each vertical PS — the 1^{st} PRIME (P_{1}) remains constant (3), the 2^{nd} PRIME (P_{2}) sequentially advances one (1) PS WIN — is matched diagonally with the 2^{nd} PS, but now the 1^{st} PRIME sequentially advances, while the 2^{nd} PRIME remains constant within a given PPset.
This matching addition of the 2^{nd} PS at the bifurcation point of the common 2^{nd} PRIMES, forms the zigzag diagonal PPset Trails that are the hallmark of the PTOP.
For every subsequent vertical PPset match, the Trail increases by one (1) PPset.
The rate of such PPset Trail growth far exceeds the PRIME Gap rate.
The zigzag diagonal PPset Trails combine horizontally on the PTOP to give the ∑# of PPsets whose ∑s = The EVEN WIN.
More than simply proving the Goldbach Conjecture, the PTOP hidden within the BIM reveals a new NPS connection of the PRIMES: PRIMES + PRIMES = 90° Rangle isosceles triangles.
The entire BIM, including the ISL—Pythagorean Triples—and, PRIMES, is based on 90° Rtriangles!
Similar to how the grid cell values of the Axis, PD and IG of the Pythagorean Triples reveal additional, intimate connections within the BIM, so too do the PPsets: the 1^{st} PRIME values of each set points to the # of STEPS from the PD that intersects the given EVEN WIN (Axis^{2}), on a straight line path back to its Axis, at the apex where its other PPset member intersects — this is no mere coincidence — and that apex is, of course, the 90° Rangle isosceles triangle that results. For example: EVEN = 24, Axis = 12, PD = 144, # of STEPS from PD towards Axis = 5 and 19 for the 5+19 PPset and 7 and 17 for the 7+17 PPset, and 11 and 13 for the 11+13 PPset that each forms the EVEN 24.
As the bifurcation concatenation of the PS — 3,5,7,11,13,17,19,..— with the same base PS — 3,5,7,11,13,17,19,…— of those EVEN WIN — that when “3” is subtracted, the remainder is a nextinthesequence PRIME — remains one of a similar split with the Pythagorean Triples: for every “Primitive” parent PT, there are multiple “NonPrimitive” child PTs and it is the PPTs (Primitive Pythagorean Triples) that ultimately form the interconnectedness of ALL PTs back to the original PPT — the 345. With the PRIMES, one raises the question: why are these “EVENS” preselected to be the “parent” EVENS forming the “beginning” or “start” of every PPset Trail with all other EVENS hitching on to that Trail further down the sequence?
Another set of STEPS (S) from the core Axis value , directly on the Axis, identifies each symmetrical pair of a given PPset that forms that given EVEN. These STEPS may by Universally calculated from the EVENS, P_{1} and P_{2} values. Examples are given further down.
As every new discovery unlocks many more questions, it follows that the details of the PTOP and BIM visualizations should both satisfy the proof offered and, more importantly, provide provocative data that will advance the field for the next researcher!
There are five sets of data to consider:
BIM (BBSISL Matrix): grid visualizations that overview the entire work
 Fig. 1PTOP: Periodic Table Of PRIMES (100, original)
 Fig. 2PTOP: Periodic Table Of PRIMES (100, annotated)
 Fig. 3PTOP: Periodic Table Of PRIMES (100, upgraded)
 Fig. 4PTOP: Periodic Table Of PRIMES (200, annotated)
 Fig. 5BIM: Symmetrical STEPS of the PPsets for EVEN 24 (original)
 Fig. 6BIM: Symmetrical STEPS of the PPsets for EVEN 24 (annotated)
 Fig. 7Table 46: Symmetrical STEPS of the PPsets for EVEN 128 annotated snapshot
 Fig. 8BIM: Symmetrical STEPS of the PPsets for EVEN 126 snapshot
 Fig. 9BIM: Symmetrical STEPS of the PPsets for EVEN 126
 Fig. 10BIM: Symmetrical STEPS of the PPsets for EVEN 128 snapshot
 Fig. 11BIM: Symmetrical STEPS of the PPsets for EVEN 128
 Fig. 12 animated gif of video (below)
 Video: PTOP rule 169170 Annotated
PTOP: the actual Table
 Table 34: Original PTOP from 2009 (EVENS 6100)
 Table 35: Upgraded PTOP (EVENS 6200)
 Table 36: Upgraded PTOP (EVENS 6400)
 Table 37: Upgraded PTOP (work in progress, EVENS 61000)
PTOP: Analysis
 Table 38: Distribution and NPS of the PPset Trails (EVENS 6404)
 Table 39: Distribution and NPS of the PPset Trails (EVENS 6914)
 Table 40: Distribution and NPS of the PPset Trails (EVENS 62360 and up and up)
 Table 41: Bifurcation Addition and PPset ∑s
Reference
 Table 42: PRIME Gaps
 Table 43: PRIME Partitions = PPsets per EVENS (42000)
 Table 44: Summaries of Table 43 (EVENS 42000)
 Table 45: Equations for PTOP Tables 3841
PRIME PPset Trails
 Table 46: PRIME PPset Trails (310007 and up)
 Table 47: PRIME PPset Trails simplified and extended Table 46 (3568201)
 Table 48: Working example: EVEN 8872, core Axis 4436 with 93 PPsets.
 Table 49: PRIME PPset Trails & EVENS divisible by 6,12 or 24 (310007 and up).
The DATA has been grouped in its own section down below. It is highly recommended that you, the reader, preview first — review, thereafter.
~
~~~FINDINGS & PROOF
The definitive proof depends on demonstrating that the PPset Trails grow and extend to cover the “next” EVENS at a rate that exceeds the Prime Gap rate. PRIMES Sequence, PS is the key.
The PS is well established. How many primes are there?
As every beginning PPset — of 3,P_{2} — forms from successively increasing the 2^{nd} PRIME by the next number in the PS (3,5,7,11,13,17,…), the Trail formed by bifurcating off from that point increases the total number sum (∑) of PPsets by one, i.e. Trail, Trail+1, (Trail+1)+1, ((Trail+1)+1)+1,…
A necessary, and sufficient, condition is that the Trail lengths — i.e. the total number of PPsets, in their overlapping aggregate, always exceed both the number of PRIMES and their Gaps for any and all numbers.
Specifically from Table 46 and Fig.__, locate a P_{2} PRIME with a large Gap, e.i. P_{2}= 23, Gap=6.
line#———PRIMES, P_{2}—∑#set/Trail—PRIME Gap—∆TrailGap— EVEN:∑#sets—EC——E_{ending}——EVEN, E
8 23 8 6 2 26: 3 11 46 26 We see that 23 has 8 PPsets in its Trail, it has a Gap of 6 to the next PRIME (for a ∆=2). It’s EVEN=26 and there are 3 PPsets (going horizontally across the PTOP) that will make 26. The 11 is the number of EVENS Covered (EC) as the PPsets zigzag diagonally down the PTOP. If we add double this EC  1 to the EVEN 26, we get the EVEN Ending of 46 as: 2(111) + 26 = 46. So this Trail alone inclusively covers the EVENS 2646, although their will be some holes. To fill the holes, we look at the Trails that started up above this 23 Trail.
For the Specific set of equations exclusively for the 3,P_{2} sets, where ∆ = P_{2}  3 (see Table 45: Equations):
2EC = P_{2} 1
Ending EVEN Covered = E_{e} = 2(EC  1) + EVEN
5 13 5 4 1 16: 2 6 26 16 6 17 6 2 4 20: 2 8 34 20 7 19 7 4 3 22: 3 9 38 22 We can readily see that the 17 and 19 Trails, when their EVENS are added to their 2EC1, respectively, will equal or exceed the EVEN of the NEXT PRIME up by the GAP=6.
Take Trail 17: EVEN = 20 with EC = 8. 2(81) + 20 = 34.
Take Trail 19: EVEN = 22 with EC = 9. 2(91) + 22 = 38.
And these Trails 17 and 19 will overlap the Trail 23 EC span, filling in any and all holes between it and the next Trail 29.
9 29 9 2 7 32: 2 14 58 32 For completeness, we can see that Trail 13, while reaching Trail 23 at it’s start, does not overlap any further.
Take Trail 13: EVEN = 16 with EC = 6. 2(61) + 16 = 26. We also can see that 13 + 13 = 26.
~~~
The PPset Trails for EVEN 26 gives 3 PPsets via the overlapping Trails 17, 19 and 23, AND, their overlapping Trails extend past the next 3 EVENS (28, 30 and 32), ensuring that they are “covered” with PPsets in the Gap jump to the next PRIME (29). All very neat and clean.
The actual PPsets ∑s equal the EVEN 26:
 3 + 23 = 26
 7 + 19 = 26
 13 + 13 = 26
which comes about more simply as the core Axis value ± steps away:
 13 ± 10 = 3 and 23
 13 ± 6 = 7 and 19
 13 ± 0 = 13 and 13
 where the core Axis value = 1/2 the EVEN, or 26/2 = 13 and we locate it on Column 2, PRIMES (line 5) for reference. It is not actually the PRIME, as we will see. Here it happens to land directly on the PRIME 13 as it is one of the PPsets. Other times it may well be somewhere between two PRIMES in this column. We will clarify this further on, but for now it is a reference to numbers on the Axis of the BIM that, being core Axis values, are key in showing that the PPsets will ALWAYS have their Pair members symmetrically located on either side of this reference marker.
But what about a larger Gap? How about Trail 113, EVEN = 116 and a Gap of 14 to the next Trail 127? (See image from Table 36.)
Trail 113 is overlapped by Trails 73, 79, 97, 103, and 109 that account for the 6 PPsets that will form the EVEN = 116 (the 3,113 PPset is included in the tally and you can follow it in Table 43). Again, from Table 46:
1 3 1 2 1 6: 1 1 6 6 2 5 2 2 0 8: 1 2 10 8 3 7 3 4 1 10: 2 3 14 10 4 11 4 2 2 14: 2 5 22 14 5 13 5 4 1 16: 2 6 26 16 6 17 6 2 4 20: 2 8 34 20 7 19 7 4 3 22: 3 9 38 22 8 23 8 6 2 26: 3 11 46 26 9 29 9 2 7 32: 2 14 58 32 10 31 10 6 4 34: 4 15 62 34 11 37 11 4 7 40: 3 18 74 40 12 41 12 2 10 44: 3 20 82 44 13 43 13 4 9 46: 4 21 86 46 14 47 14 6 8 50: 4 23 94 50 15 53 15 6 9 56: 3 26 106 56 16 59 16 2 14 62: 3 29 118 62 17 61 17 6 11 64: 5 30 122 64 18 67 18 4 14 70: 5 33 134 70 19 71 19 2 17 74: 5 35 142 74 20 73 20 6 14 76: 5 36 146 76 21 79 21 4 17 82: 5 39 158 82 22 83 22 6 16 86: 5 41 166 86 23 89 23 8 15 92: 4 44 178 92 24 97 24 4 20 100: 6 48 194 100 25 101 25 2 23 104: 5 50 202 104 26 103 26 4 22 106: 6 51 206 106 27 107 27 2 25 110: 6 53 214 110 28 109 28 4 24 112: 7 54 218 112 29 113 29 14 15 116: 6 56 226 116 30 127 30 4 26 130: 7 63 254 130 What about the Gap = 14 going to the Trail 127, EVEN = 130? Overlapping Trails 71, 83, 89, 101,107 and 113 combine with the 127 to account for the 7 PPsets that will form EVEN = 130.
Of course to account for all the EVENS between Trail 113 and 127 — EVENS 116 to 130, we have Trails 59 to and including 127 in various combinations to account for that. We can easily see this on Table 46, lines 1630, where each has an E_{e} (118–254) that meets or exceeds those EVENS (116–130). Altogether, there are some 80 PPsets that account for these 15 EVENS 116130. Easily done.
What if you want to know specifically who covers, say EVEN 128? Looking at the full table, line 18, we see that Trail 67 and above have E_{e} > 128. How can we determine that the 19+109, 31+97 and 61+67 PPsets are the ones we are looking for?
Without even knowing the values for the Column 6 (Table 46): EVEN, ∑# of PPsets, one can:
look at an expanded PTOP
look at an expanded BIM
calculate using this trick:
on Table 46, find where EVEN 128 would be located between line 17 and 18 as 1/2 of 128 = 64
see that PRIMES 61 & 67 on either side = 128 when added together and both are PRIMES
knowing from the BIM that ALL PPsets are symmetrical about the center Axis core that points to the 90° Rangled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 64 slot like the 61 & 67
the next pair out would be 31 & 97, followed by 19 & 109
this is easily calculated as the core Axis value ± steps away:
 64 ± 3 = 61 and 67
 64 ± 33 = 31 and 97
 64 ± 45 = 19 and 109
 where the steps away are literally the number of steps along the Axis to either side of the core Axis value that goes to a PPset
their ∑s equal the EVEN 128:
61 + 67 = 128
31 + 97 = 128
19 + 109 = 128.
We can confirm this on the PTOP (Table 35), and, of course, directly on the BIM. Tables 43 and 44 can also confirm ALL EVENS up to 2000. One can also use an online calculator to get the results.
it can be quite helpful to follow these examples directly on the BIM. The visualizations of how the symmetrical PPsets contributed from the separate, individual PRIME Trails line up their 90° Rangled isosceles triangles over the central core with each of their respective apexes inline and pointing towards the EVEN that they are forming. On the Axis, one can easily see the symmetrical steps from the core to each PPset.
(See the PDF versions for clear details.)
What if you want to know specifically who covers, say EVEN 126? Looking at the full table, line 18, we see that Trail 67 and above have E_{e} > 126. How can we determine that the
13+113
17+109
19+107
23+103
29+97
37+89
43+83
47+79
53+73
59+67
PPsets are the ones we are looking for?
Without even knowing the values for the Column 6 (Table 46): EVEN, ∑# of PPsets, one can:
look at an expanded PTOP
look at an expanded BIM
calculate using this trick:
on Table 46, find where EVEN 128 would be located between line 17 and 18 as 1/2 of 126 = 63
see that PRIMES 59 & 67 on either side = 126 when added together and both are PRIMES
knowing from the BIM that ALL PPsets are symmetrical about the center Axis core that points to the 90° Rangled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 63 slot like the 59 & 67
the next pair out would be 53 & 73, followed by 47 & 79.
this is easily calculated as the core Axis value ± steps away:
63 ± 4 = 59 and 67
63 ± 10 = 53 and 73
63 ± 16 = 47 and 79
63 ± 20 = 43 and 83
63 ± 26 = 37 and 89
63 ± 34 = 29 and 97
63 ± 40 = 23 and 103
63 ± 44 = 19 and 107
63 ± 46 = 17 and 109
63 ± 50 = 13 and 113
their ∑s equal the EVEN 126:
 59 + 67 = 126
 53 + 73 = 126
 47 + 79 = 126
 43 + 83 = 126
 37 + 89 = 126
 29 + 97 = 126
 23 + 103 = 126
 19 + 107 = 126
 17 + 109 = 126
 13 + 113 = 126
 The key is seeing that the PPsets are symmetrical about the core Axis value of the EVEN and applying these sequential, linear # of steps out from the core in identifying the PPsets that form that EVEN! EVERY PPset that forms an EVEN is part of a symmetrical pair of PPs, that together forms the EVEN. This is a critical and paramount finding in the behavior of the PRIMES and in the PROOF of the strong form of the Goldbach Conjecture.
 The ONLY EVENS (>4) with less than 2 PPsets are the EVENS 6, 8 and 12. EVEN 6 has only 1 PPset of 3,3 and EVEN 8 has only 1 PPset of 3,5. The EVEN 8 is the beginning of the 3 + 5 Trail, while the EVEN 6 is BOTH the start and end of the 3 + 3 Trail that begins the PTOP. The EVEN 12 is the one isolated example where the middle of a PRIME Trail — here the 3 + 7 Trail — is the only PPset to form EVEN 12, as the 3 + 5 Trail is — being the early start of Trails — not yet sufficient enough in length to cover the EVEN 12 slot. All 3 EVENS — 6,8,12 — are the exceptions precisely because they are the very beginning of the PTOP where the PRIME Trails are first being established.
 So we are claiming here — and giving as proof — that ALL EVENS ≥ 14 (and including EVEN 10), have a MINIMUM OF 2 PPsets. It is within the geometry of the number distribution of simple, natural, Whole Integer Numbers (WIN) that ALL EVEN WINs shall have their EVEN/2, Axis core value be symmetrically flanked on either side by ODD WINs, and, as we have shown here on the PTOP and the PTOP as hidden within the BIM, will have two (2) or more ODD PPs pairs within that symmetrical flanking that will, as PPsets, form said EVEN.
 The symmetry simply falls out of the inherent interconnection between the number values of each cell within the BIM. Every 1^{st} Parallel EVEN has a straight line path back to its Axis and along that path the cell values must reflect the product of the two Axis values located symmetrically from the core Axis value outward, such that at their 90° Rangle, isosceles intersection on that path, one finds that product.
 The real wonder — and part of this amazing new finding — is that when such Axis numbers are found that are PRIME, they ALWAYS have a symmetrical ODD # counterpart on the other side of their common, core Axis number: and at the minimum, two or more candidates will have their ODDs be PRIMES, thus completing two or more PPsets!
 To remember, the Axis core number may itself be either EVEN or ODD as it is simply 1/2 of the EVEN that the path points to from its 1^{st} Parallel Diagonal location. Also, the above rule does NOT mean that ALL ODD #s found on the Axis are part of the PPsets that form the EVEN. Even though symmetrical, and forming the same products on the path, BOTH PPset MEMBERS FORMING THE PAIR MUST BE PRIMES, e.i. an Axis 9, 15, 21,.. are NOT PRIME and NOT part of a PPset forming the EVEN. The geometry defines the algebra = algebraic geometry.
 The growth rate of the staggered, zigzagging diagonal Trails of the PS ensures that the number of overlapping Trails will always exceed the growth rate of the PRIME Gaps by a sizable margin.
Towards the bottom of Table 46, lines 1092  1107, there is a good run of PRIME Gaps in close proximity. Below is the table in plain form with the same column headings:
line # PRIMES, P**2≥3** ∑# of 3+P**2PPsets/Trail** Prime Gap ∆ **TrailGap** EVEN: ∑# of**PPsets** EC=# of EVENs covered E**e = Ending EVEN covered** EVEN
1080 8681 1080 8 1072 94 4340 17362 8684 1081 8689 1081 4 1077 100 4344 17378 8692 1082 8693 1082 6 1076 89 4346 17386 8696 1083 8699 1083 8 1075 96 4349 17398 8702 1084 8707 1084 6 1078 132 4353 17414 8710 1085 8713 1085 6 1079 92 4356 17426 8716 1086 8719 1086 12 1074 112 4359 17438 8722 1087 8731 1087 6 1081 101 4365 17462 8734 1088 8737 1088 4 1084 136 4368 17474 8740 1089 8741 1089 6 1083 93 4370 17482 8744 1090 8747 1090 6 1084 138 4373 17494 8750 1091 8753 1091 8 1083 104 4376 17506 8756 1092 8761 1092 18 1074 111 4380 17522 8764 1093 8779 1093 4 1089 91 4389 17558 8782 1094 8783 1094 20 1074 95 4391 17566 8786 1095 8803 1095 4 1091 125 4401 17606 8806 1096 8807 1096 12 1084 124 4403 17614 8810 1097 8819 1097 2 1095 102 4409 17638 8822 1098 8821 1098 10 1088 90 4410 17642 8824 1099 8831 1099 6 1093 93 4415 17662 8834 1100 8837 1100 2 1098 141 4418 17674 8840 1101 8839 1101 10 1091 93 4419 17678 8842 1102 8849 1102 12 1090 91 4424 17698 8852 1103 8861 1103 2 1101 93 4430 17722 8864 1104 8863 1104 4 1100 118 4431 17726 8866 1105 8867 1105 20 1085 121 4433 17734 8870 1106 8887 1106 6 1100 148 4443 17774 8890 1107 8893 1107 30 1077 94 4446 17786 8896 1108 8923 1108 6 1102 95 4461 17846 8926 1109 8929 1109 4 1105 125 4464 17858 8932 1110 8933 1110 8 1102 95 4466 17866 8936 1111 8941 1111 10 1101 101 4470 17882 8944 1112 8951 1112 12 1100 104 4475 17902 8954 1113 8963 1113 6 1107 96 4481 17926 8966 1114 8969 1114 2 1112 84 4484 17938 8972 1115 8971 1115 28 1087 104 4485 17942 8974 1116 8999 1116 2 1114 110 4499 17998 9002 1117 9001 1117 6 1111 95 4500 18002 9004 1118 9007 1118 4 1114 133 4503 18014 9010 1119 9011 1119 2 1117 96 4505 18022 9014 1120 9013 1120 16 1104 119 4506 18026 9016 Without even knowing the values for the Column 6 (Table 46) which have been inserted here: EVEN, ∑# of PPsets, one can take EVEN 8872:
look at an expanded PTOP
look at an expanded BIM
calculate using this trick: (see *below for a simple Universal calculation method as shown in Table 45: Equations.)
on Table 46, find where EVEN 8872 would be located as the core Axis value between lines 601 and 602 as 1/2 of 8872 = 4436
see that PRIMES 4423 & 4441 on either side = 8864 when added together and both are PRIMES we are close
knowing from the BIM that ALL PPsets are symmetrical about the center core that points to the 90° Rangled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 4436 slot like the 93 PPsets that equal EVEN 8872:
 5+8867
 11+8861
 23+8849
 41+8831
 53+8819
 89+8783
 131+8741
 173+8699
 179+8693
 191+8681
 263+8609
 359+8513
 443+8429
 449+8423
 503+8369
 509+8363
 599+8273
 641+8231
 653+8219
 701+8171
 761+8111
 863+8009
 953+7919
 971+7901
 1019+7853
 1031+7841
 1049+7823
 1181+7691
 1223+7649
 1229+7643
 1283+7589
 1289+7583
 1373+7499
 1439+7433
 1523+7349
 1619+7253
 1721+7151
 1871+7001
 1889+6983
 1901+6971
 1913+6959
 1973+6899
 2003+6869
 2039+6833
 2069+6803
 2081+6791
 2111+6761
 2153+6719
 2213+6659
 2273+6599
 2309+6563
 2351+6521
 2381+6491
 2399+6473
 2423+6449
 2543+6329
 2549+6323
 2609+6263
 2699+6173
 2729+6143
 2741+6131
 2819+6053
 2843+6029
 2861+6011
 2969+5903
 3011+5861
 3023+5849
 3089+5783
 3203+5669
 3221+5651
 3299+5573
 3371+5501
 3389+5483
 3491+5381
 3539+5333
 3593+5279
 3701+5171
 3719+5153
 3821+5051
 3833+5039
 3851+5021
 3863+5009
 3929+4943
 4001+4871
 4073+4799
 4079+4793
 4139+4733
 4229+4643
 4289+4583
 4349+4523
 4391+4481
 4409+4463
 4421+4451
as before, these can be simplified as:
 4436 ± 5 = 4431 and 4441. Together, 4431 + 4441 = 8872. See Table 48 for the complete list.
 4436 ± 11
 4436 ± 23
 4436 ± 41
 4436 ± 53
 4436 ± 89
 4436 ± 131
 4436 ± 173
 4436 ± 179
 4436 ± 191
 4436 ± 263
 4436 ± 359
 4436 ± 443
 4436 ± 449
 4436 ± 503
 4436 ± 509
 4436 ± 599
 4436 ± 641
 4436 ± 653
 4436 ± 701
 4436 ± 761
 4436 ± 863
 4436 ± 953
 4436 ± 1019
 4436 ± 1031
 4436 ± 1049
 4436 ± 1181
 4436 ± 1223
 4436 ± 1229
 4436 ± 1283
 4436 ± 1289
 4436 ± 1373
 4436 ± 1439
 4436 ± 1523
 4436 ± 1619
 4436 ± 1721
 4436 ± 1871
 4436 ± 1889
 4436 ± 1901
 4436 ± 1913
 4436 ± 1973
 4436 ± 2003
 4436 ± 2039
 4436 ± 2069
 4436 ± 2081
 4436 ± 2111
 4436 ± 2153
 4436 ± 2213
 4436 ± 2273
 4436 ± 2309
 4436 ± 2351
 4436 ± 2381
 4436 ± 2399
 4436 ± 2423
 4436 ± 2543
 4436 ± 2549
 4436 ± 2609
 4436 ± 2699
 4436 ± 2729
 4436 ± 2741
 4436 ± 2819
 4436 ± 2843
 4436 ± 2861
 4436 ± 2969
 4436 ± 3011
 4436 ± 3023
 4436 ± 3089
 4436 ± 3203
 4436 ± 3221
 4436 ± 3299
 4436 ± 3371
 4436 ± 3389
 4436 ± 3491
 4436 ± 3539
 4436 ± 3593
 4436 ± 3701
 4436 ± 3719
 4436 ± 3821
 4436 ± 3833
 4436 ± 3851
 4436 ± 3863
 4436 ± 3929
 4436 ± 4001
 4436 ± 4073
 4436 ± 4079
 4436 ± 4139
 4436 ± 4229
 4436 ± 4289
 4436 ± 4349
 4436 ± 4391
 4436 ± 4409
 4436 ± 4421

We can further simplify by applying these Universal (vs Specific, see Table 45:) equations:
 By definition, EVEN = P_{1} + P_{2} = PPset
let S = steps, E = EVEN = 2(core Axis value) = 2(A_{x}), as E/2 = A_{x}
S = P_{2}  E/2 = P_{2}  A_{x}
the PPset for a given EVEN:
P_{2} = S + E/2 = S + A_{x}
P_{1} = P_{2}  (2S)
2S = P_{2}  P_{1}
S = A_{X}  P_{1}
E = P_{1} + P_{2}
Example: 3,5 PPset for EVEN = 8: S = P_{2}  E/2 = P_{2}  A_{x} 1 = 5 4
P_{2} = S + E/2 = S + A_{x}
5 = 1 + 4
P_{1} = P_{2}  (2S) 3 = 5  (2*1)
And, as P_{1} = E  P_{2}, or E = P_{1} + P_{2}
P_{1} = P_{2}  (2S) = E  P_{2}
2P_{2}  E = 2S 2(5)  8 = 2(1) S = 1
As S = (2P_{2}  E)/2
S = (2P_{2}  E)/2 = P_{2}  E/2
2P_{2}/2  E/2 = P_{2}  E/2
P_{2}  E/2 = P_{2}  E/2 = S
P_{2}  A_{x} = P_{2}  A_{x} = S
The builtin symmetry of the PPsets around the core Axis value is easily calculated as these four examples show:
EVEN 24 with core Axis value (A_{x}) = 24 / 2 = 12 with 3 PPsets:
P_{1} = P_{2}  (2S)
as S = P_{2}  E/2 = P_{2}  A_{x}
11,13 1312 = 1 13  (2x1) = 11
7,17 1712 = 5 17  (2x5) = 7
5,19 1912 = 7 19  (2x7) = 5
EVEN 26 with core Axis value = 26 / 2 = 13 with 3 PPsets:
P_{1} = P_{2}  (2S)
as S = P_{2}  E/2 = P_{2}  A_{x}
13,13 1313 = 0 13  (2x0) = 13
7,19 1913 = 6 19  (2x6) = 7
3,23 2313 = 10 23  (2x10) = 3
EVEN 22 with core Axis value = 22/2 = 11 with 3 PPsets:
P_{1} = P_{2}  (2S)
as S = P_{2}  E/2 = P_{2}  A_{x}
11,11 1111 = 0 11  (2x0) = 11
5,17 1711 = 6 17  (2x6) = 5
3,19 1911 = 8 19  (2x8) = 3
EVEN 100 with core Axis value = 100 / 2 = 50 with 6 PPsets:
P_{1} = P_{2}  (2S)
as S = P_{2}  E/2 = P_{2}  A_{x}
47,53 5350 = 3 53  (2x3) = 47
43,57 5750 = 7 57  (2x7) = 43
41,59 5950 = 9 59  (2x9) = 41
29,71 7150 = 21 71  (2x21) = 29
17,83 8350 = 33 83  (2x33) = 17
11,89 8950 = 39 89  (2x39) = 11
~
~~~~~~_{~} ⁃As one moves successively along the EVENS, the Trails get longer and longer, adding one new member for each successive PS (See Tables 4446.). This rate of increase far exceeds the size and incidence rate of the PRIME Gaps (See Tables 42 and 46.), ensuring that for every EVEN ≥6, there is at least one PPset of ODD PRIMES that will form it. Actually, as we have shown, there are always a minimum of 2 sets of PPsets that form the EVENS (≥14, including 10).
Table 44 shows how ∑s of the number of PPsets/ EVEN grows. It increases such that for every 60 successive EVENS on average, the ∑ increases by 3 (as calculated from 41080).
For example:
Columns generally have 60 entries ending in multiples of 120.
∑of PPsets per column Ave/Col
Col A 262 / 59 = 4.4 = 4
Col B 516 / 60 = 8.6 = 9
Col C 718 / 60 = 12 = 12
Col D 930 / 60 = 15.5 = 16
Col E 1076 / 60 = 17.9 = 18
Col F 1267 / 60 = 21.1 = 21
Col G 1302 / 60 = 21.7 = 22
Col H 1534 / 60 = 25.6 = 26
Col I 1687 / 60 = 28.1 = 28
SUMMARY and CONCLUSION
It is worth repeating here from above:
 So we are claiming here — and giving as proof — that ALL EVENS ≥ 14 (and including EVEN 10), have a MINIMUM OF 2 PPsets. It is within the geometry of the number distribution of simple, natural, Whole Integer Numbers (WIN) that ALL EVEN WINs shall have their EVEN/2, Axis core value be symmetrically