Decomposition into weight × level + jump

Abstract

The decomposition writes each term of a strictly increasing integer sequence as \(a(n)=k(n)\,L(n)+d(n)\) — weight × level + jump — with \(k(n)\) the smallest divisor of \(\ell(n)=a(n)-d(n)\) exceeding the jump \(d(n)\). On the natural numbers the construction collapses exactly onto the sieve of Eratosthenes: the weight of \(n\) is the smallest prime factor of \(n-1\), and the level-classified naturals are precisely the shifted primes. On the prime sequence it produces a coordinate system in which several classical objects become strata — lesser twin primes are the weight-3 column, balanced primes are level (1;1) — and in which exactly one native open problem arises: Conjecture 9, the rarefaction of level-classified primes.

This third report adds four things to its predecessor. First, total verification: the decomposition of every prime below 10⁹ (50 847 531 decomposed terms) was recomputed from scratch and audited at 66 523 points against an independent implementation, against the project's PARI/GP kernel, and against the OEIS b-files of A117078 and A117563 — zero mismatches. The level-one census was extended to 2.5·10⁹ and reproduces the challenge paper's Tables 2–3 exactly. Second, two ±1 findings are resolved: the arXiv paper's Table 3 counts weights over weight-classified primes only (the difference from the all-decomposable count is exactly the nine level-classified primes of weight ≤ 23), and the challenge paper's Table 1 entry lev1(10⁸) is internally inconsistent with its Table 2 by exactly the boundary prime 99 999 989, whose \(\ell=99\,999\,971\) is prime. Third, a new native micro-result, the mod-3 comb: a level-classified prime with \(3\mid\ell\) has \(3\mid L\) outside an explicit finite window; the exception set to 10⁹ is exactly \(\{5,13,887\}\). Fourth, a sharpened literature position: targeted searches (July 2026) find no published work on the divisor distribution of the gap-coupled form \(2p_n-p_{n+1}\); the fixed-shift theory of Ford and Koukoulopoulos does not directly apply, so Species II of Conjecture 9 appears to be a genuinely unaddressed problem.

Contents

Section 1Significance of the framework

The construction does one thing, exactly and everywhere: for any strictly increasing integer sequence, each term whose successor satisfies \(a(n{+}1)<\tfrac32 a(n)\) is written uniquely as

weight times level plus jump, where the jump is the forward gap and the weight is the smallest divisor of \(\ell(n)=a(n)-d(n)\) exceeding that gap. Nothing is estimated and nothing is modelled: the map \(a(n)\mapsto(k,L,d)\) is a change of coordinates, invertible termwise, defined by the sequence itself. Its significance, as we assess it after verifying the record to 10⁹, rests on four legs — and on a clear statement of what it does not do.

It unifies. On the naturals the decomposition is the sieve of Eratosthenes (§3): weight = smallest prime factor of \(n-1\), level-classified ⟺ \(n-1\) prime. On the primes, classical objects become visible strata of one picture: the lesser twins are the weight-3 column (Lemma 3.1 of the paper; the count to 10⁹ matches the twin tables to the unit, §4.2), the balanced primes are level (1;1), the three-term arithmetic progressions anchored at consecutive primes are Species I of Conjecture 9. One construction, one scatter plot (§5), and a family of A-numbers falling out of its strata — A117078, A117563, A001223, A118534, A162174, A162175, A121155 among them.

It generalizes. The same operator applies verbatim to any monotone sequence — the project's atlas at decompwlj.com holds a thousand decomposed OEIS sequences with their 2D and 3D portraits, and the two-wing morphology of §5 recurs across unrelated sequences. The naturals are the solved prototype; the primes are the serious case; the atlas is the evidence that the coordinate system is canonical rather than bespoke.

It isolates one native problem. Nine conjectures were originally attached to the prime case. Under the honest ledger, five are exact reformulations of classical open problems (twins, Polignac-type families, balanced primes), two are elementary mod-3 facts now retired as trivial theorems, one is a finite claim now verified with its exception set complete to 10⁹ — and exactly one, Conjecture 9, is native: the level-classified primes rarefy. Its level-one species is settled unconditionally (a sieve bound, §4.4); its level-greater-than-one species is, to the best of a targeted July 2026 literature search, an unaddressed problem about divisors of the gap-coupled form \(2p_n-p_{n+1}\) (§6.3). A framework that manufactures one genuinely new, sharply posed, plausibly tractable question has earned its coordinates.

Its data layer is now verified. The project site states, with characteristic candour, "my data have not been verified." As of this report that caveat is discharged for the prime sequence to 10⁹ and for the level-one census to 2.5·10⁹: three independent implementations agree at every sampled point, the published tables are reproduced under their stated conventions, and the two residual ±1 discrepancies are diagnosed to their exact causes (§4.6) — one a population convention, one an upstream boundary erratum of a single prime.

Section 2Definitions, existence, classification

Let \(a(1) $$d(n)=a(n{+}1)-a(n),\qquad \ell(n)=\begin{cases}a(n)-d(n)&\text{if }a(n)-d(n)>d(n),\\[2pt] 0&\text{otherwise,}\end{cases}$$ $$k(n)=\min\{\,k>d(n):\ k\mid \ell(n)\,\}\ \ (\text{if }\ell(n)\neq 0),\qquad L(n)=\ell(n)/k(n).$$

Then \(a(n)=k(n)L(n)+d(n)\). The term is decomposable when \(\ell(n)\neq 0\); it is classified by level when \(k(n)>L(n)\) and by weight otherwise (ties \(k=L\) count as weight). Three remarks fix the geometry of the object.

Third, the classification is a comparison of the two factors of one number: \(\ell=kL\) with \(k\) forced upward past the gap. Level-classification \(k>L\) is equivalent to \(k>\sqrt{\ell}\), i.e. to the statement that \(\ell\) has no divisor in the window \((d,\sqrt{\ell}\,]\). That single sentence is the multiplicative content of the whole prime-side theory: everything in §4 and §6 is about when the window \((g,\sqrt{\ell}\,]\) is empty of divisors.

2.1 The first seventeen primes, recomputed

Table 2 of the paper, regenerated by the pipeline and by the PARI kernel (identical output; the kernel's selftest also checks these rows against a naive reference). VERIFIED

Two worked examples fix the mechanics. For \(p=89\): the next prime is 97, so \(g=8\), \(\ell=81\); the divisors of 81 exceeding 8 begin at 9, so \(k=9\), \(L=9\) — a tie, classified by weight. For \(p=887\): next prime 907, \(g=20\), \(\ell=867=3\cdot17^2\) with divisor set {1, 3, 17, 51, 289, 867}; the least divisor exceeding 20 is \(k=51\), \(L=17\), level-classified — and, as §4.5 will make precise, 887 is the unique level-classified prime below 10⁹ with \(3\mid\ell\), \(L>1\) and \(3\nmid L\), as well as the largest member of Conjecture 4's exception set. One small prime carries three exceptional structures at once.

Section 3The fundamental theorem and the sieve of Eratosthenes

The case \(a(n)=n\) is where the construction meets the multiplicative bedrock, and it is worth doing slowly because everything on the primes is this picture with one knob turned.

For the naturals, \(d(n)=1\) for every \(n\), so \(\ell(n)=n-1\) (for \(n\ge4\); the terms 1, 2, 3 are not decomposable), and the weight is the smallest divisor of \(n-1\) exceeding 1 — that is,

the smallest prime factor and the largest proper divisor. The fundamental theorem of arithmetic enters exactly here: it guarantees that \(\mathrm{spf}\) is well defined and that the pair \((k,L)\) is a genuine, canonical factorization of \(n-1\) — the decomposition of the naturals is FTA read through its minimal divisor. Classification becomes primality: \(n\) is level-classified \(\iff k>L \iff \mathrm{spf}(n-1)>\sqrt{n-1} \iff n-1\) is prime. PROVED (elementary; the kernel selftest checks \(k=\mathrm{spf}(n-1)\) and the equivalence to \(n=2001\); an independent 5 000-sample audit with bounded trial division confirms it at 10⁷ scale).

So the two wings of the naturals' portrait (§5, Fig. N) are: a weight wing quantized on vertical fibres \(k=p\) — the columns of the sieve of Eratosthenes, the multiples of 2, of 3 but not 2, of 5 but not 2 or 3, and so on, exactly the author's figure sieveNb.jpg — and a level wing that is the shifted primes \(\{p+1\}\): when \(n-1\) is prime, \(k=n-1\) and \(L=1\), so the entire level class of the naturals sits on the single ray \(L=1\). The sieve is one ray plus a fan.

The general construction as Eratosthenes with a moving threshold. On the naturals the window is \((1,\sqrt{n-1}\,]\) — empty iff \(n-1\) is prime. On a general sequence the window is \((d(n),\sqrt{\ell(n)}\,]\): the same sieve primitive, "least divisor above a cutoff", but the cutoff is now the local gap of the sequence rather than the constant 1. That is the precise sense in which the framework generalizes the sieve of Eratosthenes: the weight function is \(\mathrm{spf}\) relative to a sequence-adapted floor. When gaps are bounded (the naturals) the theory is classical; when gaps fluctuate like the primes' — mean \(\log x\), Poisson-like tails — the emptiness of the window becomes a genuinely new question, and §6 will show that its random-integer analogue is a theorem while its prime-sequence form is Conjecture 9. It should be said plainly that the decomposition factors \(\ell(n)\), not \(a(n)\): on the primes nothing factors \(p\) itself (nothing could be expected to), and the multiplicative object under study is the additive companion \(\ell=2p_n-p_{n+1}\).

Section 4The primes: theorems, conjectures, and the verified record

From here on \(a(n)=p(n)\), \(d(n)=g(n)\), and the sequences are weight = A117078, level = A117563, jump = A001223, \(\ell\) = A118534. The primes 2, 3, 7 are the only non-decomposable terms (\(11>\tfrac32\cdot7\)) — proved in the paper and the only zeros of A117078, as conjectured on the OEIS page and settled there. Everything below 10⁹ that follows was recomputed this session; Appendix B documents the audit.

4.1 The conjecture ledger

4.2 Weight 3 is the twin column

Verified as an identity of arrays: over all 50 847 531 decomposable primes to 10⁹ the predicate \(k=3\) agrees with the predicate \(g=2\wedge p>3\) with zero disagreements, and the common count is 3 424 505 — equal to \(\pi_2(10^9)-1=3\,424\,506-1\) from the external twin-prime tables (the −1 removes \(p=3\)). At index scale, the count over \(n\le5\cdot10^7\) is 3 370 445, of which 3 370 444 are weight-classified: the single level-classified weight-3 prime is \(p=5\) (\(k=3,L=1\)), the first entry of §4.6's nine-prime list and the reason the paper's Table 3 column reads one less than the twin count. VERIFIED, externally anchored

4.3 Mod-3 rigidity and its sharp corollaries

The theorem cuts the decomposable primes into a pool \(6\mid g\) (44.51 % of them at 10⁹; 43.04 % at 10⁷, the challenge paper's 0.4304) on which \(3\nmid\ell\), and its complement on which \(3\mid\ell\) — and it propagates through every stratum we measure. Three sharp consequences, the second new to this report:

(i) Species I lives on the pool. A level-one prime has \(\ell\) prime or \(\ell\in\{9,25\}\) (§4.4); if \(6\nmid g\) then \(3\mid\ell\), forcing \(\ell=3\) or \(\ell=9\) — that is \(p=5\) or \(p=13\). The level-one primes with \(6\nmid g\) are exactly {5, 13}, verified as such over the full range to 10⁹ (2 708 029 of the 2 708 031 level-one primes have \(6\mid g\)). PROVED (given §4.4's exception sets) VERIFIED to 10⁹

(ii) The out-of-pool level class is exactly the 3∣ℓ level class. At 10⁹ the level-classified primes with \(6\nmid g\) number 3 667 475 = 3 667 473 (the level->1 primes with \(3\mid\ell\)) + 2 (the primes 5 and 13). Theorem 5's bookkeeping closes to the unit across eight and a half million level-classified primes. VERIFIED

(iii) Conditional level-one density inside the pool. At 10⁷, \(P(\text{level-one}\mid 6\mid g)=0.1539\) against 0.0662 unconditionally — the challenge paper's Observation 7, reproduced to its printed digits (0.4304, 0.1539). VERIFIED

4.4 Conjecture 9 and its two species

Write \(f(x)\), \(f_1(x)\), \(f_{>1}(x)\) for the shares of decomposable primes \(\le x\) that are level-classified, level-one, and level-greater-than-one. C9 asserts \(f(x)\to0\). The two summands behave differently and must be kept apart — conflating them produced the first report's overclaim.

Census. With \(N_1(x)\) the number of level-one primes \(\le x\) (complete convention: every prime \(\le x\) is decomposed, using its true successor):

The census column counts decomposable primes with \(\ell\) prime; adding the two composite-ℓ exceptions {13, 31} gives N₁ — which is precisely the challenge paper's Table 2 formula "\(\#\{2p_n-p_{n+1}\ \text{prime}\}+2\)" with the count over decomposable primes. All six printed values of challenge Tables 2–3 (339 870; 2 708 031; 6 227 807; 1.0866; 1.1037; 1.1097; window constants 1.028, 1.056) are reproduced exactly. VERIFIED to 2.5·10⁹

Generations. Level (1; i) — level-one with \(\ell=p(n-i)\) — counted over \(n\le5\cdot10^7\), extending the paper's table from \(i\le7\) to \(i\le12\):

Rows \(i\le7\) equal the paper's Table 5 exactly. The ratio drifts slowly downward through ≈ 0.41 rather than sitting at a constant; a Gallagher-type Poisson model of gaps predicts geometric-like decay qualitatively, and we leave the ratio's limit unclaimed. VERIFIED HEURISTIC (decay law)

Species II (level > 1). By the level bound, \(L\le g-1\): the level of a level-classified prime is a small odd number on the scale of the gap, and \(k=\ell/L>\sqrt\ell\). The share obeys, empirically, \(f_{>1}(x)\approx c_2\log\log x/\log x\) with windowed \(\hat c_2\) flat across five decades — 0.7425, 0.8071, 0.7690, 0.7705, 0.7669, 0.7656 — while the cumulative ratio \(f_{>1}\log x/\log\log x\) still reads 0.805 at 10⁹ and is not converged (the second report's two-constants erratum). Section 6.2 derives the \(\log\log x/\log x\) shape from a smooth-part heuristic and explains why its random-integer analogue is a theorem while this one is open. HEURISTIC (shape and constant) OPEN — NATIVE (that f₍>1₎ → 0 at all)

4.5 The mod-3 comb — a native micro-result

Table 4's level histogram is conspicuously enriched at multiples of 3 (L = 3, 9, 15 fat relative to trend; §5, Fig. E). Theorem 5 explains why 3 divides ℓ on the whole out-of-pool half; the new statement is that within the level class the 3 is then forced into the level, not the weight:

The same divisor-minimality argument gives the structural bound behind Conjecture 4: if the weight \(k\) of a level-classified prime is composite, every proper divisor of \(k\) divides ℓ and is below \(k\), hence \(\le g\); so \(k\le g^2\), and \(k>\sqrt\ell\) then forces \(\ell\ell^{1/4}\approx p^{1/4}\) — excluded far beyond 10⁹ by exhaustive maximal-gap verification, which we cite without quoting unverified bounds. PROVED (bound) set complete to 10⁹

4.6 Conventions and errata: the two ±1 findings

(a) Paper, Table 3 — a population convention, not an error. Our first recount of the weight histogram at \(n\le5\cdot10^7\) exceeded the paper in seven columns by a total of +9. The deficit multiset {3, 5, 9, 11, 11, 13, 17, 17, 19} is exactly the weight multiset of the nine level-classified primes with weight ≤ 23:

This list is complete to 10⁹, and trivially finite for any weight bound \(B\): \(k\le B\) and \(k>L\) force \(\ellRESOLVED

(b) Challenge paper, Table 1 vs Table 2 at x = 10⁸ — a genuine internal ±1. Table 1 prints lev1(10⁸) = 339 869; Table 2 prints N₁(10⁸) = 339 870. Both cannot hold under one convention. The cause is a single prime: 99 999 989, the largest prime below 10⁸, has successor 100 000 007 and \(\ell=99\,999\,971\), which is prime — so it is level-one. A pipeline that only decomposes primes whose successor also lies below the mark excludes it; the Table 2 formula includes it. At the other magnitude marks the boundary prime's ℓ is composite (999 963, 9 999 963, 29 999 997, 999 999 867 = 3·333 333 289), so only the 10⁸ row is affected; f₁(10⁸) is 0.0590 either way at four decimals. The complete-convention value is 339 870. DIAGNOSED

4.7 The verified record at a glance

Challenge Table 1 reproduced (with the diagnosed 10⁸ boundary entry corrected); the 10⁹ row is new. At index scale \(n\le5\cdot10^7\): 8 553 468 level-classified (17.107 %), 41 446 529 weight-classified — the paper's Table 6 exactly, at every printed index from 10² to 5·10⁷. Diagnostics at 10⁹: \(f_1\log x=1.1037\); \(f_{>1}\log x/\log\log x=0.8046\); level share by decade 0.2969, 0.2712, 0.2278, 0.2042, 0.1846, 0.1689 — monotone decrease, C9's empirical content. VERIFIED

Section 5Analysis of the graphs

The project's figures are not illustrations of the theory; several of them are the theory, in the sense that the theorems of §4 were read off their structure before being proved. We analyse the author's graphs first — re-examined against our verified arrays — and then present six figures regenerated from this session's data. Every structural claim made about a figure below was checked numerically, not visually.

5.1 The master map

5.2 The two wings at scale, and our recomputation

5.3 The naturals: Eratosthenes as a portrait

5.4 Rarefaction, constants, combs, generations — the measured Conjecture 9

5.5 The remaining project figures

Four further author figures were examined for the second report and their readings stand unchanged against the verified arrays. 3D_natural_numbers_2.JPG and the 2018 WebGL capture (Screenshot…three_js…png, from decompwlj.com's live views) extrude the two planar portraits along \(n\): the naturals' fan of prime-indexed sheets with the level ray, and the primes' two wings with the level wing visibly starving along the third axis — Table 6's declining column as geometry. lesser_3D.png applies the atlas operation at second order — the decomposition of A001359, the lesser twins, themselves the \(k=3\) column of the primes' portrait — a self-application we record as an observed structural motif of the atlas without attaching a claim. A117078_20210214.png, the OEIS capture of the weight sequence, documents provenance and third-party review: T. Ordowski's 2013 comment states decomposability as \(2\,p_{n+1}<3\,p_n\) — the paper's Lemma 2.1 independently rediscovered on the sequence page — and the page's conjecture that 2, 3, 7 are the only zeros is the paper's theorem. decompwlj_1_20231216.jpg shows the author's sieve-style PARI/GP program; our benchmarking note from the kernel work stands: the fordiv smallest-divisor kernel is optimal per term on sparse sequences like the primes, while forfactored-style batch factorization wins only on dense sequences such as \(\mathbb N\) (9–11× there), so algorithm choice is sequence-dependent (Appendix A). The animated atlas 150decompanimAnum.gif cycles the \((\log k,\log L)\) portraits of 150 decomposed OEIS sequences; the recurring two-wing morphology across unrelated sequences is the empirical case for the construction's universality claimed in §1.

Section 6Hypothetical impacts: the additive–multiplicative bridge

The project's ambition is stated in its subtitle: a bridge between multiplicative structure and additive gap phenomena. After verification we can say precisely where the bridge stands, what traffic it can carry today, and what would have to be built for it to carry more. The claims in this section are graded carefully; the word "hypothetical" in the section title is doing real work.

6.1 The coupling, and the scale at which C9 lives

By §2, a decomposable prime is level-classified iff \(\ell(n)=2p_n-p_{n+1}\) has no divisor in \((g(n),\sqrt{\ell(n)}\,]\). Everything additive about consecutive primes (the gap) and everything multiplicative about the companion \(\ell\) (its divisors in a window) meet in this one predicate. That is the bridge — and also the obstruction, because the window's floor is the gap and the number \(\ell\) is built from the next prime: the shift varies with \(n\) and is correlated with the very factorization being interrogated. Note also the scale: C9 is decided at gap scale \(g\asymp\log x\). The Riemann Hypothesis constrains prime counts at resolution \(\sqrt x\) and is silent here; C9 and RH are logically orthogonal (neither implies the other), and the natural conditional neighbours are instead the Hardy–Littlewood prime-pair conjectures (Species I) and, via primes in progressions, GRH — which also does not settle C9. PROVED (equivalence)

6.2 The smooth-part skeleton, and a theorem the primes are asked to imitate

Suppose \(p\) is level-classified with gap \(g\). The level bound gives \(L\le g-1\), and every divisor of \(L\) is a divisor of ℓ below \(k\), hence \(\le g\). So \(\ell=L\cdot k\) with a small "sub-gap" factor \(L\) and a large factor \(k>\sqrt\ell\) having no divisor of ℓ between \(g\) and itself. In the dominant configuration \(k\) is prime and \(L\) is the entire \(g\)-smooth part of ℓ (the prime 887, whose \(k=51\) is composite, shows the configuration is dominant, not universal). Heuristically, then, a level-classified prime is one whose companion factors as (small odd smooth part) × (prime), and the density of that event, summing \(1/L\) over odd \(L\le g\) against the prime density of the cofactor and averaging over the gap distribution, has the shape

the measured Species-II law, with \(c_2\) a Mertens-type constant filtered by parity, by the mod-3 rigidity of §4.3 (which decides, per gap class, whether 3 may divide \(L\)), and by the Poisson-like distribution of \(g\). We display this as the shape's origin and claim nothing about the constant. HEURISTIC

6.3 The literature position: fixed shifts reach the door, not through it

The mature theory of divisors of shifted primes concerns \(p+a\) for a fixed shift: Ford's H(x, y, z) machinery (Annals, 2008) gives the order of magnitude of shifted primes with a divisor in a dyadic window from above, Koukoulopoulos (2010) the matching lower bounds in wide ranges; the survey literature is explicit that even there the lower-bound situation was long "bleak". Two features stop a direct application to Species II: the shift here is \(-g(n)\), different at every \(n\) and correlated with the factorization; and the relevant event is emptiness of the whole window \((g,\sqrt\ell\,]\), an intersection across \(\sim\log\ell\) dyadic blocks whose joint behaviour under the coupling is unstudied. A targeted search this month (July 2026) for work on the divisor distribution of \(2p_n-p_{n+1}\) — under its OEIS name A118534, under its arithmetic-progression description, and under gap-coupled shifted-prime phrasings — returns nothing. To the best of our search, Species II is an unaddressed problem, and we state that as a finding about the literature, not a certificate of novelty: absence of evidence, diligently sought. SEARCHED, NEGATIVE (July 2026)

6.4 The nearest published neighbour

Gafni–Tao, "Rough numbers between consecutive primes" (arXiv:2508.06463, August 2025; still a preprint as of this writing), studies gaps \((p_n,p_{n+1})\) in \([X,2X]\) containing no \(\sqrt X\)-rough number, proving the count of exceptional gaps is \(O(X/\log^2X)\) unconditionally and \(\sim cX/\log^2X\) with \(c\in(2.7,2.8)\) under Hardy–Littlewood with Montgomery–Soundararajan uniformity. It is the closest published relative of the framework's questions we can find: a multiplicative property (roughness) interrogated inside the additive window between consecutive primes, with the exceptional structure dominated by the same mod-6 skeleton — their key obstructed configurations are the small gaps \(g\in\{2,4\}\), ours is the partition by \(6\mid g\) of Theorem 5. The methodological overlap (singular series over gap classes; congruence rigidity mod small primes) suggests the two problems will eventually share tools, and we flag the paper as required reading for anyone attacking Species II. STATUS CHECKED

6.5 Impacts claimed, impacts declined

What we are prepared to claim, at the "hypothetical impact" grade the section title promises: (i) the framework poses a sharp, well-motivated test problem — extend shifted-prime divisor theory from fixed to gap-coupled shifts — with a verified 10⁹-scale dataset, clean invariants, and a proved random-integer target to imitate; (ii) it packages additive conjectures in sieve-shaped multiplicative language (twins as the weight-3 fibre; balanced primes and prime 3-APs anchored at consecutive primes as Species-I strata; Polignac families as weight fibres), which is a genuine translation service even where it is not a discount; (iii) its mod-3 rigidity and comb give a worked example of congruence structure propagating through a divisor-minimality construction — a small but native and fully proved mechanism. What we decline to claim: cryptographic relevance (the construction consumes factorizations and produces none; there is no trapdoor; the honest connection is a shared divisor-window toolbox with the analysis of Diffie–Hellman parameters); any difficulty discount for the reformulated conjectures; and any constant beyond the windows that measured it. A proof of Species II, if it comes, will most plausibly proceed stratum-by-stratum in \(L\) — for each fixed odd \(L\), showing almost all decomposable primes with \(L\mid\ell\) acquire a divisor in the window — and the uniformity in \(L\le g\) is exactly where the gap coupling will have to be broken. PROGRAMME

Section 7Conclusion and future research

7.1 What this report establishes

The decomposition into weight × level + jump is an exact, canonical coordinate system on monotone sequences; on the naturals it is the sieve of Eratosthenes and its Conjecture 9 is a theorem; on the primes its data layer is now verified to 10⁹ by three concordant implementations, its published tables are reproduced under their stated conventions with the two residual ±1 discrepancies diagnosed to their exact causes, and its conjecture list resolves under the honest ledger into five reformulations (undiscounted), two retired trivialities, one finite claim with exception set {13, 31, 113, 131, 887} complete to 10⁹, and one native open problem. Of that problem's two species, the level-one species is \(o(\pi(x))\) unconditionally with a census now extended to 2.5·10⁹ matching the challenge paper exactly, and the level-greater-than-one species stands as an apparently unaddressed question about divisors of the gap-coupled form \(2p_n-p_{n+1}\), whose random-integer analogue is a theorem. Two small native results are added: the comb proposition with exception set {5, 13, 887}, and the sharp statement that the level-one primes with \(6\nmid g\) are exactly {5, 13}.

7.2 Corrections to propagate upstream

Three concrete edits for the project's documents. In the challenge paper, either correct Table 1's lev1(10⁸) to 339 870 or state the boundary convention beside the table (and note that Table 2 already uses the complete convention). In the arXiv paper, add one sentence to Table 3's caption naming its population (weight-classified primes) and, ideally, the nine-prime list of §4.6 as a reading aid. On the project site, the sentence "my data have not been verified" can now cite this report's audit for the primes to 10⁹; the thousand-sequence atlas beyond the primes remains unverified and should keep the caveat.

7.3 Research directions, in order of leverage

The literature question, sharpened. Determine whether any current technique yields \(f_{>1}(x)\to0\) — equivalently, that almost all decomposable primes have a divisor of \(2p_n-p_{n+1}\) in \((g,\sqrt\ell\,]\). The fixed-shift theory does not directly apply (§6.3); the concrete first target is the stratum \(L=3\): show that among decomposable primes with \(3\mid\ell\) (the \(6\nmid g\) half, by Theorem 5), those with no divisor of \(\ell/3\) in the window have vanishing relative density. A single fixed stratum already contains the coupling in miniature.

A conditional Species-I asymptotic. Under Hardy–Littlewood with standard uniformity, a Gallagher-style average over the gap distribution should produce \(N_1(x)\sim c_1\,x/\log^2x\) with an explicit singular-series constant; the computation is routine-shaped but has not been written down, and the measured drift of \(\hat c_1\) (0.8431 → 1.0561 across the verified range) is the quantity it must explain. This is the most publishable self-contained piece currently on the table.

Computation. The level-one census needs only primality of ℓ, so 10¹⁰ costs roughly 12× this session's 27-second stage and is within reach of the present hardware; it would extend Fig. D's abscissa by \(\Delta T\approx0.09\). Full classification to 10¹⁰ is a ~40-minute job for the staged pipeline with disk checkpoints; 10¹¹–10¹² should abandon exhaustive sweeps for windowed sampling (exact decomposition on random windows of 10⁷ consecutive primes at several magnitudes), which measures the window constants without the memory bill. Each extension should re-run the full audit battery of Appendix B, not just the new range.

Writing. The natural venues remain the Journal of Integer Sequences and Integers for the framework-plus-census paper; Experimental Mathematics or Research in Number Theory become realistic if the submission leads with the sharpened Species-II problem statement, the negative literature finding, the comb proposition with proof, and the conditional Species-I constant — in that order, with the ledger discipline of this report intact and the two upstream errata fixed before submission.

The framework's claim to significance does not rest on solving the classical problems it re-coordinates — it does not — but on the one question it asks that no one else appears to have asked, now posed sharply enough to be attacked: does the companion of a prime, \(2p_n-p_{n+1}\), see its divisor windows fill the way a random integer's do? Everything verified to 10⁹ says yes; nothing yet proves it.

Appendix APARI/GP algorithms

The project's reference kernel (decompwlj_optimized.txt) provides decomp(n, n1) (fordiv smallest-divisor search — optimal per term on sparse sequences such as the primes), decomp_fact(n, n1) (factorization-based, for large terms: ~1 ms at \(p\sim10^{25}\)), decomp_auto (dispatch at 10⁸), decompseq(V) (batch over a sequence vector), and selftest() — all re-run and passing this session, and concordant with the Python engine at every point compared (§B). Two routines are added for this report's censuses, in the same style. Both use the complete convention: every \(p\le X\) is decomposed with its true successor.

\\ Level-one census to X, complete convention.
\\ Counts decomposable p <= X with ell = 2p - nextprime prime, then adds
\\ the two composite-ell level-one primes {13, 31}. Equals N1(X) of the text.
N1(X) =
{ my(c = 0, p = 0);
  forprime(q = 3, nextprime(X + 1),
    if(p,
      my(l = 2*p - q);
      if(l > q - p && isprime(l), c++));
    if(q > X, break);
    p = q);
  c + 2; }

\\ Window constant  c1hat  on (a, b]:  (N1(b)-N1(a)) / (pi(b)-pi(a)) * log sqrt(ab).
c1win(a, b) = (N1(b) - N1(a)) / (primepi(b) - primepi(a)) * log(sqrt(a * b));

\\ Mod-3 comb exceptions to X: level-classified p with 3 | ell and 3 ∤ L.
\\ Uses the kernel's decomp(); returns the exception list ({5, 13, 887} to 1e9).
combx(X) =
{ my(p = 0, r, v = List());
  forprime(q = 3, nextprime(X + 1),
    if(p,
      r = decomp(p, q);                    \\ r = [k, L, d]
      if(r[1] && r[1] > r[2]               \\ level-classified
         && (p - (q - p)) % 3 == 0         \\ 3 | ell
         && r[2] % 3,                      \\ 3 ∤ L
        listput(v, p)));
    if(q > X, break);
    p = q);
  Vec(v); }

Checks: N1(10^6) = 5953, N1(10^8) = 339870, matching the Python census; combx(10^4) = [5, 13, 887]. The heavy 10⁹ sweep was executed by the Python pipeline of Appendix B for memory-bandwidth reasons; the PARI routines are the specification and were used as the cross-implementation at 10⁶-scale and on samples.

Appendix BReproducibility: the verification pipeline

Environment: Python 3.12 (numpy, sympy), PARI/GP 2.15.4, one CPU core, 4 GB memory ceiling, hard per-call execution budget. The design constraint that shaped everything: no single stage may hold more than ~1.6 GB or run more than ~4 minutes, and every stage must be resumable from disk. Total wall time for the full record, about four minutes.

The audit lattice. Correctness is not argued from one implementation but from agreement of unrelated ones: (1) pipeline vs sympy at 66 523 points — 0 mismatches; (2) pipeline vs the PARI kernel — Table 2's seventeen rows, level count 18 353 and level-one 5 953 at 10⁶, k = 3 count 173 on the first 999 primes, C4/§4.5 exception sets; (3) pipeline vs OEIS — 94 sampled b-file rows of A117078 through n = 10 000 and 85 header terms of A117563, 0 mismatches; (4) pipeline vs external tables — π(10⁹), π(2.5·10⁹), maximal gap 282, and the twin anchor weight-3(10⁹) = π₂(10⁹) − 1; (5) pipeline vs the published papers — every printed value of paper Tables 2–6 and challenge Tables 1–4 reproduced under its stated convention, the two ±1 findings diagnosed rather than absorbed. Structural invariants were swept globally, not sampled: Lemma 3.1 (0 disagreements in 5.08·10⁷), Theorem 5 (0 violations), the level bound (max L−g = −1), and the closure of the mod-6 bookkeeping to the unit.

Failure discipline. The three implementation failures of this session (memory kill, time-budget kill, unbounded naive search) are disclosed in the corrections log; none produced output. The rule that caught last session's checkpoint bug — anchor every long computation to an independently known constant before trusting anything downstream of it — caught nothing this session, which is the intended steady state.