Decomposition into weight × level + jump · a pedagogical report
The Weight–Level–Jump Primer
What the construction is, why it is canonical, what has been proved in it, what remains conjectural, how to read its pictures — and what it could and could not change in number theory. Written for a graduate student or researcher meeting the construction for the first time.
Session of 3 September 2026 · framework: Rémi Eismann, Decomposition into weight × level + jump and application to a new classification of primes, arXiv:0711.0865 · atlas: decompwlj.com
Reference text: the pedagogical treatise, tenth edition (19 August 2026), whose numbering of lemmas, propositions and theorems this report keeps so that the two can be read side by side.
Every number on this page was recomputed from an empty directory this session: a C census engine (primes and two random controls to , 11.0 s on one core), a Python divisor audit sharing no code with it, PARI/GP 2.15.4 running the four project kernels and a fifth that transcribes the founding definition, and a streaming factoriser for fourteen further OEIS sequences.
5,761,452
decomposable primes below — every prime but 2, 3 and 7
1,078,707
level-classified, a share of 0.1872 — falling in every decade
440,311
primes of weight 3 = lesser twin primes below , to the unit (A007508)
0
violations of any elementary statement of Part II, over the whole census
Before startingHow to read this report
The construction takes a strictly increasing sequence of integers and writes each term as weight × level + jump, with nothing chosen and nothing fitted. On the natural numbers it is the sieve of Eratosthenes. On the primes it produces a two-class partition whose statistics can be proved, predicted and — the part this report insists on — tested against a random control.
Nothing beyond divisibility is needed for Parts I and II, and every proof there is complete on the page. Part III states the two analytic results, explains how their proofs work, and gives the proofs in full in Appendix B. Part IV is the reader's guide to the figures. Part V is the ledger — proved, conditional, heuristic, open, and (the column usually missing) classical in disguise. Part VI is the algorithms. Part VII, the closing section, is the speculative one: what the framework could change in number theory if its open problems fell, labelled by plausibility and paired with what it provably cannot change.
The two-hour route§1–§4, then §5 with the proofs open, then the figures of §11. That is the whole construction and everything certain about it.
The analytic route§6 and §7 with Appendix B open beside them; then §8–§10 to see what the constants mean and what the control takes away.
The sceptic's route§10 and §12 first — they say what the framework is not; then §14 to see what a fair reader could still hope for.
The visual route§3, §4 and §11: every picture explained, feature by feature, with the theorem that produces it.
Claims are tagged.proved a complete proof appears here or is cited to a published source. proved* the argument has passed internal audit across editions of the project's analysis but has not been refereed. conditional names its hypotheses. heuristic a derivation with a step that is not justified. open open. verified attaches to numbers, not to theorems: recomputed this session by at least two engines that share no code. classical marks a statement that is a known hard problem in disguise; native one that cannot be formulated without these coordinates.
Numbering. Lemmas 1–5, Propositions 1–3, Theorems 1–2 and Corollary 1 carry the numbers they have in the tenth-edition treatise. Statements that are new to this report, or that the treatise leaves implicit, carry letters: Propositions A, B and C. The founding paper's conjectures are cited as Conjecture 1–9 of that paper and located in the ledger of §12.
Notation. Throughout, is the -th term of a strictly increasing integer sequence, its jump, , the weight and the level. On the primes , the jump is the prime gap , and . Appendix D lists every symbol.
Part I · §1The definition, led by the founding form
The weight was first written down, in the OEIS entry A117078, not as a divisor but as a remainder: is the smallest such that . That is the form to lead with, because it derives the one inequality that carries the whole construction instead of imposing it. The divisor form, which every later statement uses, follows from it in three lines.
Definition 1weight × level + jump, in the founding form
Let be a strictly increasing sequence of positive integers, and let be its jump. Write for the Euclidean remainder, the integer in . The admissible set of the -th term is
If , the weight is , the level is the quotient , and by the definition of Euclidean division
If we set and call not decomposable.
A decomposable term is level-classified if and weight-classified if ; ties count as weight.
Read the definition dynamically and it says: the sequence advances by its own residue, , in the smallest modulus for which that step is legal. The decomposition is this recurrence rearranged. What makes it a piece of mathematics rather than bookkeeping is that the modulus is forced.
Proposition Athe founding form and the divisor form are the same definitionproved
Put . For every integer ,
Consequently , which is non-empty if and only if ; when it is, is its largest element, the weight is the least divisor of exceeding , and is an integer.
Proof
means for an integer with ; that is, and . Conversely, if and , then with is the Euclidean division of by , so . The set is non-empty exactly when its largest candidate qualifies, i.e. when ; and then the minimum is the least divisor of exceeding . ∎
So the construction, in the form used from here on, reads:
Definition 1, divisor form
; if , otherwise ; = the smallest divisor of that exceeds , if , otherwise ; if , otherwise . Then whenever .
Three remarks fix the mechanics before any theorem.
The weight exists and is unique. The set of divisors of exceeding is non-empty whenever , because divides itself; so is a minimum over a non-empty finite set, is an integer, and is an identity, not an approximation. Minimality pins : there is no window, threshold or cut-off anywhere in the definition. This is what makes the objects invariants of the sequence.
Why the minimum, and why a remainder. Proposition A says always, and it is the largest element; a "largest admissible " would return , for every term and carry no information. The minimum is the only non-degenerate canonical choice — the tightest modulus in which is still a legal remainder. And if the founding line were read as a congruence rather than a remainder function, the constraint would vanish, would solve it for every , and the weight would be identically . The remainder carries the inequality for free, and that inequality is the entire construction.
The decomposition sees only two consecutive terms. and depend on and and on nothing else. That locality is the source of both the construction's tractability and its central difficulty: on the primes, is the next prime, and "next" is a global condition wearing local clothes. §6 turns on the moment when it can be dropped.
Two worked examples, by hand
Take . The next prime is , so and . The divisors of are ; those exceeding are and ; so , , and . Since the prime is level-classified. Figure 1 shows the same computation in the founding form: the sawtooth meets the line exactly at the admissible moduli.
Figure 1. The founding definition as a picture. The graph of . On the branch where the map is the line of slope ; it takes the value at , an integer exactly when . So the admissible are the lattice points of the hyperbola to the right of the cut , the level is the branch index, and taking the smallest is taking the highest branch. The grey band can never qualify, because there .
Now take two primes of the same size that fall on opposite sides of the classification: (next prime , , ) and (next prime , , ). For the first, the divisors of exceeding begin with , so and : weight-classified. For the second, nothing divides between and , so the weight is forced all the way up to : , level-classified. Figure 2 draws both on a logarithmic divisor axis, and it is the picture to keep: the weight is the first divisor to the right of the band , and the whole classification is whether that divisor lands inside the window or jumps clean over it.
Figure 2. The whole construction, twice. Top: ; the divisor of lands inside the window and the term is weight-classified. Bottom: ; the window is empty of divisors of , the weight is forced up to , and the term is level-classified. Whether the first dot right of the grey band falls inside the coloured window is the entire classification (§3).
Table 1 is the same computation for the first seventeen primes. It is finger-work, and it is worth doing two or three rows with a pencil before going on. The rows are Table 2 of the founding paper; all five algorithms of §13 reproduce them exactly.
Table 1. The decomposition of the first primes. is the smallest divisor of exceeding ; the identity is checked in the last column. Rows with are not decomposable. verified
divisors of
class
2
3
1
—
—
0
0
—
—
3
5
2
—
—
0
0
—
—
5
7
2
3
1, 3
3
1
level
5
7
11
4
—
—
0
0
—
—
11
13
2
9
1, 3, 9
3
3
weight (tie)
11
13
17
4
9
1, 3, 9
9
1
level
13
17
19
2
15
1, 3, 5, 15
3
5
weight
17
19
23
4
15
1, 3, 5, 15
5
3
level
19
23
29
6
17
1, 17
17
1
level
23
29
31
2
27
1, 3, 9, 27
3
9
weight
29
31
37
6
25
1, 5, 25
25
1
level
31
37
41
4
33
1, 3, 11, 33
11
3
level
37
41
43
2
39
1, 3, 13, 39
3
13
weight
41
43
47
4
39
1, 3, 13, 39
13
3
level
43
47
53
6
41
1, 41
41
1
level
47
53
59
6
47
1, 47
47
1
level
53
59
61
2
57
1, 3, 19, 57
3
19
weight
59
Three things are visible already, and all three become theorems. Every row with has — Lemma 4, which says that the lesser member of a twin pair is exactly a prime of weight . The rows with are exactly the rows where is prime (), plus two rows, and , where is a prime square — the near-equivalence of Lemma 5, which drives §7. And the rows and show that the classification depends on only through the cut: and have the same and the same divisor pair ; moving the jump from to moves the cut past , the weight climbs from to , and the same two numbers swap roles. The weight of is the level of .
Part I · §2When the decomposition exists
Lemma 2existence criterionproved
is decomposable if and only if . For the primes, the non-decomposable terms are exactly , and .
Proof
By Proposition A, is decomposable iff , i.e. , i.e. . For the primes apply Nagura's theorem: for the interval contains a prime, so once . Below one checks by hand: , and fail (), while pass. ∎
So on the primes the construction is defined everywhere except on a set of size three, and "for all decomposable primes" means "for all primes but ". The boundary is easy to misremember: is decomposable (, giving ), while is not (). The set , which has appeared in project notes, is an error; the census confirms at every decade, and turns out to be the unique level-classified member of the twin column (§5.2). verified
The barrier is a real restriction on the domain, not a technicality. On the decomposition is defined nowhere. On a polynomial sequence of degree it is defined from some index on, because . One degenerate branch must be excluded explicitly for arbitrary sequences: if then for every , so the founding form alone would return with ; the convention removes it. On the primes it never occurs (Bertrand).
Part I · §3The classification plane
Every decomposable term yields a point of the positive quadrant lying on the hyperbola . The diagonal cuts the quadrant in two, and the two halves are the two classes. That is the whole of the classification, and it is worth saying plainly what it measures.
What the classification is really testing
is level-classified precisely when has no divisor in the window . Indeed , and is the smallest divisor exceeding ; so says exactly that contains no divisor of . The tie is , a divisor exactly at the right end of the window.
The classification is therefore a statement in the theory of divisors in short intervals — Erdős's multiplication-table problem, Ford's , Tenenbaum's -function — evaluated at the arithmetically special argument , with the window's left endpoint tied to the prime's own gap. That is the correct address of this subject, and §14 returns to it.
Plotting on a square canvas with equal aspect makes the ratio readable directly as distance from the diagonal, which is why every figure in this report carrying both coordinates is square. Figure 3 is the entire census of the primes below in that plane; §11 reads it feature by feature. For now, three structures: above the diagonal, a weight sheet striated into vertical columns; below it, a level class that resolves into horizontal lines; and a dotted anti-diagonal that is not a guide for the eye but the exact boundary , beyond which no point can lie.
Figure 3. The classification plane of the primes: all 5,761,452 decomposable . Square canvas, equal aspect, so the ratio is faithful; shading is the number of primes per cell on a logarithmic scale, blue for the weight class and orange for the level class. Each of the three visible structures is a theorem or a section of this report: the columns are §9.1, the leftmost column is the twin primes (Lemma 4), the level lines are odd only (Lemma 3), and the lowest line is Species I, carrying 339,870 primes. verified
Part I · §4The base case is the sieve of Eratosthenes
Before the primes, run the machine on the natural numbers. What comes out is not an analogy with the sieve of Eratosthenes; it is the sieve of Eratosthenes. This is the single fact that makes the construction more than an arbitrary bookkeeping scheme.
Proposition 1Eratosthenes collapseproved
For one has , , and for all
where is the smallest prime factor. Moreover is level-classified if and only if is prime, and then ; and is a tie, , if and only if is the square of a prime.
Proof
Here and . The smallest divisor of exceeding is its smallest prime factor, so . If is prime then and : level-classified. If is composite, write with ; every prime factor of is , so and the term is weight-classified, with equality exactly when . ∎
In the founding form the statement is even shorter: , so the equation is , and the weight of is the smallest modulus in which leaves remainder — the -shifted analogue of the sieve's own test . Eratosthenes tests ; the decomposition tests with supplied by the sequence itself. Once the least admissible need not be prime ( all occur on the primes), because the cut shadows the small prime factors of ; this is the mechanism behind the non-monotone column profile of §9.
The dictionary, entry zero
On : the weight columns are the residue classes struck at stage of the sieve of Eratosthenes; the level class is the shifted primes , the numbers the sieve never strikes; the ties are , exactly where the sieve begins striking at stage ; and iterating the construction on the level — decomposing in turn — recovers the complete factorisation of , smallest prime factor first.
So the weight is a generalised smallest prime factor and the level class a generalised set of primes, for an arbitrary increasing sequence. That is the claim the framework stakes, and Proposition 1 is the only place where it is a theorem rather than an analogy. Figure 4 is the statement drawn, and Figure 5 puts the base case and the primes on identical axes — the most useful single image in this report, because everything that is hard about the primes is visible in the difference between the two panels. verified at : 9,999,997 decomposable , of which 664,578 are level-classified (every one on the line ), 446 are ties, and the columns hold 4,999,998, 1,666,667, 666,667 and 380,953 terms — the densities of the sieve.
Figure 4. Proposition 1 as a picture. The integers in rows of , coloured by weight. Because , the colour depends only on for the first three stages: the dark cells are the odd (struck at stage 2), the next shade is what remains of , and so on. What survives uncoloured — the white cells — are precisely the with prime, confined to the eight columns coprime to , shifted by one. The sieve of Eratosthenes and the weight–level classification of are the same picture.
Figure 5. The same construction on two sequences, both , identical square axes. Left, : the weight sheet is an arrowhead of columns and the region strictly below the diagonal is provably empty except for the single line , because forces every composite onto or above the diagonal. The whole level class of is that one line, and it is the shifted primes. Right, : the same columns survive (odd ones only), but the level class has opened into a family of lines occupying a two-dimensional region. The reason is exactly that is now the least divisor exceeding rather than exceeding : the window has become wide enough to be empty sometimes. Part II is the study of how often. verified
What the base case does and does not license
It is easy to overclaim here, so the strength of Proposition 1 should be stated exactly. What is true: the construction restricted to is a faithful re-encoding of trial division, and the classification is a primality test for . What is not true, and is nowhere claimed, is that the construction applied to another sequence inherits any of the sieve's power. On the primes the analogous statement — "level-classified means is prime" — is only almost true (Lemma 5, with a two-element exception set), and the almost is doing real work. The honest summary is that Proposition 1 identifies the construction's type: and are multiplicative data attached to an additive quantity, and the level/weight split is the smallest-divisor-in-a-window question. Everything downstream is an instance of that type at a harder argument.
Part II · §5The primes: five rigidity results and a census to
Everything in this section is elementary, complete and proved. Taken together it is the framework's most secure asset: a small set of rigidity results that constrain the level class so tightly that the analytic work of Part III becomes possible at all.
Throughout, is a decomposable prime, , (an object with its own OEIS entry, A118534), and . Two consequences of the formula for are visible before any computation. First, , and are three terms of an arithmetic progression with common difference : the decomposition is secretly about a triple, and §7 is the study of that triple. Second:
Proposition Cparityproved
For every decomposable prime , the gap is even, is odd, and therefore every divisor of — in particular the weight and the level — is odd.
Proof
and are odd, so is even and is odd. ∎
Trivial as it is, Proposition C is used constantly and silently below, and it is what makes the level class a comb of odd lines and the weight sheet a fence of odd columns in Figure 3 (66 occupied values of and 872,436 of below ).
5.1 Divisor localisation and the level bound
Lemma 3divisor localisation; level boundproved
Let be decomposable and level-classified. Then
no divisor of lies in the open interval — equivalently, every divisor of smaller than is at most ;
;
.
Proof
(i) is the minimality of restated: a divisor in would exceed and be smaller than , contradicting the definition. (ii): level classification is , and , so . (iii): divides and , so by (i) . Since is odd (Proposition C), is odd, while is even; so and therefore . ∎
The parity step in (iii) is worth pausing on. Without it one gets only , and is genuinely attained on general sequences — the witness , gives , , , . The sharpened bound is a fact about the primes specifically, and it is what makes the exponent bookkeeping in Theorem 1 come out.
verified over all 1,078,707 level-classified primes : zero violations. The largest level attained anywhere below is , at , whose gap is — so the bound is met there with 48 units to spare. There : its divisors below the gap are , and the next one is the prime . That prime is also the last entry of the maximal-gap ladder below (A002386): the record level sits at a record gap, as the bound suggests it should. Figure 6 is the lemma made visible.
Figure 6. Lemma 3, drawn. The joint occupancy of over all 1,078,707 level-classified primes below , on a square canvas with equal aspect so the wedge's slope reads as . Every cell above the line is empty — not sparse, empty. The inset resolves the small corner, where the parity of the bound shows as a checkerboard: only odd occur, only even . The single point at the top right is the record at . verified
5.2 The twin column
Lemma 4the twin columnproved
For a decomposable prime : . That is, a prime has weight precisely when it is the lesser member of a twin pair.
Proof
() Suppose , so and are both prime and . Among the three integers — consecutive in the arithmetic progression of common difference — exactly one is divisible by , because they occupy the three residue classes mod ( is invertible mod ). Both and are primes exceeding , hence not divisible by ; so . Since , minimality gives .
() If then by definition of the weight, and is even and positive, so . ∎
In the founding form the lemma is one line: means with ; for the gap is even, so and — the lesser of a twin pair.
verified to : , with zero one-sided cases in either direction. The column contains exactly one level-classified member: forces , hence and . Every other member is weight-classified, so the column splits as .
An external anchor
Adding back the one twin-pair lesser member that is not decomposable — — gives twin pairs with lesser member below . The census returns at : exactly the published terms of OEIS A007508, eight decades to the unit, by a route that never mentions twins. verified
A warning that governs the whole report
Lemma 4 makes the twin-prime conjecture equivalent to the statement "the weight column is infinite". This is a translation, not a reduction. Nothing in the weight–level coordinates gives purchase on that column that was not already available in the language of gaps. The same applies to every fixed weight column, every fixed level line, and their Polignac-type analogues. We call this the difficulty-conservation principle and invoke it, explicitly, wherever a reformulation might be mistaken for progress.
5.3 Mod-3 rigidity
Proposition 2mod-3 rigidity (the founding paper's Conjectures 7 and 8)proved
For every decomposable prime : .
Proof
Since , reducing mod gives . If then , so , whence . If then, being even, , so . Both and are primes exceeding , hence neither is . Were , we would get , impossible. Therefore , i.e. . ∎
verified: zero violations over all 5,761,452 decomposable primes below . Proposition 2 settles Conjectures 7 and 8 of the founding paper. It should be stated, as here, as an elementary observation and never presented as a contribution: it is a two-line congruence argument, and the OEIS entry records it struck through for exactly that reason. Its real use is downstream: combined with Lemma 5 it forces for every level-one prime except and , which is why the singular series of §7 is supported on multiples of , and why the constant that emerges there is the twin-prime constant rather than something else.
5.4 The cube bound for composite weights
Proposition 3cube lemma (the founding paper's Conjecture 4)proved
If is level-classified with composite weight, then
with equality if and only if (, , , ). Moreover the level-classified primes with composite weight are exactly , with weights , for all .
Proof
Write with ; then and are proper divisors of , hence divisors of smaller than . By Lemma 3(i) they are at most ; since is odd its divisors are odd, while is even, so in fact . Therefore , and with from Lemma 3(iii), . (When one gets the stronger .) Equality forces , so with an odd prime power; the census finds the unique instance .
For the exhaustive list, the argument has to be run in two ranges, and the census of this report is exactly deep enough. Such a prime satisfies . If then , so the prime lies inside the census, which finds exactly the five listed. If , the exhaustive gap tables of Oliveira e Silva, Herzog and Pardi (every gap below is at most ) give ; but the first prime gap of size or more occurs at (A002386), so every prime below has gap at most — a contradiction. ∎
The closure argument printed in the tenth-edition treatise (" for every ") is not correct as stated — the first prime with gap is — and the two-range argument above, first given in the project's illustrated edition, replaces it; the conclusion is unchanged. verified to by two independent engines: the composite-weight level set is and there are zero violations of the cube bound. Of the 1,078,707 level-classified primes below , therefore, 1,078,702 have a prime weight and five do not. The closure to is carried from the refereed gap tables, not recomputed here.
5.5 The level-one reduction
Lemma 5level-one reduction and exception setsproved
Let be a decomposable prime.
If , then if and only if is prime.
The decomposable primes with (the "empty window") are exactly , for all .
The level-one primes with composite are exactly (with ), for all .
Consequently, for , .
Proof
(i) means , i.e. has no divisor in the open interval . If is prime this holds vacuously. If is composite it has a divisor ; then is also a divisor, and by hypothesis, so and .
(ii) The condition with gives . For this is , inside the census, which finds exactly the six listed primes; a gap does not occur below , where the maximal gap is .
(iii) Among those six: and are level-one with prime; has , , ; has , , ; and () and () are level-one with composite . The counting identity (iv) follows. ∎
verified to : the empty-window set is exactly and the composite- level-one set is exactly , with no further members. Independently, the census counts 339,868 decomposable primes below with prime against 339,870 with — a difference of exactly , which is the arithmetic content of (iii) stated as a subtraction.
Lemma 5 is the hinge of the whole analytic story. It converts a divisor question about into a primality question about , and thereby converts "level-one prime" into "the triple consists of three primes, the upper step of which is a prime gap". Everything in §7 is the study of that triple. Two corollaries fall out at once.
Corollary 1normal formproved
Let be level-classified with and . Then the weight is prime: with prime and odd. The number of level-classified with and is .
Proof
The first claim is Proposition 3 in contrapositive, together with Lemma 3. For the second, each pair with , determines at most one prime , and there are at most such pairs. ∎
Corollary 2balanced primes lie on the level line proved
A balanced prime (A006562) is a prime , . Every balanced prime is decomposable and level-one.
Proof
Balance means , so is prime. It is decomposable, since is the statement , Bertrand's postulate. Then prime with leaves itself as the only divisor exceeding , so and . ∎
verified: all 167,031 balanced primes below (beginning , the DATA line of A006562) are level-one, with zero exceptions. Corollary 2 is what makes Problem 2 of §6 — is the level class infinite? — contain the balanced-prime problem as a special case.
5.6 The census, decade by decade
Table 2 is the whole classification of the primes below , computed from source this session. Rows to agree digit for digit with the tenth-edition treatise's two engines, which reach with different algorithms; the treatise's last two rows are quoted in §6 as carried, not recomputed.
Table 2. Complete classification census. is the prime count, "dec" the decomposable count, the level-classified count, its level-one stratum (Species I), the rest (Species II); ; "ties" counts , the points exactly on the diagonal. verified
decomposable
ties
25
22
12
7
5
0.5455
2
168
165
75
24
51
0.4545
3
1,229
1,226
390
135
255
0.3181
6
9,592
9,589
2,658
880
1,778
0.2772
8
78,498
78,495
18,353
5,953
12,400
0.2338
12
664,579
664,576
138,049
44,011
94,038
0.2077
28
5,761,455
5,761,452
1,078,707
339,870
738,837
0.1872
79
The prime counts are the standard values of ; the census reproduces them exactly, which is the cheapest available check that the underlying sieve is not losing terms. The decomposable count is always , by Lemma 2. The single number the classification computes for each prime is the ratio — level or weight is the sign of its logarithm — and Figure 7 is the distribution of that number. The level share decreases in every decade: at , 0.1872 at , and (carried from the treatise) at and at . That the decrease continues forever is Theorem 1; that most of it is not a fact about primes is §10.
Figure 7. The classification as one number. The distribution of over all 5,761,452 decomposable primes below ; the classification is the sign of this quantity. The weight class spreads smoothly over eight orders of magnitude; the level class is a much narrower distribution pushed hard to the right, because confines it. Exactly 79 primes land on the dividing line (). verified
5.7 The invariant panel
Every structural statement above was re-tested against the full stream of 5,761,452 decomposable primes below , by the C engine and, below , independently by the Python audit. The result in each case is zero violations. Table 3 lists the tests and their complete witness sets — the whole exceptional set found, not a sample.
Table 3. The invariant panel at .verified
Statement
Test
Violations
Witness set
Lemma 2
non-decomposable primes
0
exactly
Definition 1
on every term
0
—
Proposition C
and odd
0
—
Lemma 3(iii)
on the level class
0
max at
Lemma 4
0
440,311 each side
Proposition 2
0
—
Proposition 3
, composite weight
0
Lemma 5(ii)
empty window
0
Lemma 5(iii)
level-one with composite
0
Corollary 2
balanced primes
0
167,031 balanced primes
A002386
maximal-gap ladder
0
ends 132 at 1,357,201, 148 at 2,010,733, 154 at 4,652,353, 180 at 17,051,707, 210 at 20,831,323, 220 at 47,326,693
A007508
twin-pair counts via
0
eight decades, to the unit
founding paper, Table 6
level/weight counts among the first primes
0
, , , ,
Part III · §6Rarefaction — Theorem 1
This is the framework's native question: it cannot even be asked without the coordinates, and it has an answer. Everything else in the programme is either elementary (§5), classical in disguise (§5.2), or conditional (§7).
6.1 The statement
Write for the number of level-classified primes up to , for the level-one count, and for the level share among decomposable primes up to . The founding paper observed that decreases in every decade it could reach, and conjectured (its Conjecture 9). That is now a theorem.
Theorem 1rarefaction of the level class (the founding paper's Conjecture 9)proved*
As ,
In particular : the level-classified primes have relative density zero among the primes.
The asterisk means one thing only: the argument has passed internal audit across editions of the project's analysis but has not been externally refereed. It is not a hedge about the mathematics; it is a statement about the mathematics' social status, and submission removes it. The proof is given in full in Appendix B.1; this section explains how it works.
6.2 Why the proof can work at all
Before the argument, the obstruction it dodges. A level-classified prime is one whose has no divisor in . The quantity is the next prime after , so the event couples a divisor condition on to the compositeness of every one of . That coupling is what puts an asymptotic for out of reach: there is no known technique for conditioning a sieve on "no primes in between".
The key move
For an upper bound the coupling can simply be discarded. The level event implies a triple-prime event; a triple-prime event can be sieved; and throwing away a constraint only increases an upper bound. This is legitimate, it is the whole trick, and it is also exactly why the same method gives nothing in the lower-bound direction — which is why Problem 2 below remains open and, by difficulty conservation, contains the balanced-prime problem.
6.3 The proof in four steps
Fix a large and set the truncation parameter . Split the count into three pieces according to the size of the gap and the size of :
where counts level-classified with ; those with and ; and is everything else, which by Corollary 1 is in normal form. Each piece is handled by a different mechanism, and the interest is in which.
The tail : large gaps are rare for free. The gaps telescope — — and a sum of non-negative terms bounded by can have at most terms exceeding . Hence . No arithmetic whatsoever is used, only that gaps sum to the range.
The degenerate range : absorbed. By Corollary 1, , smaller than any power saving. This is where Proposition 3, the cube lemma, earns its place: without it the composite-weight terms would not be confined to a polylogarithmic set, and the normal form of the next step would be unavailable.
Normal form. Each prime counted by has with prime, odd, . Writing and turns the prime into a triple of primes
and the map is injective. This is the step where consecutiveness is dropped: nothing afterwards demands that be the immediate successor of .
Sieve, then sum. For fixed the three linear forms , , have degree one, coefficients and shifts of size , and pairwise resultants of the same size. These are exactly the hypotheses under which the dimension-3 Selberg sieve (Halberstam–Richert, Theorem 5.7) gives an upper bound uniform in the parameters — and that uniformity is precisely what the truncation was bought for. Summing the resulting bound over and odd costs a factor from the first sum and from the second, giving at , which is exactly the value balancing against . ∎
6.4 Reading the proof
Three things are worth extracting, because they are what a reader should carry away.
The exponent is an artifact. It comes from the balance between the gap truncation and the sieve saving, and the sieve saving is because the normal form produces a triple. The conjectured truth is . Closing the gap — proving unconditionally — is Problem 1, and it is the most concrete unconditional target the framework offers.
The is the price of the -sum. Restricted to , the same argument gives with no . Species II is what costs the extra factor.
Nothing here is prime-gap technology. No bound on is used beyond telescoping, and no equidistribution input beyond a standard upper-bound sieve. The proof would survive a fairly brutal weakening of what we know about prime gaps. That robustness is a feature — and §10 shows it is also a warning.
6.5 The finite- content
A bound of the shape makes a checkable finite prediction: the envelope statistic must stay bounded, and if the shape is right it should decrease. It does, at every decade, with no sign of a plateau.
Table 4. The Theorem-1 envelope. Monotone decreasing with no plateau — consistent with the bound being true but not sharp, i.e. with the conjectured and the exponent being an artifact of the dimension-3 normal form. The last two columns are carried from the tenth-edition treatise's census. verified to
(carried)
(carried)
0.2772
0.2338
0.2077
0.1872
0.1709
0.1575
0.3849
0.3310
0.3000
0.2758
0.2567
0.2410
Figure 8. Three bounded diagnostics, three panels. Left: the Theorem-1 envelope, decreasing through 0.2758. Middle: the Species-II normalisation (§8), cumulative and by decade window. Right: the windowed Species-I density , rising toward the dashed asymptote of Theorem 2 but far from reaching it. The honest reading is bounded decrease and bounded increase — none of the three has plateaued, and no constant should be extracted from any of them by extrapolation. verified
Problem 2the remaining native questionopen
Is the level class infinite? Theorem 1 is compatible with finiteness. Infinitude contains the balanced-prime problem — by Corollary 2, balanced primes occupy the level-one stratum — so by difficulty conservation this is not expected to be easy. But unlike the twin-prime restatement of §5.2, it is a question that only exists in these coordinates.
Part III · §7The constant of Species I — Lemma 1 and Theorem 2
Theorem 1 gives an upper bound with no constant. For the dominant stratum one can do better and identify the density constant exactly — at the cost of two hypotheses. The constant turns out to be , twice the twin-prime constant, and that identification is itself unconditional.
7.1 The reduction, restated
By Lemma 5, for ,
So the level-one primes are — up to the two exceptions and — the primes for which is prime, being 's own gap to the next prime. Equivalently: sits in the middle of a three-term arithmetic progression of primes whose upper step is a prime gap. Two conditions are in play and they must be handled separately: a triple condition — all prime — which is Hardy–Littlewood territory; and a consecutiveness condition — no primes strictly between and — which is gap-distribution territory.
7.2 Singular series, in one paragraph
Notation
For an admissible tuple the Hardy–Littlewood singular series is , where . Write for the pair series and for the triple series of the progression. The twin-prime constant is , and .
These objects are not abstractions imported for the occasion; the pair series is directly visible in the census. The number of primes below with a given gap oscillates violently with — gaps divisible by are roughly twice as common as their neighbours, gaps divisible by more common still — and that oscillation is (Figure 9).
Figure 9. The sawtooth is the singular series. Left: the observed gap counts (blue) and the same counts divided by (orange). Right: residuals about a smooth three-parameter envelope , before and after that division. The raw histogram departs from any smooth curve by 38.5% RMS over the 64 gaps with at least observations; after dividing by the pair singular series the departure is 6.0% — no fitting, no free parameter, the singular series supplied from the Euler product alone. (At the treatise finds 33.4% and 3.7%.) This isolates the factor of hypothesis (H2) below: the singular series accounts for the oscillation, and what remains is the smooth envelope that (H2) models by . verified
7.3 The line average — where the twin constant enters
Lemma 1line average of the triple singular seriesproved
unless , in which case
and
The vanishing is immediate: if the tuple meets both classes mod , and if it covers all three residues mod , killing the local factor. The evaluation of the average is a Dirichlet-convolution computation given in Appendix B.2. What is worth extracting here is the single step that makes the twin constant appear, because everything else in the computation is bookkeeping.
The identity at the heart of it
For every prime ,
The left side is a triple-series local factor times a divisor-average local factor; the right side is a pair-series local factor. Both equal ; the left simplifies because the awkward factors and cancel. Verified this session in PARI/GP symbolically — the difference simplifies to exact — and numerically to 38 digits. verified
7.4 The constant , and a correction that stands
A value has circulated in project material for this constant. It is wrong from the eighth significant digit. Two independent routes agree on the correct value:
where is the classical Hardy–Littlewood prime-triplet constant . The identification is not a coincidence: is the same Euler product over with prefactor instead of . It makes the value independently checkable against the literature rather than against one's own code. Route one this session: PARI/GP's zeta-accelerated prodeulerrat at 40 digits — the same routine that reproduces the published exactly. Route two: the direct Euler product over the primes below , agreeing to nine digits, the truncation error being . The misprint is numerically harmless downstream (relative in every prediction using ), but the printed digits should read . One attribution should also be corrected: the founding arXiv paper contains neither this constant nor the per-gap window claim of §7.6 — its Section 5 is Conjectures 1–9 and nothing analytic — so neither should be cited to it. verified
7.5 Theorem 2 and its hypotheses
Theorem 2conditional asymptotic for Species Iconditional on H1 + H2
Assume:
(H1) the Hardy–Littlewood conjecture for the admissible triples , with uniformity for : ;
(H2) the Gallagher-type model for consecutive gaps: with ,
uniformly for , with negligible tail.
Then
The derivation is short and is given in Appendix B.3: (H1) relative to the pair density assigns the extra event " prime" the conditional density , (H2) supplies the number of gap- pairs to weight it with, and Lemma 1 evaluates the resulting average of against an exponential weight. The constant is exactly the line average of Lemma 1, which is why its identification does not depend on either hypothesis.
Where the argument is not a proof
The step "the interior constraints do not affect the singular-series ratio at first order" is a decoupling assumption, not a deduction. It treats the left point as independent of the compositeness conditions on . The data say this decoupling is wrong by a stable to — see §7.6 — and §9.3 identifies the mechanism. Theorem 2 should therefore be read as: the constant is , and the first-order model is right to about one percent, with the percent itself an object of study rather than an error bar.
7.6 The gap-by-gap test — isolating (H1)
Removing the gap-distribution layer isolates (H1). For each with , feed the observed per-gap logarithmic mass through the ratio and compare with the observed count of gap- primes with prime. No gap model, no free parameter, nothing fitted.
Table 5. Gap-by-gap test at , -prime convention (the two composite- exceptions and are excluded, since the prediction models the primality of ; with the convention stated the counts are exact, and would otherwise read one higher). Singular series computed with the corrected . verified
observed
predicted
obs / pred
6
96,133
96,746
0.9937
12
69,185
67,579
1.0238
18
48,737
48,188
1.0114
24
32,906
32,174
1.0228
30
42,632
41,662
1.0233
36
13,767
14,180
0.9709
42
13,983
13,441
1.0404
48
6,309
6,173
1.0221
54
4,396
4,157
1.0576
60
5,484
5,269
1.0408
66
2,149
2,162
0.9938
72
1,134
1,040
1.0908
78
928
919
1.0096
84
749
717
1.0440
all
339,863
335,664
1.0125
Figure 10. The full gap-by-gap test at , with Poisson error bars. Blue: the 12 gaps with at least level-one observations, of which 10 lie inside the shaded window . Orange: small-count gaps, where the scatter is Poisson noise. The systematic excess of the observations over the parameter-free prediction — 1.25% in total here, at in the treatise — is the decoupling error of §7.5, and §9.3 explains where it comes from. verified
A claim, and what each census says about it
Project material states that per-gap agreement "lies in at every gap ". The statement depends on the census height. At (this report) of the gaps with at least observations lie inside the window and the rest is Poisson scatter; at the ninth-edition treatise found seven gaps outside; at the tenth edition finds all gaps with inside. The correct statement is therefore: the claim is not supported by a or census and is supported by a one, the earlier outliers having been Poisson scatter that more data resolved.
7.7 The constant, honestly — and what the control says about it
A reader who wants to see emerge from the raw data will be disappointed, and it is important to say so plainly. The cumulative normalised count stands at 1.0866 at , and the windowed density of Figure 8 rises through 0.98 in its last window, approaching the asymptote only logarithmically. The slowness is exactly what the model predicts: at the weight has barely begun to sample the large gaps whose enhancements raise the line average to its limit. Taken literally, with for all primes below , the model of Theorem 2 predicts a windowed density of 0.90 at , 1.00 at and still only 1.15 at — the constant is, for practical purposes, unobservable as a raw density. The observed density runs some 11% above that crude prediction; this is a finite-height error of the exponential gap model (H2) rather than of the singular-series ratio, since the gap-by-gap test above, which uses the observed gap counts in place of (H2), agrees to about a percent. The identification of the constant rests on Lemma 1's exact identity plus that test — not on the raw density at accessible heights.
There is a sharper way to look at the same data. Instead of asking how close the Species-I density is to , ask how it compares with the same statistic computed for a random sequence of identical density. A random sequence has no singular series: its analogue of the constant is , not . So the two curves should separate, and the direction and the slowness of that separation are both predictions. Figure 11 shows them separating.
Figure 11. The arithmetic separating from the generic. Left: the normalised Species-I count for the primes (limit under Theorem 2) and for a random sequence of density (limit ). Right: the ratio of the two censuses. The aggregate level count settles onto the control and stays there within a percent — that is the deflationary result of §10 — while the Species-I count starts 13.0% below the control at , crosses it at , and is 1.8% above it at , on its way — if Theorem 2 is right — to the asymptotic factor . The treatise's census, at a different seed, crosses at ; the crossing point is seed-sensitive at this height. verified
This is the cleanest available statement of where the arithmetic lives in this construction. The aggregate rarefaction is a property of density; Species I is a property of the primes; and the crossing near is the point below which a census is too small to see the difference. Any study of this framework conducted below would have concluded, wrongly, that the primes are slightly poorer in level-one terms than a random set.
Part III · §8Species II, and what is not known about it
The stratum carries about 68% of the level class at and has no identified constant. Two empirical normalisations are stable.
Table 6. Species-II diagnostics, decade windows and cumulative. normalises against the working law , evaluated at the window's geometric midpoint; normalises against the Mertens mass , the heuristic probability that an integer of size has no prime factor in . verified
window
windowed
0.7690
0.7705
0.7669
windowed
0.4445
0.4458
0.4432
cumulative at upper end
0.8312
0.8204
0.8108
cumulative at upper end
0.4469
0.4460
0.4436
The windowed sits in a band around across these three decades (the treatise extends it to five: ); the cumulative diagnostic nonetheless drifts steadily downward, which is what refuted an earlier working conjecture . That conjecture is retracted and the retraction stands. Identifying either constant is Problem 3, and the natural technology is the theory of divisors of shifted primes — Ford's and Koukoulopoulos's work — since is literally a Mertens-normalised count of integers with no divisor in a short interval. Whether the slow downward drift of terminates is, on the present data, undecidable, and saying so is the only defensible position.
Part III · §9Exact structure: weight columns, the congruence lock, interior credit
This section contains the framework's cleanest results: statements that are exact, provable in a paragraph, and false in any model that ignores arithmetic. They are also the clearest instances of the dictionary the construction advertises — a multiplicative weight column turns out to be a finite union of additive, residue-conditioned gap events.
9.1 Column identities
Column identity for proved
For every : . Equivalently, the column consists exactly of the lesser members of cousin-prime pairs with (OEIS A074822).
Proof
() If then , and is even and positive, so . By Lemma 4, forces . Hence , and . () Suppose and . The weight is the least divisor of exceeding ; the interval contains no integer, so that divisor is . For the second formulation: for odd , and means is the next prime, i.e. is a cousin pair. ∎
Column identity for proved
.
Proof
forces even, so , and is claimed by (Lemma 4). For : the interval is empty, so . For : the integers in are and ; is odd so ; hence and . ∎
verified at , on the nose: , and every one of those primes is with next prime ; .
These identities also yield a clean measurement of an independence question. Among the 440,256 gap- primes below , the joint divisibility is observed 29,364 times against the product prediction (a gap- prime is confined to three admissible classes mod , of which the lock class is one) — a deviation of , or with (binomial). The two congruence locks are statistically independent at to the available resolution. verified
Figure 12. The weight columns at . Left: occupancy of the columns . The profile is strikingly non-monotone, and the reason is arithmetic rather than noise: by the identities above, the column draws on a single gap value while draws on two, so the two are not comparable a priori; and a weight is the least divisor of exceeding , so small primes shadow one another — the ten largest columns are , with (170,245) outranking (146,593). Right: all 872,436 occupied columns on log–log axes. The prime weights lie on one smooth trend; the composite weights fan out beneath it into families indexed by their smallest prime factor and stop dead at : a composite weight is either in the weight class, where , or one of the five primes of Proposition 3. Every column right of is a prime weight of a level-classified prime, and those columns hold one to eight primes each. verified
Figure 13. The level lines at . Only odd occur (Proposition C), and the levels divisible by stand systematically above their neighbours — the mirror, on the level side, of the mod-3 rigidity that shapes the weight side: by Proposition 2, whenever , and a level-classified with prime must then carry the factor in . The line is Species I with 339,870 primes; carries 296,479 and carries 81,804, more than (55,919). An occupancy law for individual level lines is open and is not pursued here. verified
9.2 The congruence lock
The lockproved
For an odd prime and a decomposable prime with gap ,
immediately, since .
So a divisibility condition on — a multiplicative fact, and one of the inputs to the weight — is identical to a residue condition on . This is the founding identity of §1 read one prime at a time. If one now asks how often the lock closes, a flat prediction suggests itself: among gap- primes, is confined to the residue classes mod that are neither nor , and if it were equidistributed among them the lock class would occupy a share whenever . Measured at , that flat law is right to within a small but structured bias: exact when , and deficient by up to as . The bias has an explanation.
9.3 The interior-credit law
The mechanism
Fix and an odd prime , and condition on being consecutive primes. Consecutiveness means every interior even offset has composite. Now observe: if , then , and the compositeness of that interior point is guaranteed for free. That residue class needs one fewer accident to achieve gap exactly , so it is over-represented among gap- primes. The lock class earns no such credit — its offset is odd, hence never in — so it is under-represented relative to the credited classes. That is the observed deficit.
First-order bookkeeping makes this quantitative with no free parameter. Let
be the Hardy–Littlewood conditional probability that the interior point is prime given the prime pair — the "prime risk" that the credit removes — where is measured within the gap- population. Weight each admissible residue by
the factor renormalising the risk of the non-credited points once is known. Each lands in exactly one admissible class (as ), so expanding at the lock class , where , gives
The structural zeros, explained.: , so exactly. : and the pattern covers all residues mod , so , giving and again — the interior point of a pair is always divisible by , so no prime can earn credit there. Measured at : gives , gives , gives . verified
The parameter-free test at . Over all 392 cells with even , prime , at least gap- primes and lock hits, the law predicts the measured deficit with no fitted constant, reproducing the full shape of the bias surface — its growth in and its per- fine structure — with magnitudes running about 22% low. One cell, , is not a measurement at all: mod a gap- prime has a single admissible class and it is the lock class, so the deficit there is zero by Lemma 4 rather than by observation, and it is excluded. Over the remaining 391 cells a single global factor
(weighted least squares through the origin) absorbs the second-order enhancement and gives ; without it, . At , then, the first-order law is not yet decisively distinguishable from the data — the treatise needed the census for that ( parameter-free, ). Two remarks belong here. A leave-one-gap-out jackknife over the 20 gap groups gives , slightly smaller than the binomial error, so at this height the correlation of cells sharing a gap does not inflate the uncertainty. And the linearisation above is itself a choice: fitting instead to the exact-in-model deficit , with the right-hand side above, gives — about three points of the "second-order factor" are the trivial renormalisation of the denominator, and the remaining twenty are physics. verified
Table 7. The interior-credit law at representative cells, . "Free" is the parameter-free first-order prediction ; "" applies the single global second-order factor. verified
measured
free
(2, 5)
1.001247 ± 0.002132
1.000000
1.000000
(4, 5)
0.998917 ± 0.002131
1.000000
1.000000
(4, 7)
1.002087 ± 0.003017
1.000000
1.000000
(6, 7)
0.965162 ± 0.002251
0.970636
0.964305
(6, 11)
0.984384 ± 0.003204
0.984619
0.981303
(12, 13)
0.949783 ± 0.004211
0.956733
0.947405
(20, 23)
0.965854 ± 0.009765
0.954472
0.944657
(30, 31)
0.932671 ± 0.010838
0.949746
0.938913
(40, 41)
0.942891 ± 0.024302
0.950163
0.939419
Figure 14. The interior-credit law over all 391 fitted cells at : measured deficit against the parameter-free prediction, on a square canvas with equal aspect so a slope of reads as . Colour encodes ; marker size encodes precision. The structural zeros at cluster at the origin; the dark line is the fitted second-order factor , the dashed line is . The scatter about the fitted line is consistent with the quoted binomial errors. verified
Status of §9.3
The derivation is first-order local calculus with an independence step, so the law itself is heuristic; its predictive success is verified as stated; the structural zeros at are proved, given pattern inadmissibility. What is open is the second-order constant: deriving from the next order of the same calculus — pairwise interior patterns, i.e. , plus the tilt that conditioning on "gap exactly " induces on the environment. The same machinery summed over residues rather than resolved by them is the natural route to the consecutiveness excess of §7.6; the two are the same phenomenon seen from two sides.
Part III · §10The deflationary control
Any framework that produces curves must be asked the following question, and most are not: would a random sequence of the same density produce the same curves? Here it is asked, and part of the answer is uncomfortable.
Two random models were run through the identical classifier this session, to , with seed 20260903 — different from every previous edition's seed, so agreement with the record is not a seed artifact. Model A, the classic Cramér sequence: each is included independently with probability . Model B, parity-matched: each odd is included with probability , so that gaps are even and is odd, exactly as for the primes.
Table 8. The primes against two random controls, same classifier, same run. is the level share, the level-one share, both among decomposable terms. Model A produced 5,761,985 decomposable terms below against the primes' 5,761,452 — the models are matched in density to within . verified
primes
model A
model B
primes
A
B
0.4545
0.3901
0.5141
0.1455
0.1978
0.3390
0.3181
0.3102
0.4041
0.1101
0.1308
0.2463
0.2772
0.2769
0.3439
0.0918
0.1032
0.1972
0.2338
0.2365
0.2896
0.0758
0.0801
0.1583
0.2077
0.2097
0.2543
0.0662
0.0671
0.1332
0.1872
0.1879
0.2267
0.0590
0.0579
0.1158
Figure 15. Rarefaction, with its controls. Upper family: the level share ; lower family: the level-one share . Both fall monotonically from on. Dashed: model A, which tracks the primes' to within 1.1% at every decade past and to 0.4% at ; dotted: the parity-matched model B, which overshoots badly. The lower panel is the ratio, and it is the reason this section exists: the aggregate ratio sits at ; the Species-I ratio climbs through it. verified
Figure 16. The same plane, twice: the primes and a random sequence of the same density, both to . The gross anatomy is identical — a weight sheet above the diagonal, a level class below it, the same boundary, and level shares agreeing to 0.37% at this decade. Everything that distinguishes them is fine structure: the primes' level class occupies odd lines only (Proposition C) and its weight sheet is striated into columns at (Lemma 4 and §9.1); the control has neither, filling every level and every weight. Not one statement of §9 is true in the right panel. verified
The verdict, and it cuts both ways
The aggregate level share of the primes is generic. Model A tracks to within 1.1% at , 0.9% at and 0.4% at (the treatise: 0.07% at , 0.45% at , at its own seed). It follows that the rarefaction rate of Theorem 1 must not be presented as prime-specific structure: the decay shape is what any -density sequence does under this construction. Theorem 1 is a theorem about the construction at least as much as about the primes.
The composition is arithmetic through and through. The parity-matched model overshoots the level share by 21% and doubles Species I, because the real is prime with the Hardy–Littlewood conditional density , not the bare of a random odd sequence. Every one of §9's exact statements is false in both models.
And the first moment is generic while the second is not. Model A does not merely fail to reproduce Species I; it fails in a direction and at a rate that the singular series predicts. At the primes have 5.3% fewer level-one terms than the control; the deficit shrinks decade by decade, vanishes near , and becomes a 1.8% excess at (Figure 11), on its way — if Theorem 2 is right — to the asymptotic factor . Meanwhile the aggregate ratio stays flat within a percent across the same range.
So the correct summary of the control experiment is not "the framework's curves are generic". It is: the first moment is generic and the second is not, the crossover happens above , and any census stopping short of that height cannot tell the difference. That is a statement about what to measure, and it is the main methodological product of this section.
Part IV · §11Graph analysis: the primes, , a random sequence, and twelve fingerprints
The figures of this report are not illustrations of the theorems; they are the theorems' data, drawn. This section is the reader's guide: for each feature of the pictures, the statement that produces it, and for each statement, where to look. The conventions are fixed throughout — a square canvas with equal aspect whenever both coordinates and appear, so that the ratio is a distance from the diagonal; blue for the weight class, orange for the level class; density on a logarithmic scale.
11.1 Reading the classification plane of the primes (Figure 3)
The boundary. Every point lies on its hyperbola , and ; so the whole cloud lies strictly below the anti-diagonal , which is drawn dotted. It is not a fit and not a guide: nothing can cross it. As the census grows, the cloud grows outward along that line.
The diagonal and the ties. is the classification boundary; the 79 primes exactly on it have with the least divisor of exceeding — for instance , and — the only tie below with composite — shows that need not be prime, only free of divisors in . On the naturals the ties are exactly the prime squares (Proposition 1); on the primes they are rarer than one per terms and grow like the number of that happen to sit a prime gap below a prime.
The weight sheet and its columns. Above the diagonal the points are striated into vertical lines at odd : each column is the set of primes whose has as its least divisor beyond the gap. The leftmost column is the lesser twin primes (Lemma 4) and is the tallest, reaching up to . The columns thin out as grows for two reasons at once: a larger least divisor is rarer, and by the founding form , so a weight can only occur for primes whose gap is below . The sheet's upper edge is the boundary, its lower edge is the diagonal, and the fence of columns is dense near and sparse by .
The level class and its lines. Below the diagonal the points resolve into horizontal lines at odd , each ending at the boundary. The line (Species I) is the longest and densest: by Lemma 5 it is, to two exceptions, the primes with prime, and its weight is that prime. The lines are confined to (Lemma 3), which is why the level class is a thin horizontal band while the weight sheet is a tall triangle; on the square canvas the band's height is the width of the prime-gap distribution.
The empty wedge. Between the sheet's lower edge and the level band, near the diagonal, the plane is nearly empty on both sides. On the weight side a point near the diagonal needs : the least divisor beyond has to be almost as large as , i.e. has no divisor in yet one just below — rare. On the level side a point near the diagonal needs while : impossible for large . The wedge widens as the census grows.
11.2 Reading the base case (Figures 4 and 5)
The left panel of Figure 5 is Proposition 1 as a picture. The columns are at every prime (not only odd ones) and their heights are in the sieve's proportions — of the terms. The region strictly below the diagonal is empty except for the single line : composite has and so lands on or above the diagonal, and prime lands on . The ties are , prime, 446 of them below , exactly where each column begins — the sieve starting to strike at stage . Put the two panels side by side and the entire difficulty of the primes is the difference: replace the jump by the prime gap , and the window in which a divisor makes the term weight-classified becomes wide enough to be empty sometimes; the single line opens into a comb of lines ; and the even columns disappear because is now odd.
11.3 Reading the random control (Figure 16)
The right panel of Figure 16 is the same construction applied to a random sequence of density , and it is deliberately hard to tell from the primes at a glance: the same boundary, the same triangle above the diagonal, the same band below it, the same share of orange to within a percent. What the control lacks is everything §5 and §9 prove: its gaps are odd as often as even, so its is even half the time, and every level and every weight occurs — no odd-line comb, no odd-column fence, no twin column, no column identities. The lesson for reading these pictures is that the gross anatomy of the plane is a property of density under this construction and carries no arithmetic information; the fine structure — which lines and columns exist, and in what proportions — is where the number theory is. §10 makes this quantitative.
11.4 Twelve fingerprints: the construction on other sequences
Because the definition is parameter-free, any strictly increasing sequence can be fed to it, and its plane is a fingerprint. To compare sequences whose last term ranges from to on one frame, normalise: with , plot and , so that becomes and becomes . Every frame then shares the same diagonal and the same boundary. Figure 17 shows twelve sequences, the first terms of each, and Table 9 gives their censuses.
Figure 17. Twelve fingerprints. The first terms of twelve OEIS sequences on one normalised frame; each panel carries its level share as a blue/orange bar. Reading across: the triangular numbers are entirely orange and stop a factor of two short of the diagonal; the palindromes are a self-similar staircase; the numbers not divisible by are two vertical lines; the odd numbers are the naturals' fence with the even columns removed; the lesser twin primes are the primes' plane pulled toward the level side by their large gaps; the digit-addition trajectory has a level class of six horizontal lines; the sorting numbers have a weight fence that steps right at every power of two; the lucky numbers look like the primes with a larger level share; the Flavius Josephus sieve is almost entirely level; the Ulam and Harshad numbers, near the primes' density, look like the primes without their columns; and the binary numerals read in decimal are a family of clean diagonal lines. verified — identity checked on every record; censuses agree cell for cell with the project's 50-sequence atlas of 28 August.
Table 9. The atlas sample: fourteen sequences, first terms each, sorted by level share. "crit" is the share of decomposable terms with , the forced-level criterion of Proposition B. verified
OEIS
sequence
decomposable
level
share
ties
crit
gaps
A000217
triangular numbers
5,000,050,000
99,995
99,995
100.00%
0
2,642
100.00%
6…100,000
A000960
the Flavius Josephus sieve
7,854,038,953
99,996
96,560
96.56%
2
13,258
77.43%
6…521,092
A002113
palindromes in base 10
900,010,009
99,995
71,652
71.66%
16
4,699
9.21%
1…11,000
A001359
lesser of twin primes
18,409,199
99,995
47,766
47.77%
0
16,811
0.18%
6…2,190
A000959
lucky numbers
1,429,431
99,997
32,314
32.31%
25
19,084
0.01%
2…144
A001855
sorting by binary insertion
1,568,929
99,995
32,016
32.02%
46
12,230
0.00%
3…17
A000040
the prime numbers
1,299,709
99,996
22,999
23.00%
13
7,458
0.01%
2…114
A002858
Ulam numbers
1,351,223
99,996
21,855
21.86%
19
7,615
0.02%
1…587
A005349
Harshad numbers
1,033,781
99,996
20,334
20.33%
4
2,956
0.00%
1…88
A005408
odd numbers
199,999
99,997
17,982
17.98%
85
17,982
0.00%
2…2
A004207
the digit-addition trajectory
2,609,882
99,995
14,193
14.19%
1
3
0.01%
1…50
A007088
n in binary, read in decimal
1.10 × 1016
99,983
12,035
12.04%
1
3,673
0.43%
1…888,888,888,888,889
A000027
the natural numbers
100,000
99,997
9,592
9.59%
65
9,592
0.00%
1…1
A001651
numbers not divisible by 3
149,999
99,997
3
0.00%
2
2
0.00%
1…2
Two mechanisms account for nearly everything in the gallery, and both are elementary.
Proposition Bthe forced-level criterionproved
If a decomposable term satisfies , it is level-classified. Consequently a strictly increasing sequence with at every decomposable term is entirely level-classified. For one has if and if ; so every polynomial sequence of degree at least , and every quadratic one with leading coefficient , is level-classified from some index on.
Proof
The weight exceeds by definition, so . For the jump is , so , which tends to for and to for . ∎
The triangular numbers are the model case: and , so and every term with is level-classified; sharper, , so the cloud stops a full factor of two short of the diagonal — the clear channel of width visible in the first panel. The squares, the pentagonal numbers, the cubes, and the sums of the first primes are all-level for the same reason. The Flavius Josephus sieve grows like , so typically and of its terms are forced; the remaining terms are level of the time because the window is only wide multiplicatively — usually too short to catch a divisor. The palindromes split by the parity of their digit length: the typical gap is at even lengths and times smaller at odd ones, so the criterion fires for of even-length terms and of odd-length ones, and the level share oscillates between and from one digit length to the next.
The second mechanism: forced divisors of , measured against the cut
Where is large the criterion is dead and the classification is decided by whether happens to own a divisor in the window. A sequence that forces a divisor into at every term shifts the answer, and the direction depends on where the forced divisor sits relative to . The numbers not divisible by have gaps and at every term; since the forced divisor is always eligible, always, the plate is two vertical lines in the ratio , and the level class is annihilated (three terms in ). The odd numbers go the other way: is odd, the divisor is unavailable, and the level share nearly doubles against the naturals on the same range (17.98% against 9.59%). The digit-addition trajectory carries at every term () but , so the is never eligible and only its multiples bite: the weight fence sits on multiples of , and — since by the analogue of Lemma 3 while the factor must sit in once is coprime to — the level class of A004207 is six horizontal lines (plus once and twelve strays below ), which is exactly what the sixth panel shows.
Figure 18 puts the first mechanism on one axis: across the fourteen sequences the level share is a decreasing function of the median (Spearman ; over the project's fifty), and the points that break the trend — A001651 at and A005408 at , both near the naturals' median — are exactly the forced-divisor cases. Density alone, at fixed construction, sets the level share to first order: the primes, the Ulam numbers and the Harshad numbers — three unrelated constructions of similar density — have shares within three points of one another here (, , ), and in the project's matched census to , where their densities are , and , their shares , and are monotone in density. This is the atlas-wide version of the control experiment of §10.
Figure 18. The level share is governed by . Left of the dashed line , Proposition B forces every term into the level class. To its right the share falls with the median ratio, with two marked exceptions — the numbers not divisible by 3 (a forced eligible divisor) and the odd numbers (a forbidden one) — that are the second mechanism at work. verified
11.5 A remark on three-dimensional plots
The atlas at decompwlj.com also draws in three dimensions. Since , one has , and the last term vanishes whenever the jump is small beside the term. Every decomposable sequence with therefore lives on the plane , and the thickness of that plane is the gap: for the skip-take sequence A004202 the deviation is below decades for of terms. Seen edge-on, millions of points collapse to a line; seen down the normal, the plate is the ordinary classification plane, sheared by the projection. The three axes are not independent, and the 3-D picture carries exactly one extra bit over the 2-D plate, namely . This is why the square 2-D canvas is the reference picture of this report.
Part V · §12The ledger: theorems, conjectures and problems — and difficulty conservation
What is proved, what is conditional, what is open — and, the column most often missing from expositions of a private framework, what is a restatement of something already hard. "Native" marks statements that cannot be formulated without the weight–level coordinates; "classical" marks statements that are known problems in disguise.
Table 10. The complete status ledger, as of this report. Numbering follows the tenth-edition treatise; lettered items are stated in this report.
Item
Statement
Kind
Status
Prop. A
the founding (remainder) form and the divisor form of the weight coincide;
native
proved
Prop. 1
Eratosthenes collapse on : , level class shifted primes, ties shifted prime squares
native
proved
Lem. 2
decomposable ; on the primes the exceptions are
native
proved
Prop. C
on the primes every and every is odd
native
proved
Lem. 3
divisor localisation; ;
native
proved
Lem. 4
(the twin column)
native
proved
Prop. 2
(founding Conj. 7, 8)
native
proved — elementary, never a contribution
Prop. 3
cube bound , sharp at ; composite-weight level set to (founding Conj. 4)
native
proved (closure via refereed gap tables)
Lem. 5
level-one reduction ; exception sets to
native
proved
Cor. 1, 2
normal form ; balanced primes lie on
native
proved
Prop. B
level; polynomial sequences of degree (and quadratics with ) are all-level
native
proved
Lem. 1
line average of equals ;
classical tools
proved
§9.1
column identities for (= A074822) and
native
proved
Thm. 1
rarefaction (founding Conj. 9)
native
proved* — awaiting refereeing
Thm. 2
native + HL
conditional on H1, H2
§9.3
interior-credit law for the lock bias
native
heuristic; fit verified at and
§9.3
second-order constant –; is it constant in ?
native
open
Prob. 1
unconditionally
native
open
Prob. 2
is the level class infinite?
native
open — implied by the balanced-prime conjecture
Prob. 3
identify or (Species II)
native
open (Ford–Koukoulopoulos technology)
Prob. 4
the consecutiveness excess of the gap-by-gap test
native
open; mechanism identified in §9.3
Conj. A
rarefaction universality: for the Cramér model, with the same for the primes
native
open — six decades of evidence
—
working claim
—
retracted
Conj. 1
infinitely many primes
classical
open — is the twin-prime conjecture
Conj. 2, 3, 5, 6
infinitude and structure of fixed columns and lines
classical
open — Polignac-type
Difficulty conservation
Recasting a classical problem in weight–level coordinates does not lower its difficulty. Twin primes remain twin primes when called "the column ". Polignac remains Polignac when called "the column ". Goldbach is not visible here at all. Nothing in this report bears on the Riemann Hypothesis. The framework's yield is the native row set — Theorem 1, Propositions 1–3 and A–C, Lemmas 1–5, the column identities and the lock structure — and it should be judged on that and on nothing else.
Part VI · §13Algorithms: the reference kernels and the census engine
Five algorithms compute the same function and are used as mutual checks. All are in PARI/GP, following the project's convention that the reference implementation is the readable one. The census engines are written for throughput and agree with all five.
13.1 The reference kernels
\\ The definition, transcribed. O(tau(l)) after one factorisation.
decomp_fact(n, n1) = {
my(d, l, D);
d = n1 - n; l = n - d;
if (l <= d, return([0, 0, d])); \\ not decomposable
D = divisors(l); \\ sorted ascending; D[#D] = l > d
for (i = 1, #D, if (D[i] > d, return([D[i], l/D[i], d])));
}
\\ Trial division with a parity wheel and the descending level search.
\\ Theta(sqrt l) worst case, but no factorisation is required.
decomp(n, n1) = {
my(d, l, s, k0, step);
d = n1 - n; l = n - d;
if (l <= d, return([0, 0, d]));
s = sqrtint(l); \\ exact integer square root
if (l % 2, k0 = d + 1 + !((d+1) % 2); step = 2, k0 = d + 1; step = 1);
forstep (k = k0, s, step, if (l % k == 0, return([k, l/k, d])));
\\ Level branch. If l has no divisor in (d, sqrt l], then k > sqrt l and
\\ L = l/k is the LARGEST divisor le of l with le <= d AND l/le > d. The
\\ cofactor clause is essential: divisor pairs lying entirely below d occur
\\ exactly for the empty-window primes 13, 31, 113 (e.g. l = 9, d = 4: the
\\ divisor 3 has cofactor 3 <= d and must be skipped; the answer is k = 9).
forstep (le = if (l % 2, d - !(d % 2), d), 1, -if (l % 2, 2, 1),
if (l % le == 0 && l/le > d, return([l/le, le, d])));
}
\\ One-line fordiv variant (project file decompwlj_fordiv.txt).
decomp_fordiv(a, b) = { my(d = b-a, l);
if (a <= 2*d, return([0,0,d]));
l = a - d; fordiv(l, k, if (k > d, return([k, l/k, d]))) }
\\ The founding form, transcribed: the least k with a mod k = d. O(a); for checking only.
decomp_mod(a, b) = { my(d = b - a);
if (a <= 2*d, return([0,0,d]));
for (k = d+1, a, if (a % k == d, return([k, (a - d)/k, d]))) }
decomp_auto(n, n1) = if (n < 10^8, decomp(n, n1), decomp_fact(n, n1));
dclass(r) = if (!r[1], 0, if (r[1] > r[2], 2, 1)); \\ 0 none, 1 weight, 2 level
The complexity crossover is real and worth stating: trial division costs on every level-classified term, whereas factoring a random is almost always far cheaper. At the two are comparable; at trial division would need iterations while decomp_fact answers instantly. Deep witnesses computed this session, far beyond the census range, reproducing the treatise's values: verified
All five kernels agree with each other on every prime below (the founding form below ), reproduce Table 1 exactly, and produce the same census below as the C engine: decomposable, level, level-one, ties, with , with , zero identity violations, zero mod-3 violations, non-decomposable set . Verified on PARI/GP 2.15.4. verified
13.2 The census engines
To (this report). A complete smallest-prime-factor table to (400 MB of 32-bit entries, 3.5 s to build) gives the factorisation of every by repeated lookup; the divisors of are enumerated from it, the least exceeding is the weight, and the identity is asserted on every term. The primes, the two random controls and the naturals stream through the same classifier: 11 s in all on one core. The design is the simplest correct one, and its limit is memory: at the table is 4 GB, at it is out of reach.
To (the treatise). The tenth edition's engine processes the range in segments of , sieves the odd numbers of each segment with the base primes , collects the values of the segment's primes — which all lie in a window — and factors them all simultaneously by walking the multiples of each base prime through the window and dividing it out to the full power; each residual cofactor is or a single prime, since two prime factors exceeding would multiply to more than . No primality test and no trial division are used anywhere. The cost is with a small constant: 244 s at in 120 MB. That engine and this one share no logic, and they agree digit for digit on every entry of Table 2 up to , on every invariant and on every exception set.
Arbitrary sequences. The atlas engine reads a file of 64-bit terms and factors by trial division on the primes below , then Pollard rho–Brent with a Miller–Rabin test, asserting on every record. One caution that has already caught an error in this project: non-decomposable terms are skipped, not written, so the record index is not the sequence index, and any statistic keyed on must be rebuilt from .
Part VII · §14Hypothetical impacts on the future of number theory
Everything in this section is conditional on something that has not happened. It is written because the question "what would this change?" deserves a precise answer, and because the precise answer is smaller, and more interesting, than either an enthusiast or a sceptic would guess.
Rules of this section
Each scenario names its trigger — the theorem, computation or transplant that would have to happen — and carries two labels. Plausibility: program (a concrete direction with an identifiable existing technique and a plausible route to a theorem), conjecture (a precise statement believed on stated evidence), or speculation (an idea with no evidence beyond its own coherence). Reach: local (a result inside the framework), adjacent (a result in the analytic theory of prime gaps, sieves, or divisors that happens to be reached through the framework), or broad. Nothing here is a result, and the last two scenarios are the ones a reader should weigh most heavily: the first says what the framework cannot change, and the second lists the predictions that would refute the rest.
14.1 The baseline: what is already on the table
Before the hypotheticals, the actuals, so that the hypotheticals can be measured against them. The construction is a canonical, parameter-free map from increasing sequences to pairs of multiplicative invariants; on the natural numbers it reproduces the sieve of Eratosthenes exactly (Proposition 1), and on the primes it produces a partition with a proved rigidity theory (§5), a proved rarefaction theorem (Theorem 1), an exactly identified conditional constant (Lemma 1, Theorem 2), exact column identities (§9.1), and a heuristic lock law that predicts a bias surface of nearly four hundred cells with no free parameter (§9.3). It has also produced a methodological result that is not about primes at all: a framework's headline curve can be entirely generic while its second moment is entirely arithmetic, and a census must reach a computable height — here — before the difference is visible (§10). That baseline is modest and it is real. Every scenario below is an increment on it.
Trigger. An unconditional upper bound of the conjectured order for the primes with prime. Theorem 1 loses a factor against this target, and §6.4 locates the loss exactly: the truncation that buys sieve uniformity, and the sum over .
What it would change. The event " is prime, where is 's own gap" is a prime-triple event in which one of the shifts is not a fixed parameter but a random variable determined by the prime itself, and the current sieve literature handles fixed shifts. A method that obtains the right order here — a weighted sieve that sums over the shift with the singular series as weight, or a Bombieri–Vinogradov-type input that makes the sum over in step 4 of §6.3 cost no factor of — would be a small but genuine addition to the toolkit for gap-indexed prime constellations, of which balanced primes, the Lemke Oliver–Soundararajan consecutive-prime biases and the interior-credit law are all instances. The result itself would confirm that the exponent was an artifact, as §6.4 asserts.
What it would not change. It is an upper bound. It says nothing about whether the level class is infinite, and nothing about twin primes, since the twin column is not a level-class object at all.
Scenario IIProblem 2 falls: the level class is infiniteplausibility · speculation, with a named route · reach · adjacent
Trigger. A proof that infinitely many primes have free of divisors in — by Corollary 1, that with and prime, infinitely often.
Where it sits. Corollary 2 places the balanced primes inside the level class, so the balanced-prime conjecture (A006562, open) implies Problem 2; Problem 2 is strictly weaker, and that weakness is the interesting part. A level-one prime is the middle of a three-term progression of primes whose upper pair is consecutive; a balanced prime needs both pairs consecutive. The union over weakens the target further to " has a prime factor larger than ". Statements of the shape "infinitely often, at least one of a bounded family of events occurs among consecutive primes" are exactly what the Maynard–Tao sieve, and its refinement to consecutive primes by Banks, Freiberg and Turnage-Butterbaugh, are built to prove; whether that machinery can be pointed at a progression-shaped event with a gap-indexed shift is not known, and the honest label is speculation. If it can, the by-product would be a new unconditional statement about three-term progressions of primes with a consecutiveness constraint, which no current result provides.
What it would not change. A proof of Problem 2 that does not pass through balanced primes proves nothing about balanced primes. Difficulty conservation applies in both directions.
Scenario IIIThe is derived: consecutiveness conditioning as a portable toolplausibility · program · reach · adjacent to broad
Trigger. A second-order derivation of the interior-credit calculus of §9.3 — pairwise interior patterns () plus the tilt that conditioning on "gap exactly " induces on the environment — that produces instead of fitting it, and, summed over residues rather than resolved by them, produces the consecutiveness excess of the gap-by-gap test. The tenth edition's three independent decade windows () and this report's at are the data such a derivation must reproduce.
What it would change. Standard Hardy–Littlewood heuristics predict the frequency of an event ; they do not predict conditioned on exactly, and that conditioning is ubiquitous — Gallagher's model, the Montgomery–Soundararajan moments, every empirical study of gaps — and almost always assumed harmless. It is not harmless, its first-order effect is computable from singular series alone, and it was confirmed here against cells at and in the treatise against cells at . The phenomenon is of the same family as the biases in consecutive primes modulo found by Lemke Oliver and Soundararajan, which are also explained by conditioning Hardy–Littlewood on consecutiveness; the lock deficit is that mechanism resolved by a single gap value, in a setting where the first-order prediction has no free parameter and the second-order correction is measured to three decimals. A derivation of the second order would be a device whose natural consumers are not in this framework at all: anyone who has ever conditioned a prime statistic on an exact gap. This is the scenario with the best ratio of reach to plausibility.
Scenario IVConjecture A is proved and the universal constant is computedplausibility · program · reach · local, methodological
Trigger. For the Cramér model — each included independently with probability — a proof that almost surely , and the value of . A random model has no consecutive-prime coupling, so the lower bound that is out of reach for the primes should be a second-moment computation; the upper bound is Theorem 1's argument with the sieve replaced by independence.
What it would change. It would turn §10 from an experiment into a theorem: the aggregate rarefaction is a property of density, with a known constant, and the primes' residual against it — Species I above the crossing, Species II, the columns and lines — is what remains to be explained arithmetically. Every diagnostic of §7 and §8 could then be re-expressed as a ratio to the universal prediction, and the arithmetic factor, built from and , is what is left. It is the cheapest high-value computation in the programme, and it would also settle, by comparison with Table 4, whether the primes' is as conjectured in §6.4.
Scenario VThe construction over plausibility · program · reach · adjacent
Trigger. Order the monic irreducibles of by degree and then lexicographically, so that "next irreducible" is defined; Definition 1 transfers verbatim, with computed in the polynomial ring and the weight the least divisor exceeding the jump in the same order. The obstruction throughout this report has one name — the word next — and over , conditioning on "no irreducibles in between" is a computation on a moduli space rather than a hypothesis, while the Hardy–Littlewood analogues are theorems in the large- limit (Bary-Soroker; Sawin–Shusterman for twin primes over ).
What it would change. In order of value: the analogue of Proposition 1 (does the construction collapse to the sieve there too?); Theorem 1 unconditionally and with the right exponent; Theorem 2 with its constant; and the exact analogue of the consecutiveness excess, which over is a number nobody can compute. It is the least glamorous-sounding item on this list and the one most likely to produce a refereed theorem, because every input it needs already exists.
Scenario VISpecies II is identified: divisors of shifted primes with a moving windowplausibility · program · reach · local to adjacent
Trigger. A value for or (Problem 3). Level classification is exactly " has no divisor in ", and the theory of integers with no divisor in a short interval is mature — Erdős, Tenenbaum's -function, Ford's , Koukoulopoulos on divisors of shifted primes. What the framework contributes is a perverse normalisation: the integer is a shifted prime and the shift is the prime's own gap, so the window's left endpoint is a random variable correlated with the shift. The relevant technology is Buchstab-type.
What it would change. A worked instance of the divisors-in-short-intervals theory at a gap-indexed argument, which is a new kind of argument for that theory; and, inside the framework, a complete first-order description of the level class (Species I by Theorem 2, Species II by this). Whether the slow downward drift of — about per decade in the treatise's data — terminates is undecidable on present evidence, and a proof would decide it.
Scenario VIIFingerprints: the construction as taxonomy, testbed and teaching deviceplausibility · already happening · reach · broad but soft
Trigger. None needed; this is extrapolation of §11. The construction assigns every increasing sequence a canonical plane, and §11 shows that the plane's gross shape is set by the regime of (all-level for polynomial growth of degree , sieve-like near density , annihilated by an eligible forced divisor) while its fine structure encodes the sequence's arithmetic (columns for forced divisors, combs for parity, staircases for digit rules).
What it could change. Three soft things. A taxonomy: the OEIS holds hundreds of thousands of sequences, and a two-number summary — level share and median — with a fingerprint plate is a cheap, canonical way to sort them into families that current metadata does not capture; Proposition B and the forced-divisor mechanism already classify a majority of the fifty-sequence atlas by proof rather than by inspection. A testbed: the framework provides an explicitly computable, exactly verifiable, parameter-free instance of the additive–multiplicative divide on which heuristics can be tested against exact samples in four minutes of compute; in a field where most heuristics are tested against a handful of small cases or none, that is worth having, and §7.6, §9.3 and §10 are three examples of what it catches. A teaching device: the sieve of Eratosthenes, trial division, prime gaps, the twin-prime constant and the Hardy–Littlewood singular series appear in one construction, in that order, each as the answer to a question the previous one raises; a graduate course could be built along the two-hour route of this report.
Scenario VIIIWhat will not change, whatever happensplausibility · certain · reach · —
The classical problems. By difficulty conservation (§5.2, §12), no theorem in this framework will prove the twin-prime conjecture, any case of Polignac's conjecture, or the infinitude of balanced primes, except by proving those statements directly under another name. Goldbach's conjecture does not appear in these coordinates at all: the decomposition is a statement about consecutive terms of one sequence, and Goldbach is a statement about pairs summing to a target. Nothing here bears on the Riemann Hypothesis or on the error term of the prime number theorem; the only analytic inputs used anywhere are an upper-bound sieve and, conditionally, the Hardy–Littlewood conjecture and a gap model.
Algorithms. The weight of requires , and the map is not invertible without it; the construction gives no purchase on factoring an isolated integer, on primality testing beyond trial division, or on any cryptographic primitive. Its computational content is a census, not an algorithm, and the reference kernels of §13 are transcriptions of the definition.
The size of the yield. In the scenario where everything above succeeds, the sum is: two or three refereed theorems in the sieve theory of gap-indexed constellations, one portable heuristic device, one function-field transplant, and a classification tool. That is a real but modest body of work. In the scenario where nothing above succeeds, what survives is the rigidity of §5 (small, sharp, permanent), Theorem 1, the column identities and the lock calculus. Any presentation of this framework that cannot say both of these things is not to be trusted.
14.2 Predictions that would refute the scenarios
A section of hypotheticals is only honest if it says what would falsify it. The following are checkable with the next decade or two of census, at the engine speeds of §13.
Rarefaction universality (Conjecture A, Scenario IV). At the level share of the primes and of a fresh Cramér sequence of matched density agree to within 1%, and the Theorem-1 envelope of Table 4 continues to decrease. A persistent divergence of the aggregate shares, in either direction, refutes Conjecture A and reinstates the aggregate rarefaction as arithmetic.
Species I keeps separating (Scenario III, Theorem 2). The ratio , which crosses near and stands at 1.018 at , continues to rise at every decade. Under (H1)+(H2) with the crude uniform- gap model the windowed Species-I density is 0.90 at , 0.96 at and 1.00 at ; the observed value runs about 11% above that crude model at (0.980 against 0.886), the excess being (H2)'s finite-height error, since the gap-by-gap test isolates (H1) to about a percent. The prediction is therefore a windowed density near at — the treatise measures near — and near at , with the limit not approached to within 10% below (crude model: 1.15 there). A plateau below , or a value above before , refutes the model.
The second-order constant is constant (Scenario III). A disjoint-window determination of at lies within of . A monotone drift of like across four windows refutes the "pairwise-pattern at fixed order" explanation and supports a decaying correction.
The per-gap window (§7.6). At every gap with and at least level-one observations has with the -prime convention, and the total ratio lies in . A total ratio drifting toward would mean the consecutiveness excess is a finite-height effect; one drifting away would mean the decoupling assumption of Theorem 2 fails at first order, and the constant would then be the wrong constant.
The exception sets (§5). No new composite-weight level prime beyond , no new empty-window prime beyond and no new balanced prime off the line at any height — these are theorems, so a counterexample would indicate an engine error, not new mathematics, and they are the census's built-in self-test.
14.3 The largest claim this report is willing to make
The decomposition takes an additive datum — the gap to the next term — and uses it to select a multiplicative datum — a divisor of the term. It does so canonically, with no parameter, for every increasing sequence, and on the natural numbers it reproduces the sieve of Eratosthenes exactly. Constructions with that property are rare. It is not going to knock down a wall; what it provides is an explicitly computable, exactly verifiable, parameter-free instance of the additive–multiplicative divide, a native theorem (Theorem 1) answering a question only these coordinates can pose, a small permanent rigidity theory, and a place where the Hardy–Littlewood heuristics can be conditioned on consecutiveness and checked to the third decimal. If the future of number theory is touched by it at all, it will be through Scenario III and Scenario V — a correction term everyone needs and a transplant that makes the conditional unconditional — and not through any reformulation of a problem that was hard before the coordinates existed and remains exactly as hard in them.
Appendix AVerification record
Everything numerical in this report was recomputed from source this session. The directory was empty at the start; no data, no binary and no intermediate file was reused from a previous edition of the project's analysis.
A.1 Engines
C census engine (the project's decompwlj_census_engine.c, rebuilt). Smallest-prime-factor table to ; complete factorisation of every by lookup; exact for every decomposable term; the identity asserted on every record. Streams the primes, Cramér model A, parity-matched model B (seed 20260903) and the naturals through one classifier. 11.0 s in all on one core; 5.3 s to the end of the primes.
Python audit (written this session, shares no code with the C engine). Primes from a plain sieve; the complete divisor set of every built by literal trial division; the weight read off that set exactly as Definition 1 says. All decomposable primes below compared tuple by tuple against the C engine's records: zero mismatches. Every structural test of §5 independently re-derived over the same range — divisor localisation, the level bound, the twin column, mod-3 rigidity, the cube bound, the level-one reduction, parity — with zero violations; Proposition 1 re-verified for ; the founding remainder form checked against the divisor form for every prime below .
PARI/GP 2.15.4. Five kernels — decomp_fact, decomp, decomp_fordiv, decomp_mod, decomp_auto — agree on the 17 rows of Table 1 and on every prime below (the remainder form below ); an independent census to equal to the C stream digit for digit. Constants at 40 digits by prodeulerrat; the Lemma-1 per-prime identity simplified symbolically to exact ; the three deep witnesses of §13.1 reproduced.
Streaming factoriser (the project's decompwlj_stream.c, rebuilt) for the fourteen atlas sequences, terms each, generated from their OEIS definitions and checked against the DATA lines; the identity asserted on all records.
NumPy analysis for every table and figure, from the records; singular series from Euler products over the primes below with the exact local factors at the primes dividing the pattern's discriminant.
A.2 External anchors reproduced
for , ending — the published values, produced by the census sieve itself (Table 2).
The twin-pair counts A007508 at : , each equal to (Lemma 4).
The maximal-gap ladder A002386, entry for entry, ending at below .
The cousin-prime column: lesser cousin primes below (A074822).
The founding paper's own classification table (its Table 6, indexed by prime rank rather than size): among the first primes, , , , and are level/weight-classified — reproduced to the unit at all five ranks.
The balanced primes A006562: below , beginning , all on .
(the published twin-prime constant), (the published prime-triplet constant),
The tenth-edition treatise's census: every entry of Table 2 for –, the record level at , , and its two-term decomposition, the exception sets, the envelope of Table 4, and the cumulative and windowed and of Table 6 — all digit for digit.
The project's illustrated edition (21 August 2026) at : the gap-by-gap totals against (ratio ), gaps with observations of which inside the window, and with jackknife — reproduced.
The project's 50-sequence atlas (28 August 2026): the fourteen censuses of Table 9, cell for cell.
A.3 Notes arising this session
The non-decomposable primes are . A project note still lists . is decomposable and is the unique level-classified prime of weight ; has and is not. Confirmed by every engine at every decade.
Proposition 3's closure. The tenth edition's closing sentence (" for every ") is false as stated, since ; the two-range argument of §5.4 replaces it and closes the set to from a census. The same argument closes Lemma 5(ii).
The lock law's normalisation. The first-order law is stated in linearised form; fitting the exact-in-model deficit instead moves from to . Both are reported; comparisons with the treatise use the linearised form.
The jackknife does not inflate 's error at . Leave-one-gap-out over 20 gap groups gives against the binomial ; the treatise's caution that cells sharing a gap are correlated is correct in principle and does not bite at this height.
Attribution. The founding arXiv paper (0711.0865v4) contains neither the constant nor the per-gap window claim; its Section 5 is Conjectures 1–9. Both statements are cited here without a source claim, as the illustrated edition recommended.
The crude gap model underpredicts Species I at finite height. Using (H2) literally with gives a windowed Species-I density of 0.886 at against the observed 0.980; the gap-by-gap test, which uses the observed gap counts, is off by only . The discrepancy is therefore in the finite-height accuracy of the exponential gap model, not in the singular-series ratio.
Figure conventions. Every figure carrying both and is drawn on a square canvas with equal aspect; densities are binned in log–log at one bin per output pixel band, occupied cells given a three-cell mark before a light Gaussian so that one-cell-wide lines survive rasterisation, and the two classes composited as two hues on white.
A.4 What was not done here
Theorem 1's proof was audited for internal consistency, not refereed. The asterisk stands.
No decade beyond was computed; the and rows quoted in Table 4 and §6 are carried from the tenth-edition treatise, which computed them with two independent engines.
The closure of the exception sets of Proposition 3 and Lemma 5 to rests on the refereed gap tables of Oliveira e Silva, Herzog and Pardi and on A002386 beyond .
The universal constant of Conjecture A was not computed. Nothing was checked against the atlas at decompwlj.com beyond the OEIS entries named above.
Appendix BThe analytic proofs in full
§6 and §7 state these results and explain how the arguments work. Here they are written out, following the tenth-edition treatise. Nothing in this appendix is needed to read the rest of the report.
B.1 Theorem 1
Theorem 1.. proved*
Proof
Let be large and set . Split , where counts level-classified with ; those with and ; and is the remainder, which by Corollary 1 is in normal form.
Step 1 — the tail . The gaps telescope: for large. A sum of non-negative terms bounded by can have at most terms exceeding . Hence
No arithmetic is used — only that gaps sum to the range. This is the crudest step and it is not the bottleneck.
Step 2 — the degenerate range . By Corollary 1, , which is smaller than any power saving and is absorbed into the final bound. This is where Proposition 3 earns its place: without it, the composite-weight terms would not be confined to a polylogarithmic set and the normal form of Step 3 would be unavailable.
Step 3 — normal form. By Corollary 1 each prime counted by satisfies with prime, odd, . Writing and ,
so that the triple consists of three primes, with . The map is injective — a prime determines its gap and hence its decomposition, and conversely — so, writing ,
Inequality (B.1) is where the consecutiveness requirement is dropped: nothing now demands that be the immediate successor of .
Step 4 — the sieve, and the two mean values. For fixed the three linear forms , , have degree one; their coefficients and shifts are of size ; and their pairwise resultants divide , again of size . These are exactly the hypotheses under which Halberstam–Richert's Theorem 5.7 gives an upper-bound sieve estimate uniform in the parameters:
where is a non-negative multiplicative function whose local factors are at primes and elsewhere. If some local density vanishes, the triple is inadmissible and , which is better still. Uniformity in is essential and is precisely what the truncation buys.
Since we have , so (B.1)–(B.2) give
Both sums are standard mean values of non-negative multiplicative functions. Writing with supported on squarefree integers and ,
Therefore
by the choice , which is exactly the value balancing against . Summing the three contributions proves the theorem. ∎
B.2 Lemma 1
Lemma 1. unless , in which case with ; and . proved
Proof
Vanishing. If then meets both classes mod with , killing the local factor. If then is invertible mod and covers all three residues, so and the factor vanishes. So .
Evaluation for . Then , giving local factors at and at . For : if , contributing ; otherwise , contributing . Collecting the constants and dividing out the generic factor at the primes dividing ,
The average. Write with supported on squarefree integers composed of primes and ; then and converges. Hence
the interchange justified by non-negativity together with . Therefore
The evaluation. Using ,
since the factor of is . ∎
Every step above is bookkeeping except the per-prime collapse displayed in §7.3, which turns a triple-series local factor times a divisor-average local factor into a pair-series local factor. That identity is the whole reason the twin-prime constant appears in a statement about triples.
B.3 Theorem 2
Theorem 2. Under (H1) and (H2), . conditional
Proof
By Lemma 5, up to an additive constant counts consecutive pairs with for which is also prime. Given such a pair, (H1) relative to the pair density assigns the extra event " prime" the conditional density , uniformly in the stated range. The interior primality constraints that make consecutive affect the pair count — which is what (H2) supplies — and not, at first order, the singular-series ratio. Hence, with ,
By Lemma 1 and partial summation, — the exponential weight has total mass and averages the line against its mean. Therefore . ∎
The step flagged in §7.5 is the sentence "and not, at first order, the singular-series ratio". It is a decoupling assumption, the data contradict it by about one percent, and §9.3 identifies the mechanism. What survives unconditionally is the constant: is the line average of Lemma 1, proved above with no hypothesis at all.
Appendix CExercises, with solutions
Eight exercises in the order of the text. The first four need only Part I; the last four use Part II. Solutions are folded below each.
Exercise 1 · the definition by hand
Decompose (next prime ) and (next prime ). Classify each, and say which of Proposition 3's statements they witness.
Solution
: , , divisors ; the least exceeding is , so , level-classified with composite weight — the largest member of Proposition 3's five-element set. : , , divisors ; , , level-classified, and is the unique case of equality in the cube bound.
Exercise 2 · the remainder form
Without factoring, find the weight of (next prime ) from the founding definition: the least with . Explain why no can work and why the search may stop at if no has been found — and what to do then.
Solution
For the remainder is , never . By Proposition A a solution is a divisor of exceeding ; the divisors come in pairs , so if none lies in then the weight exceeds and the term is level-classified with the largest divisor of that is whose cofactor exceeds : here (cofactor ), giving . This is exactly the two-branch logic of the kernel decomp in §13.1, including the cofactor clause.
Exercise 3 · the swap
Find all pairs of decomposable primes below with the same , explain the mechanism, and show that in each pair the classification is decided by the cut alone.
Solution
Listing for the primes below (, , , , , , , , , , , , , , , , , , , , , ), the coincidences are exactly , and . Mechanism: consecutive primes , share iff , i.e. ; for a twin this is a prime triple , and , , are the three below (the next is ). In each pair the twin has by Lemma 4, while has , so falls below its cut and the weight is the next divisor of the same above : , and respectively. Same divisor set, different cut: against ; against ; against . In all three the twin is weight-classified (or a tie) and its partner is level-classified.
Exercise 4 · the base case iterated
On , decompose . Then decompose the term of (whose is ), and continue until the level is . What have you computed?
Solution
: , , . The term : , , . The term : is prime, , — level-classified, and the chain stops. The successive weights are the prime factorisation in increasing order: iterating the construction on the level (re-entered as a term of through , so that its is ) is trial division, and the chain ends exactly when the remaining cofactor is prime, i.e. at the first level-classified step. This is the third entry of the dictionary of §4.
Exercise 5 · Lemma 3 on a general sequence
Show that for an arbitrary strictly increasing sequence, a level-classified term satisfies , and exhibit a sequence attaining infinitely often.
Solution
The proof of Lemma 3(iii) gives before the parity step. Equality needs with and no divisor of in : take with prime, , so the sequence must contain followed by . For and an odd prime, the pairs — e.g. for — can be spliced into any increasing sequence; a single sequence containing all of them attains infinitely often.
Exercise 6 · the column
Prove a column identity for in the style of §9.1, and check it against the census figure below if you have the records.
Solution
forces , even: ; is claimed by . For : the integers in — none — so . For : contains and ; ( odd); so . For : contains ; so . But by Proposition 2, requires , so is impossible with : .
Exercise 7 · the structural zero at
Show directly that for a gap- prime the interior point is divisible by , and deduce that no prime can earn interior credit at .
Solution
and are primes exceeding , so neither is . If then , impossible; so and . The interior point is always composite for a reason that has nothing to do with , so the class gains no free compositeness, (equivalently ), and the deficit vanishes — as measured at the cells and of Table 7.
Exercise 8 · Theorem 1's exponent
Suppose one could prove (B.2) with replaced by for the two-form pattern obtained by dropping the prime (i.e. counting only , prime). What bound on results, and why is it worse?
Solution
With a dimension-2 saving the sum balances against at , giving — better, not worse, and exactly the conjectured order; but dropping prime discards the level condition entirely (every prime with gap satisfies the remaining two conditions), so the count is no longer of level-classified primes and the bound is vacuous. The dimension-3 saving is the price of keeping 's large prime factor in the sieve, and Problem 1 asks for the dimension-2 rate while keeping the dimension-3 event.
Appendix DNotation
symbol
meaning
the -th term of a strictly increasing integer sequence; on the primes,
,
the jump ; on the primes, the prime gap (A001223)
when that exceeds , else ; on the primes (A118534)
the admissible moduli
the weight: least divisor of exceeding (A117078)
the level: (A117563)
level / weight
/ ; equivalently, has no / has a divisor in
, ,
level-classified count up to ; its and strata (Species I and II)
, ,
the corresponding shares among decomposable terms
, the gap histogram
, ,
Hardy–Littlewood singular series of ; of ; of
, ,
twin-prime constant; prime-triplet constant;
,
the two Species-II normalisations of Table 6
,
predicted lock deficit; its fitted second-order factor
, , ,
the truncation parameter and the three pieces of Theorem 1's proof
![The divisors of l on a log axis for p = 2557 and p = 2113, with the window (d, sqrt l]](data:image/png;base64,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)
