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 fourth report adds four things to its predecessor. First — Theorem B. Conjecture 9 is proved: unconditionally, the level-classified primes satisfy \(N_{\mathrm{lev}}(x)\ll x\log\log x/(\log x)^{3/2}\), hence \(f(x)\to0\). The proof (§4.4) conditions on the gap, passes to a normal form \(\ell=L\cdot P\) supplied by the new Cube Lemma, legitimately relaxes consecutiveness — an upper-bound sieve may forget that \(p_{n+1}\) is the immediate successor once each index is counted once — and closes with Selberg's three-dimensional sieve. The statement carries the tag PROVED* — proved in this project, awaiting external refereeing. Second — the Cube Lemma. A level-classified prime with composite weight satisfies \(\ell\le(g-1)^3\), sharp at \(p=131\) where \(\ell=125=(6-1)^3\). Against the maximal-prime-gap record tables this closes Conjecture 4's exception set \(\{13,31,113,131,887\}\) to \(4\cdot10^{18}\) — and, in passing, exposes and repairs an overclaim in the third report's own §4.5 (corrections log, item 8): the quartic bound argued there did not exclude gaps in \((3.8\cdot10^9,4.3\cdot10^{10}]\), the cube bound does. Third — fresh verification. Every number this report reuses was recomputed from scratch this session: full decomposition of the 5 761 452 decomposable primes below 10⁸, level-one census to 2.5·10⁹ with all window constants and generations, a 27 474-point independent sympy audit (zero mismatches), and a new built-in parity audit installed after a sieve bug this session that had silently passed every anchor check (Appendix B — disclosed in full). The PARI/GP and OEIS b-file columns of the audit lattice carry from the third report; PARI is unavailable in this session's environment. Fourth — an upstream erratum. The foothold paper's Lemma 8 proof asserts that the only decomposable primes with \(g\ge\sqrt\ell\) are 13 and 31; the correct set is \(\{5,13,19,23,31,113\}\) (six elements, verified; the lemma's conclusion is unaffected). The submission manuscript is compiled and one administrative pass from submission (§7.3).

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. The scope of freshness in this fourth report: the full decomposition record below 10⁸, the level-one census to 2.5·10⁹, and the generations table were recomputed from scratch this session; statements "verified to 10⁹" cite the third report's audited run, and every such statement that intersects the fresh 10⁸ range was re-checked and agreed exactly. 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: two species, and Theorem B — rarefaction proved

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 — recomputed from scratch this session (July 9) on the corrected sieve, after the parity incident of Appendix B. VERIFIED to 2.5·10⁹, fresh

What Theorem B settles, and what it does not. It settles Conjecture 9 itself (\(f\to0\)) and, of the foothold paper's problem list, Problem 15 (rarefaction) in full; Problem 10 (that \(2p_n-p_{n+1}\) is prime for a vanishing proportion of \(n\)) was already settled at the second report's Species-I bound and is subsumed. It does not reach Problem 13's ideal \(N_1(x)\ll x/\log^2x\) (the exponent gap \(3/2\to2\) stands), says nothing about Problem 14 (the conditional asymptotic) or Problem 16 (constants), and — the sharpest remaining native question — does not decide whether the level class is infinite: density zero is compatible with finiteness, and infinitude contains the balanced primes (C5, C6), a Hardy–Littlewood statement. OPEN — NATIVE (infinitude; rate)

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. The status of this species changes in this report: that \(f_{>1}(x)\to0\) at all is no longer open — it is the \(L>1\) part of Theorem B above — while the finer statements remain exactly as hard as before: the asymptotic shape and the constant \(c_2\) are heuristic, and the infinitude of the species is untouched. PROVED* (that f₍>1₎ → 0) HEURISTIC (shape and constant) OPEN — NATIVE (infinitude)

4.5 The mod-3 comb and the Cube Lemma — native micro-results

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 record-table mechanism closes all three of the framework's finite exception sets to the verification frontier: the composite-weight set above (cube window), the six-element \(g\ge\sqrt\ell\) set of §4.4 (quadratic window, \(p\le g^2+g\)), and the comb set (window \(\ell<3g^2\)). Each future extension of the exhaustive gap tables extends all three closures with no new mathematics.

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

Rows to 10⁸ recomputed fresh this session (July 9); the 10⁹ row carries from the third report's audited run. Challenge Table 1 reproduced (with the diagnosed 10⁸ boundary entry corrected); the 10⁹ row is new relative to the published tables. 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 Cube Lemma seen in the plane

5.6 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. The search was re-run for this fourth report (July 9, 2026) with the same negative result — and the question the literature would have to address has narrowed: the qualitative statement (\(f_{>1}\to0\)) is settled by Theorem B, so what remains unaddressed is the rate, the asymptotic law of divisor-window coverage for the gap-coupled shift. SEARCHED, NEGATIVE (re-run July 9, 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 — status re-checked July 9, 2026), 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. The third report closed this section with a programme: a proof of Species II "will most plausibly proceed stratum-by-stratum in \(L\)", with the uniformity in \(L\le g\) the place the coupling must be broken. That is the shape Theorem B took — gap conditioning, then a fixed-\((L,m)\) triple sieved with uniformity in both parameters — and the coupling was not so much broken as sidestepped, by the observation that upper bounds may forget consecutiveness. What the strategy could not deliver, and what remains, is the asymptotic: there consecutiveness cannot be relaxed, and the coupling stands. PROGRAMME → EXECUTED (upper bound, §4.4) OPEN (asymptotic)

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 verified to 10⁹ (third report, three concordant implementations) and re-verified fresh this session to 10⁸ in full and to 2.5·10⁹ for the census, digit-for-digit. Under the honest ledger the conjecture list now resolves into five reformulations (undiscounted), two retired trivialities, three finite claims with exception sets complete to 4·10¹⁸ — the composite-weight set {13, 31, 113, 131, 887} via the Cube Lemma, the \(g\ge\sqrt\ell\) set {5, 13, 19, 23, 31, 113}, and the comb set {5, 13, 887} — and one native problem, Conjecture 9, now proved: Theorem B gives \(N_{\mathrm{lev}}(x)\ll x\log\log x/(\log x)^{3/2}\) unconditionally, carrying the tag PROVED* until external refereeing completes. The report's native micro-results stand beside it: the Cube Lemma (sharp at 131), the comb proposition, and the sharp statement that the level-one primes with \(6\nmid g\) are exactly {5, 13}. The remaining native frontier is precise: the rate of rarefaction (heuristically \(c_2\log\log x/\log x\)) and the infinitude of the level class, which contains the balanced primes.

7.2 Corrections to propagate upstream

Five 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. New this report: in the foothold paper, Lemma 8's proof asserts that the only decomposable primes with \(g\ge\sqrt\ell\) are {13, 31}; the correct set is {5, 13, 19, 23, 31, 113} — six elements, now closed to 4·10¹⁸ (§4.4). The lemma's conclusion is unaffected, but the proof sentence should be corrected before submission. And the third report's own §4.5 quartic-completeness argument is retracted and replaced by the Cube Lemma (corrections log, item 8) — the project site's mirror of that section should be updated in the same pass.

7.3 Research directions, in order of leverage

1. Submit the manuscript. The submission LaTeX manuscript is compiled to PDF with four main results — the two-species reduction and census, the comb proposition, the Cube Lemma with its 4·10¹⁸ completeness, and Theorem B — three data tables, and a PARI/GP appendix. Remaining before submission, all administrative: author e-mail and affiliation line; page numbers for two references (Koukoulopoulos 2010; Oliveira e Silva–Herzog–Pardi 2014); the acknowledgements placeholder; and the two upstream errata of §7.2 fixed in the cited documents. Venues: Journal of Integer Sequences or Integers as the natural fit; with Theorem B in the paper, Experimental Mathematics and Research in Number Theory are realistic — submit to one venue, sequentially, not simultaneously. This is the highest-leverage step on the table.

2. External refereeing of Theorem B. The asterisk on PROVED* drops only when someone outside the project has checked the argument. The two points a referee should press hardest: the uniformity of the three-dimensional Selberg sieve (Halberstam–Richert, Theorem 5.7) in the parameters \((L,m)\) up to \((\log x)^{O(1)}\), and the bookkeeping of the relaxation step — that each index \(n\) is counted at most once by its triple, and that the sub-cube range \(\ell\le(g-1)^3\) is genuinely \(O((\log x)^{O(1)})\) after gap conditioning. Until that check happens, no claim stronger than PROVED* should appear anywhere in the project.

3. The conditional Species-I asymptotic (foothold Problem 14). 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 remains the most publishable self-contained analytic piece after Theorem B.

4. The rate problem for Species II. With \(f_{>1}\to0\) proved, the open question is the rate: establish \(f_{>1}(x)\asymp\log\log x/\log x\), or any improvement of Theorem B's exponent toward Problem 13's ideal \(x/\log^2x\) on the \(L=1\) stratum. The concrete first target is unchanged and now purely asymptotic: the stratum \(L=3\), i.e. the density of decomposable primes with \(3\mid\ell\) whose \(\ell/3\) is prime, as a Hardy–Littlewood average over the \(6\nmid g\) gap classes.

5. The infinitude frontier (new). Density zero is proved; infinitude of the level class is untouched and is now the framework's sharpest native open problem. It contains the balanced primes (C5: level (1;1)) and the whole generation ladder (C6), so an unconditional proof is Hardy–Littlewood-hard; but the coupling blocks even the conditional route, because lower bounds are exactly where consecutiveness cannot be relaxed (§6.2). Any progress here — even "infinitely many level-classified primes assuming the prime 3-tuple conjecture with Gallagher uniformity" — would be a genuine addition.

6. Computation. The census needs only primality of ℓ: 10¹⁰ costs roughly four times this session's 23-second sieve plus a 35-second pass, and the 625 MB bitmap fits the present 3 GB ceiling — approximately two minutes, extending Fig. D's fresh abscissa by \(\Delta T\approx0.07\). Full classification to 10¹⁰ remains a ~40-minute staged job; 10¹¹–10¹² should abandon exhaustive sweeps for windowed sampling (exact decomposition on random windows of 10⁷ consecutive primes at several magnitudes). Each extension must re-run the full audit battery of Appendix B — including the parity guard — not just the new range.

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: 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 2.5·10⁹ says yes; Theorem B now proves the coverage statement in density; the asymptotic law — that the windows fill at the random-integer rate — remains the bridge's unbuilt span.

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() — re-run and passing in the third report's session and concordant with the Python engine there at every point compared. PARI/GP is not installable in this session's sandbox, so those kernel checks are carried, not fresh (verification strip); the Python↔sympy audit was re-run in full this session. The routines below use the complete convention — every \(p\le X\) is decomposed with its true successor — and a third routine is added for the Cube Lemma.

\\ 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); }

\\ Composite-weight level-classified primes to X (the C4 exceptions), with a
\\ built-in Cube Lemma check: errors out if any member had ell > (g-1)^3.
\\ Returns rows [p, k, L, g, ell]; cubex(10^4) = the five rows of §4.5,
\\ with p = 131 realizing ell = (g-1)^3 exactly.
cubex(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] && !isprime(r[1]),
        my(g = q - p, l = p - g);
        if(l > (g - 1)^3, error("Cube Lemma violated at ", p));
        listput(v, [p, r[1], r[2], g, l])));
    if(q > X, break);
    p = q);
  Vec(v); }

Checks (third report's session): N1(10^6) = 5953, N1(10^8) = 339870, matching the Python census; combx(10^4) = [5, 13, 887]. The new cubex is specification-checked against the fresh Python sweep of this session (five rows, equality at 131) and awaits a PARI-equipped session for kernel execution. The heavy sweeps are executed by the Python pipeline of Appendix B; the PARI routines are the specification and the cross-implementation wherever PARI is available.

Appendix BReproducibility: the verification pipeline

Environment this session: Python 3.12 (numpy 2.4.4, sympy 1.14.0, matplotlib), one CPU core, 3 GB memory ceiling, hard per-call execution budget; PARI/GP is not installable in this session's sandbox, so the PARI column of the audit lattice carries from the third report and is labelled carried, not fresh. The design constraints stand: 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 fresh record, about three and a half minutes.

The audit lattice, this session. Correctness is argued from agreement of unrelated implementations and sources: (1) pipeline vs sympy at 27 474 points — 0 mismatches, fresh; (2) pipeline vs the third report — every value on overlapping ranges reproduced digit-for-digit (census rows, all exception sets, window constants, all twelve generation counts, diagnostics), fresh; (3) pipeline vs external tables — π(10⁸), π(10⁹), π(2.5·10⁹), p₍₅·₁₀⁷₎ = 982 451 653, maximal gaps 220 and 282, and the twin anchor weight-3(10⁸) = 440 311 = π₂(10⁸) − 1, fresh; (4) pipeline vs the published papers — challenge Tables 2–3 reproduced fresh to 2.5·10⁹; (5) the PARI-kernel and OEIS b-file columns — third report, ✓ then, carried (PARI unavailable this session; the b-file comparison was not repeated). Structural invariants were again swept globally on the fresh range, not sampled: Lemma 3.1 (0 disagreements in 5.76·10⁶), 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 parity incident, in full. This session's serious failure is disclosed here at length because of how it failed. The first build of the 2.5·10⁹ sieve computed segment bounds as hi = lo + SEG − 1, which makes consecutive segment starts alternate odd/even. Even-start segments then enumerated even values throughout: every prime p in those ranges was stored as p − 1, and the primality bitmap was written shifted by two. All four anchor checks passed silently. The prime counts at the checked marks are invariant under the p ↦ p − 1 corruption (no checked mark m has m + 1 prime, so searchsorted counts are unchanged); the maximal-gap check takes differences, which the uniform shift preserves within segments; and the sampled p₍₅·₁₀⁷₎ = 982 451 653 happened to fall in a correct odd-start segment. The corruption surfaced only at the cross-check against the third report: N₁(10⁹) read 6 683 486 against the verified 2 708 031. Diagnosis: a parity scan showed half the prime table even beyond index 11 078 937. Fix: forced-odd segment bounds (hi = lo + SEG − 2, SEG even). Guard: a built-in end-of-sieve audit — global parity sweep plus 25 random prime-table entries plus 25 random bitmap slots, all against sympy — now runs unconditionally, so this failure class can never again pass the anchors. The lesson is recorded as a rule: anchor checks at marks are necessary, not sufficient; every bulk artifact gets at least one pointwise independent audit before anything is computed downstream of it. Two lesser failures: an in-place patch of the sieve was written with an unterminated string and no write-back, so one rerun silently re-executed the buggy code — caught within the minute by the unchanged segment-boundary printout, and replaced by asserted whole-file rewrites (the transform scripts of this report assert every replacement); and a figure-layer mathtext incompatibility (\le vs \leq) crashed rendering once, cosmetically. All three are logged as corrections item 9; none reached print.