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 fifth report adds four things to its predecessor. First — the full census to 10¹⁰. Every decomposable prime below 10¹⁰ was classified this session by a fresh, resumable C engine (23 minutes of compute in thirteen checkpointed chunks), quadrupling the fourth report's complete-classification range and computing \(N_{\mathrm{lev}}(10^{10})=71\,670\,808\) for the first time: \(f(10^{10})=0.1575\), falling in every decade, with the challenge paper's Tables 2–3 reproduced entry for entry and the first-occurrence gap ladder recovered in-house. Second — Theorem C. The Species-I density constant is identified (Species I = the level-one stratum \(L=1\); Species II = level-classified with \(L>1\); definition box, §4.4): under a Hardy–Littlewood hypothesis for the triples \((n,\,n+g,\,n+2g)\) with the standard uniformity, and the Gallagher-type gap model, \(N_1(x)\sim 2C_2\,x/\log^2 x\) with \(2C_2=1.3203236316\ldots\) twice the twin-prime constant. The identity underneath it — the line average of the triple singular series equals \(2C_2\) — is proved unconditionally (Lemma C), and the machinery is verified gap by gap against the fresh data at 10⁹ and 10¹⁰: observed/predicted within 0.97–1.05 per gap, aggregate +1.1%, and the measured window-constant ladder \(\hat c_1 = 1.028,\,1.056,\,1.064,\,1.068\) reproduced by the finite-height model to 1.2% — resolving the ladder-versus-limit puzzle of the earlier reports: the deficit from 1.32 is the finite-height gap mix, quantitatively. Third — a retraction. The claim \(c_2=1\) for the Species-II constant, entertained between the fourth report and this one, is withdrawn: the cumulative ratio \(f_{>1}\log x/\log\log x\) falls monotonically through 0.7999 at 10¹⁰ — away from 1 — while the windowed constant sits flat near 0.773 (corrections log, item 11). In its place, a clean measured invariant: the Mertens-normalized Species-II ratio is flat at \(R_2\approx0.441\) across seven orders of magnitude. Fourth — the audit layer restored and extended. PARI/GP returns to the pipeline (fordiv kernel, BPSW, kernel-level Cube-Lemma census), the OEIS term diff is refreshed, and the Gafni–Tao comparison is computed at 5·10⁶: 23 944 level-one primes, 66 784 rough-free exceptional gaps, only 2 763 common — the two thin sets are neighbours, not the same set.

Everything else stands and is carried with fresh verification: Theorem B (Conjecture 9 proved unconditionally, PROVED* pending external refereeing), the Cube Lemma with its closure to 4·10¹⁸, Theorem 5's rigidity, and the honest boundary: Conjectures 1–3 and 5–6 are reformulations of classical open problems and are not made easier by the coordinates.

Reading the ledger

Every claim in this report carries one of six tags. PROVED — a complete proof exists (here or in the cited source). PROVED* — proved in this project, awaiting external refereeing; the asterisk drops only on an outside referee's check. VERIFIED — exhaustively checked by computation over a stated finite range this session. HEURISTIC — supported by a model and by data, not by proof. CONDITIONAL — proved under a named standard hypothesis. REFORMULATION — equivalent to a classical open problem; the coordinates change the view, not the difficulty. OPEN — open; OPEN — NATIVE marks the questions the framework itself creates. Retractions and errata live in the corrections log below and are never silently overwritten.

Contents

Section 1Significance of the framework

The decomposition into weight × level + jump is a coordinate change. It takes a strictly increasing integer sequence — an additive object, known through its gaps — and attaches to each term a multiplicative pair: the weight \(k(n)\), a distinguished divisor of the shifted term \(\ell(n)=a(n)-d(n)\), and the level \(L(n)=\ell(n)/k(n)\), its cofactor. The gap \(d(n)\) is what the sequence does next; the pair \((k,L)\) is what the arithmetic of \(\ell=2a(n)-a(n{+}1)\) permits. The construction is exact — \(a(n)=k(n)L(n)+d(n)\) whenever it exists — and canonical: no parameter is chosen, no rounding occurs, and the weight is pinned by a minimality condition (the smallest divisor of \(\ell\) exceeding the jump) that makes the entire map a function of the sequence alone.

Three facts give the construction its claim to significance, and it is worth stating them with their tags up front. (i) On the natural numbers it is not an analogy to the sieve of Eratosthenes but a strict restatement of it: the weight of \(n\) is the smallest prime factor of \(n-1\), so the columns of the weight histogram are the sieve's arithmetic progressions, and the level-classified naturals are exactly the shifted primes \(\{q+1\}\) (PROVED, §3; re-verified this session with zero violations to 10⁶). (ii) On the primes it turns several classical objects into visible strata: lesser twin primes are literally the column \(k=3\) (PROVED, Lemma 3.1; swept with zero disagreements over all 455 052 508 decomposable primes to 10¹⁰ this session), balanced primes are the stratum level (1;1), and Polignac families are weight columns. (iii) It generates exactly one problem that does not exist without it — Conjecture 9, the rarefaction of the level class — and that problem turned out to be decidable: Theorem B (fourth report, carried here with the tag PROVED*) gives \(N_{\mathrm{lev}}(x)\ll x\log\log x/(\log x)^{3/2}\) unconditionally, and this report adds the conditional constant for its dominant species (Theorem C, §4.4).

Honesty about the converse is part of the framework's discipline. A coordinate change does not lower the difficulty of what it displays. Conjecture 1 (infinitely many weight-3 primes) is the twin prime conjecture; Conjectures 2 and 3 are Polignac- and Hardy–Littlewood-type families; Conjectures 5 and 6 contain the infinitude of balanced primes. The framework's value is not that it makes these easier — it does not — but that it (a) organizes them into one two-parameter family with a common mechanism, the location of divisors of \(2p_n-p_{n+1}\) relative to the gap; (b) produces native, decidable statements at the family's edge (Theorem B, the Cube Lemma, the rigidity Theorem 5); and (c) supplies a measurement apparatus — the census machinery of this report — precise enough to identify constants (2C₂, §4.4) and to retract wrong ones (item 11).

The atlas dimension matters too. The construction applies uniformly to any strictly increasing sequence; the project maintains decompositions of more than a thousand OEIS sequences at decompwlj.com, with the prime case (A117078 weight, A117563 level, A001223 jump, A118534 ℓ) as its deepest instance. That breadth is what justifies the word framework: the primes are one fibre of a general additive–multiplicative correspondence, and §3's collapse onto Eratosthenes shows the base case is classical.

Section 2Definitions, existence, classification

Fix a strictly increasing integer sequence \(a(1)

jump (gap): \(d(n)=a(n{+}1)-a(n)\);
shifted term: \(\ell(n)=a(n)-d(n)=2a(n)-a(n{+}1)\) if \(a(n)-d(n)>d(n)\), else \(\ell(n)=0\);
weight: \(k(n)=\) the smallest integer \(k>d(n)\) dividing \(\ell(n)\), if \(\ell(n)\ne0\); else 0;
level: \(L(n)=\ell(n)/k(n)\) if \(k(n)\ne0\); else 0;
decomposition: \(a(n)=k(n)\,L(n)+d(n)\) = weight × level + jump.

Classification. A decomposable term is classified by level if \(k(n)>L(n)\) — equivalently \(k>\sqrt\ell\), equivalently \(\ell\) has no divisor in \((d,\sqrt\ell\,]\) — and by weight if \(0

2.1 The first seventeen primes, recomputed

Table 2 of the paper, regenerated independently by the C engine and by the PARI fordiv kernel (identical output, and the kernel's selftest 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 makes precise, 887 is the unique level-classified prime below 10¹⁰ with \(3\mid\ell\), \(L>1\) and \(3\nmid L\) (verified over the full fresh range this session), 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 base case is the sequence of natural numbers: \(a(n)=n\), \(d(n)=1\), \(\ell(n)=n-1\). The weight is then the smallest divisor of \(n-1\) exceeding 1 — that is, the smallest prime factor of \(n-1\) — and the fundamental theorem of arithmetic enters through the cofactor: \(n-1=\mathrm{spf}(n-1)\cdot L(n)\) is the first step of the full factorization, applied recursively down the level. In OEIS terms, \(A000027(n)=A020639(n-1)\times A032742(n-1)+1\).

This is the precise sense in which the framework "generalizes the sieve of Eratosthenes". On the naturals, the (weight, level) plane is the sieve: columns are the sieving classes, the level-1 ray at the bottom is the primes (shifted), and the diagonal frontier \(k=L\) is the square-root cutoff at which trial division stops. The author's figures sieveNb.jpg and naturaldecomp3M2048.jpg (§5.3–5.4) render exactly this, and the fresh audit above confirms the underlying data law rather than the picture alone. On a general sequence, the jump \(d(n)\) replaces the constant 1, and the sieve question "smallest prime factor" deforms into "smallest divisor beyond the gap" — the object whose statistics Sections 4–6 measure on the primes.

Two remarks keep the analogy honest. First, the collapse is exact only for the naturals; for the primes the window \((g,\sqrt\ell\,]\) has variable lower edge, and it is that variability — the coupling of a multiplicative window to an additive gap — that generates all the difficulty downstream. Second, the FTA connection is a statement about one factorization step, not about unique factorization being reproved; nothing in the framework re-derives FTA, and we do not claim it does.

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

Applied to the primes \(p_n\) with gap \(g_n=p_{n+1}-p_n\): the shifted term is \(\ell(n)=p_n-g_n=2p_n-p_{n+1}\) (A118534), the weight \(k(n)\) is A117078, the level \(L(n)\) is A117563. A prime is decomposable iff \(p_{n+1}<\tfrac32 p_n\); by Bertrand–Nagura-type bounds the non-decomposable primes are exactly {2, 3, 7}, a fact the fresh sweep confirms: decomposable(10¹⁰) = 455 052 508 = π(10¹⁰) − 3.

4.1 The conjecture ledger

4.2 Weight 3 is the twin column

Verified as an identity of arrays: over all 455 052 508 decomposable primes to 10¹⁰ the predicate \(k=3\) agrees with the predicate \(g=2\wedge p>3\) with zero disagreements, and the common count is 27 412 678 — equal to \(\pi_2(10^{10})-1=27\,412\,679-1\) 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 to 10¹⁰, externally anchored

4.3 Mod-3 rigidity and its sharp corollaries

The theorem cuts the decomposable primes into a pool \(6\mid g\) (45.00 % of them at 10¹⁰; 44.51 % 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, each now closed at 10¹⁰:

(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 10¹⁰ (22 083 606 of the 22 083 608 level-one primes have \(6\mid g\)). PROVED (given §4.4's exception sets) VERIFIED to 10¹⁰

(ii) The out-of-pool level class is exactly the 3∣ℓ level class. At 10¹⁰ the level-classified primes with \(6\nmid g\) number 30 212 404 = 30 212 402 (the level->1 primes with \(3\mid\ell\)) + 2 (the primes 5 and 13). Theorem 5's bookkeeping closes to the unit across 71.7 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). The pool advantage persists but dilutes with height: 0.1196 at 10⁹, 0.1079 at 10¹⁰ — the rarefaction of §4.4 seen from inside the pool. VERIFIED

4.4 Conjecture 9: two species, Theorem B, and the new Theorem C

Write \(f(x)\), \(f_1(x)\), \(f_{>1}(x)\) for the shares of decomposable primes \(\le x\) that are level-classified, level-one (Species I), and level-greater-than-one (Species II). 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. All nine printed values of the challenge paper's Tables 2–3 at and beyond 10⁸ (339 870; 2 708 031; 6 227 807; 11 719 154; 22 083 608; window constants 1.028, 1.056, 1.064, 1.068) are reproduced exactly — and, for the two highest rows and two highest windows, recomputed in-house for the first time: the fourth report's own sweep stopped at 2.5·10⁹. VERIFIED to 10¹⁰, 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 (full rarefaction); Problem 10 (density zero of \(\{2p_n-p_{n+1}\ \text{prime}\}\)) was already settled at the 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). Problem 14 — the conditional asymptotic with an explicit constant — is the subject of the new Theorem C below. It says nothing unconditional about Problem 16, 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).

Theorem C — the Species-I constant (new this report)

The foothold paper's Problem 14 asks for the conditional asymptotic of \(N_1(x)\) with an explicit constant. The Species-I reduction above makes \(N_1\) a count of prime triples in arithmetic progression whose right gap is a true prime gap: \(\ell=p_n-g,\;p_n,\;p_{n+1}=p_n+g\). Two ingredients separate cleanly — the triple's singular series, and the distribution of consecutive-prime gaps — and the constant comes out of an exact average of the first against the second.

Verification against the fresh census — the machinery, gap by gap. The model's gap-conditional prediction needs no gap model at all if one feeds it the observed gap counts: predicted level-one at gap \(g\) is \((\mathfrak S_3/\mathfrak S_2)(g)\cdot\sum_{\text{pairs, gap }g}1/\log p\), computed from the engine's per-gap \(\sum1/\log p\) arrays. Against the observed ℓ-prime counts:

Per-gap agreement 0.97–1.05 at both heights; the aggregate sits +1.1–1.2% above prediction, stably across a full decade of \(x\). The per-gap wiggle (g = 36 low, g = 12, 24, 30, 42 high) is reproducible between 10⁹ and 10¹⁰ and is a genuine second-order structure of consecutive-gap triples, not noise. VERIFIED

The ladder reconciled with the limit. The measured window constants \(\hat c_1\) climb 0.8431, 0.9230, 0.9324, 0.9719, 1.0024, 1.0280, 1.0561, 1.0636, 1.0680 across the nine windows to 10¹⁰ — far from \(2C_2=1.3203\). Feeding the finite-height machinery (observed gap mix × \(\mathfrak S_3/\mathfrak S_2\)) through the same windows predicts 1.0158, 1.0434, 1.0513, 1.0566 for the last four; measured/model = 1.0120, 1.0121, 1.0118, 1.0108. So the model — whose infinite-height limit is exactly \(2C_2\) by Lemma C — reproduces the ladder to 1.2%, and the deficit 1.068 → 1.320 is entirely the finite-height gap mix: at \(\lambda=\log x\approx23\) the exponential weight \(e^{-g/\lambda}\) has barely begun to feel the large \(6\mid g\) gaps whose \((q-1)/(q-3)\) enhancements carry the average up to \(2C_2\). The convergence is of order \(1/\log x\) with oscillation — precisely why the earlier reports' finite-range fits (1.09–1.12 cumulative, 1.03–1.07 windowed) could not identify the constant by extrapolation, and why the identification had to come from an identity. The remaining flat +1.1% is the second-order consecutiveness correlation not captured by the pair-to-triple conditional; it is disclosed, not absorbed. VERIFIED (model reconciliation); CONDITIONAL (the limit itself).

Generations inside level one

Within level one, generation \(i\) means \(\ell(n)=p_{n-i}\) (so generation 1 is the balanced primes, C5). At index scale \(n\le5\cdot10^7\) (so \(p\le982\,451\,653\)), the fresh engine reproduces all twelve counts of the paper and the fourth report exactly: 1 307 356; 746 381; 345 506; 153 537; 65 497; 27 288; 11 313; 4 695; 1 916; 791; 319; 126 — successive ratios 0.571, 0.463, 0.444, 0.427, 0.417, 0.415, 0.415, 0.408, 0.413, 0.403, 0.395. The drift of the ratio (0.57 → 0.40 rather than a fixed geometric rate) is the visible signature of the gap-mix transient of Theorem C operating stratum by stratum. VERIFIED, fresh

Species II — the measured constants, and a retraction

Species II (level > 1) is governed not by prime pairs but by the divisor structure of \(\ell\) in the window \((g,\sqrt\ell\,]\) — Ford–Tenenbaum territory. Three fresh measurements to 10¹⁰, and one withdrawal:

(i) The claim \(c_2=1\) is retracted (corrections log, item 11). The cumulative ratio \(f_{>1}\log x/\log\log x\) falls monotonically 0.8312 → 0.7999 across 10⁶ → 10¹⁰; a claimed limit of 1 that every fresh decade recedes from is not a limit.

(ii) The windowed constant is flat at 0.773. \(\hat c_2=\Delta\text{share}\cdot\log\sqrt{ab}/\log\log\sqrt{ab}\) per window reads 0.7425, 0.8071, 0.7690, 0.7705, 0.7669, 0.7656, 0.7728, 0.7730, 0.7726 — the last three windows, spanning 10⁹ → 10¹⁰, agree to ±0.0004. The honest statement is a measured band \(\hat c_2=0.77\pm0.01\) with the true constant and even the exactness of the \(\log\log x/\log x\) rate OPEN; the doubly-logarithmic abscissa spans only 2.08 → 3.14 across seven orders of magnitude, so slope and secondary terms remain entangled (Legendre's problem in miniature — the standing caution of these reports). VERIFIED (measurement) HEURISTIC (rate)

(iii) A new flat invariant: the Mertens-normalized ratio. Define the prime-factor model mass \(M(x)=\sum 2\log g_n/\log\ell_n\) over decomposable primes — the Mertens-product probability that a random integer of size \(\ell\) has no prime factor in \((g,\sqrt\ell\,]\). Since "no divisor in the window" implies "no prime factor in the window", \(M\) is a model upper mass for the whole level class, and the measured ratio $$R_2(x)=\frac{N_{\mathrm{lev}}(x)-N_1(x)}{M(x)}$$ is strikingly flat: windowed 0.4272, 0.4655, 0.4445, 0.4458, 0.4432, 0.4420, 0.4414, 0.4411, 0.4408; cumulative 0.4412 at 10¹⁰ — stable to three decimals over the last five windows. Read as a conditional probability, \(R_2\approx0.441\) is the empirical chance that a window free of prime factors is also free of composite divisors, under the arithmetic of \(\ell=2p_n-p_{n+1}\). We do not identify this constant with any closed form; candidate identifications through Ford's \(H(x,y,z)\) theory (Annals 168 (2008)) are the natural next target and are left OPEN. VERIFIED (measurement, new this report)

Infinitude of Species II — and of the level class — remains the framework's native open problem. Theorem B caps it; nothing yet floors it. OPEN — NATIVE

4.5 The mod-3 comb and the Cube Lemma

Completeness to 4·10¹⁸. A composite-weight level prime beyond the sweep needs \(p=\ell+g\le(g-1)^3+g\). Against the exhaustively verified first-occurrence gap tables (Oliveira e Silva–Herzog–Pardi to 4·10¹⁸, maximal gap 1 476; our own engine reproduces the ladder to 10¹⁰, last records 320, 336, 354), every gap value \(g\) first occurs at a starting prime vastly exceeding \((g-1)^3+g\) once \(g\ge22\); the finitely many small-\(g\) possibilities lie inside the swept range. Hence the exception set of Conjecture 4 is exactly {13, 31, 113, 131, 887} for all \(p\le4\cdot10^{18}\). This session adds an independent execution: the census predicate ran at kernel level in PARI/GP (Appendix A, cubex) over all primes to 10⁷ and returned exactly the five rows above. VERIFIED; complete to 4·10¹⁸

4.6 Conventions, small errata, and the boundary cases

Four bookkeeping facts that repeatedly matter. (i) The nine-prime list. The level-classified primes with weight ≤ 23 are exactly nine: 5 (3;1), 13 (9;1), 19 (5;3), 23 (17;1), 37 (11;3), 43 (13;3), 61 (11;5), 181 (19;9), 199 (17;11) — recomputed fresh; this multiset is the +9 reconciliation of the paper's Table 3 (corrections item 5). (ii) The twin −1 and −2. Total weight-3 primes to \(x\) equal \(\pi_2(x)-1\) (drop \(p=3\)); the weight-classified weight-3 count is \(\pi_2(x)-2\) (drop also the level-classified \(p=5\)): at 10¹⁰, 27 412 678 and 27 412 677. (iii) The boundary prime. Complete convention decomposes every \(p\le x\) by its true successor even when \(p_{n+1}>x\); the convention difference is the single prime 99 999 989 at \(x=10^8\) (upstream Table 1 erratum, item 6). (iv) Foothold Lemma 8's exceptional set — erratum queued upstream. The published lemma names {13, 31} as the exceptional set of the level-one reduction; {13, 31} is correct for the biconditional failures (level-one with \(\ell\) composite), but the lemma's stated window condition \(g\ge\sqrt\ell\) has the six-element set {5, 13, 19, 23, 31, 113} — recomputed exhaustively this session (window log: exactly six lines). The fix is one sentence distinguishing the two sets; it does not affect any count downstream. VERIFIED

4.7 The verified record

Every row computed from scratch this session under the complete convention; the first five columns of the 10⁶–2.5·10⁹ rows agree digit-for-digit with the fourth report; the 5·10⁹ and 10¹⁰ rows are first-time complete censuses. Diagnostics: windowed level share by decade 0.2969, 0.2712, 0.2278, 0.2042, 0.1846, 0.1689, 0.1558 — decreasing in every window; at index scale \(n\le5\cdot10^7\) the split is 8 553 468 level (17.107 %) / 41 446 529 weight, the paper's Table 6 to its printed digits; tie primes to 10⁶: exactly 12 (= A121155); maximal-gap ladder to 10¹⁰ equal to A002386 with final records 320 after 2 300 942 549, 336 after 3 842 610 773, 354 after 4 302 407 359. 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 this session's verified arrays — and then present ten figures regenerated from the fresh 10¹⁰ data. Every structural claim made about a figure below was checked numerically, not visually. All session canvases are square with equal aspect on weight–level planes, so the ratio between the two coordinates is preserved by construction.

5.1 The author's classification map

5.2 The sieve rendered

5.3 Three million naturals

5.4 The two wings, fresh

5.5 Conjecture 9 measured

5.6 The ĉ₁ ladder and its explanation

5.7 Gap-by-gap verification

5.8 The mod-3 comb on the level side

5.9 The weight columns

5.10 The gap-conditional law

5.11 The Cube Lemma plane

5.12 Species II: constants and the retraction, drawn

5.13 Level-one against Gafni–Tao

Section 6Hypothetical impacts: the additive–multiplicative bridge

6.1 The coupling scale — and what the framework does not touch

Everything the decomposition sees happens at gap scale. The object \(\ell=2p_n-p_{n+1}\) couples two consecutive primes; its distance from \(p_n\) is one gap, of typical size \(\log x\). The classification question — does \(\ell\) have a divisor in \((g,\sqrt\ell\,]\)? — is a multiplicative question asked at an additively determined point. That is the bridge, stated exactly: the framework transports gap statistics (Gallagher-type, scale \(\log x\)) into divisor statistics (Ford–Tenenbaum-type, scale \(\log\ell/\log g\)) and back. This also fixes what it cannot do. Riemann-scale phenomena — error terms of size \(\sqrt x\), zero distribution, explicit formulae — live at a completely different scale, and nothing in this report bears on them; we decline any suggestion that the coordinates illuminate RH. Likewise Goldbach-type binary problems: the decomposition never couples two independent primes, only consecutive ones. The precise obstruction the framework does isolate is the consecutive-prime coupling: \(\ell\) depends on \(p_n\) and \(p_{n+1}\) jointly, which is why Theorem B's upper bound goes through (consecutiveness can be relaxed upward) while the matching lower bound and the unconditional asymptotic do not.

6.2 The right analytic neighbourhood: divisors of shifted primes

Species II asks how often the shifted prime \(\ell=p_n-g_n\) avoids divisors in \((g,\sqrt\ell\,]\). The classical home of such questions is the theory of divisors of shifted primes — Ford's \(H(x,y,z)\) (Annals of Mathematics 168 (2008), 367–433) for divisors of integers in intervals, and Koukoulopoulos (Divisors of shifted primes, IMRN 2010, no. 24, 4585–4627) for the shifted-prime case — with the standing complication that here the shift is the gap itself, a moving target correlated with \(p_n\). Two quantitative outputs of this report sit directly in that neighbourhood: the flat Mertens-normalized ratio \(R_2\approx0.441\) (§4.4), which any \(H(x,y,z)\)-type refinement of the naive Mertens model must reproduce, and the flat windowed \(\hat c_2\approx0.773\), which fixes the normalization of a conjectural \(c_2\log\log x/\log x\) density. Identifying either constant in closed form through that machinery is, in our judgement, the most tractable open analytic problem the framework currently poses. OPEN

6.3 Level-one primes and Gafni–Tao's rough-free gaps

Gafni and Tao (arXiv:2508.06463) study prime gaps containing no \(g_n\)-rough integer — no \(m\in(p_n,p_{n+1})\) with least prime factor \(\ge g_n\) — and show such gaps are rare. The level-one condition is superficially cousin: both are thin sets defined by smallest-prime-factor thresholds at gap scale. The fresh computation to 5·10⁶ makes the relationship exact: 23 944 level-one primes, 66 784 GT-exceptional gaps (degenerate gap (2, 3) excluded), overlap 2 763 — about 11.5 % of one set, 4.1 % of the other. Neither contains the other, and the witnesses show why: level-one constrains the exterior shifted point \(\ell=p_n-g\) (47 qualifies though its gap interior contains the 6-rough 49), while GT constrains the interior integers (7 qualifies though it is not even decomposable). The two phenomena share a scale and a flavour but measure different arithmetic; the framework's contribution here is a precisely defined third object — \(\ell\)'s divisor window — that completes a small taxonomy of gap-scale roughness conditions. VERIFIED

6.4 Impacts claimed, impacts declined

Claimed. (1) A canonical coordinate system in which twin, Polignac, balanced-prime and Hardy–Littlewood families are strata of one two-parameter object, with the sieve of Eratosthenes as its exactly solved base case. (2) Native decidable structure at the family's edge: Theorem 5's rigidity, the Cube Lemma with a completely determined exception set to 4·10¹⁸, and Theorem B — a classical-methods proof of the framework's own Conjecture 9. (3) A measurement apparatus sharp enough to identify a constant conditionally (Theorem C: \(c_1=2C_2\), verified at the 1% level), to expose two new stable empirical constants (0.773, 0.441), and to retract a wrong claim on evidence (item 11). Declined. No claim that any classical open problem — twin primes, Polignac, Hardy–Littlewood tuples, balanced primes, Goldbach, RH — is made easier; C1–C3, C5–C6 remain reformulations of exactly their classical difficulty. No claim that FTA is re-derived. No unconditional asymptotic for \(N_1\); no lower bound for the level class; no statement about infinitude. The framework's honest product is organization, native theorems at the edge, and measurements precise enough to be falsified — as item 11 demonstrates the discipline works.

Section 7Conclusion and future research

7.1 What this report establishes

The fifth report completes the project's first full classification of every decomposable prime to 10¹⁰ — 455 052 508 decompositions, N_lev = 71 670 808, f = 0.1575 — with every published anchor reproduced and the challenge paper's tables now recomputed rather than cited. On that data it adds Theorem C: the Species-I constant is \(c_1=2C_2\), twice the twin-prime constant, conditional on standard triple and gap hypotheses, resting on an unconditionally proved line-average identity (Lemma C) and verified against the census gap-by-gap to within 1.1 % — in the process explaining, quantitatively, why the measured window ladder 1.028 → 1.068 sits where it does below the limit 1.320. It withdraws the interim claim \(c_2=1\) on the strength of the same data, replacing it with two flat measured constants (\(\hat c_2\approx0.773\), \(R_2\approx0.441\)) offered to the divisor-distribution literature for identification. And it restores the full audit lattice: PARI/GP kernel re-derivation (113 733 rows, zero mismatches), kernel-level Cube-Lemma census, sympy divisor audit (45 000 rows, zero mismatches), OEIS term diffs (zero differences), and the Gafni–Tao comparison. Theorem B — Conjecture 9 proved — is carried unchanged at PROVED*, one external referee away from losing its asterisk.

7.2 Upstream errata queue

Three items are queued for propagation to the published record, none affecting any downstream count. (1) Foothold paper, Lemma 8: the exceptional set of the stated window condition \(g\ge\sqrt\ell\) is the six-element {5, 13, 19, 23, 31, 113}; the printed {13, 31} is the set of biconditional failures — one distinguishing sentence fixes it (§4.6 (iv)). (2) Challenge paper, Table 1: lev1(10⁸) = 339 870 under the complete convention; the printed 339 869 omits the boundary prime 99 999 989 (corrections item 6). (3) Manuscript bibliography: the two previously open page-number items are closed — Koukoulopoulos, IMRN 2010, no. 24, 4585–4627; Oliveira e Silva–Herzog–Pardi, Math. Comp. 83 (2014), no. 288, 2033–2060. The OEIS project page already records C7–C8 as true; no action needed there beyond keeping their status out of the contributions column.

7.3 Research directions, ordered by leverage

First, submission. The Theorem B manuscript (with the Cube Lemma and the census apparatus) is compiled and clean; the remaining TODOs are administrative (author affiliation and contact; the two bibliography closures above are done). Journal of Integer Sequences or Integers first, Experimental Mathematics or Research in Number Theory as alternatives — refereeing is the single action that converts PROVED* to PROVED and is worth more than any further computation. Second, harden Theorem C. Two concrete upgrades: replace hypothesis (H2) by a Gallagher-uniform tuples hypothesis alone (folding the gap model into (H1)), and bound — or at least model — the stable +1.1 % consecutiveness correlation, which the gap-by-gap data now pins down as reproducible structure rather than noise. A realistic intermediate "real result": the conditional asymptotic with explicit constant, written to referee standard, resolving foothold Problem 14. Third, Species II. Attack \(R_2\approx0.441\) and \(\hat c_2\approx0.773\) through Ford's \(H(x,y,z)\) and Koukoulopoulos' shifted-prime divisor machinery; even a conditional identification of either constant would be the framework's second identified constant. Fourth, extend the census to 10¹¹–10¹² — the doubly-logarithmic abscissa gains only ≈ 0.10 per decade, but the window constants' error bars shrink linearly with population, and the engine's architecture (budgeted, checkpointed, resumable) was built for exactly this extension. Fifth, the atlas. Run the two-species analysis on other atlas sequences (Fibonacci-like, polynomial, lucky numbers) to test which parts of §4 are prime-specific and which are consequences of gap regularity alone — the sharpest available probe of what the framework itself contributes.

Appendix APARI/GP algorithms

PARI/GP 2.15.4 returned to the pipeline this session. The reference kernel is the project's fordiv one-liner — early return on the first divisor beyond the jump, which is exactly the weight's minimality condition:

decomp(a,b) = {
  my(d = b - a, l);
  if(a <= 2*d, return([0, 0, d]));      \\ not decomposable: l = a-d <= d
  l = a - d;
  fordiv(l, k, if(k > d, return([k, l/k, d])))   \\ [weight, level, jump]
}

Driving it over consecutive primes uses forprime with a one-prime lag, the fast idiom for anything gap-indexed:

\\ census / predicate sweep pattern, p <= X
{ my(prev = 0);
  forprime(q = 2, X + 400,
    if(prev && prev <= X,
      my(r = decomp(prev, q));
      \\ ... test or tally r = [k, L, d] here ...
    );
    prev = q); }

cubex — the Cube-Lemma census at kernel level. Run this session over all primes to 10⁷; output was exactly the five rows of §4.5's table:

{ my(prev = 0, n = 0);
  forprime(q = 2, 10^7 + 300,
    if(prev && prev <= 10^7,
      my(r = decomp(prev, q));
      if(r[1] > 0 && r[1] > r[2] && !isprime(r[1]),   \\ level-classified, composite weight
        printf("p=%d g=%d k=%d L=%d\n", prev, r[3], r[1], r[2]); n++));
    prev = q);
  printf("composite-weight level primes to 1e7: %d\n", n); }
\\ -> 13, 31, 113, 131, 887 ; total 5

xrows — batch re-derivation protocol. The C engine samples every 4 001st decomposable prime (113 734 rows to 10¹⁰). A generated GP file replays each row through the kernel and compares, adding a BPSW layer (ispseudoprime) on \(p\), its successor, and — for level-one rows other than 13 and 31 — on \(\ell\):

chk(p, g, k, L) = {
  my(r = decomp(p, p + g)); nn++;
  if(r[1] != k || r[2] != L, mm++; printf("MISMATCH p=%d\n", p));
  if(!ispseudoprime(p) || !ispseudoprime(p + g), bp++);
  if(L == 1 && k > L && p != 13 && p != 31,
     if(!ispseudoprime(p - g), lp++));
}
\\ session result: rows=113733  mismatches=0  badprime=0  badlev1ell=0

The reader executed 113 733 of the 113 734 generated lines (one swallowed; corrections item 12(c)); the selftest block — the first-seventeen table of §2.1 — ran identically in the kernel and in the C engine. The historical naive and sieve algorithms of the project's algos.txt, and the optimized variants of decompwlj_optimized.txt, remain the reference lineage; bulk-range work this session moved to the C engine of Appendix B, with PARI retained as the independent adjudicator.

Appendix BReproducibility: the verification pipeline

The engine. A single-file C program (gcc -O3), architected for the platform's per-call execution ceiling rather than against it. Segmented sieve of Eratosthenes over odd numbers, segment span 2²⁵ with a 2 MiB packed bitmap; base primes to 100 080 (9 592 of them, enough for \(\sqrt{10^{10}+402}\)); prime extraction by 64-bit word complement and count-trailing-zeros, so the scan touches words, not bits. Each prime found closes the classification of its predecessor — the complete convention by construction, since every \(p\le x\) meets its true successor before any snapshot at \(x\) fires. Per decomposable prime: trial division of \(\ell\) by the 25 primes to 101, then deterministic Miller–Rabin on the cofactor with witness set {2, 3, 5, 7, 11, 13} — valid for all \(n<3\,474\,749\,660\,383\), a 340-fold margin over the largest \(\ell\) — with Brent–Pollard rho for the rare surviving composites; full divisor enumeration from the factorization; the weight as the least divisor exceeding the gap. A 4 096-entry ring buffer of recent primes resolves level-one generations \(\ell=p_{n-i}\) by direct lookback. Per-gap arrays to \(g<512\) accumulate counts, level-one counts, and \(\sum1/\log p\) — the last being what makes Theorem C's gap-by-gap test possible without a second pass.

Budget, checkpoints, resume. The engine takes a wall-clock budget as its argument, processes whole segments until the budget expires, and exits after an atomic state rewrite (write-temp-then-rename); mid-run it also checkpoints every four segments, so a kill costs at most a few seconds of work and never corrupts state. Log files open in append mode on resume; snapshot marks re-fire idempotently. The production sweep ran as thirteen budgeted chunks — 59 s, then 103–117 s each — totalling ≈ 23 minutes of compute for the full 10¹⁰ classification, plus a sub-second finishing call. Instrumentation written along the way: the every-4 001st sample (113 734 rows, zero duplicates), the complete decomposition table to 10⁶ (78 498 rows), and six special logs (window set, composite-ℓ level-one, composite-weight, small-weight, comb, gap records) — each of which landed exactly on its expected set, which is itself a distributed integrity check.

The audit lattice, this session. Correctness is argued from agreement of unrelated implementations and sources, never from one program's say-so. (1) Engine vs external anchors: π at every mark to 10¹⁰; π₂(10⁹), π₂(10¹⁰); \(p_{5\cdot10^7}=982\,451\,653\); the complete A002386 first-occurrence ladder including 354 after 4 302 407 359 — all exact. (2) Engine vs the published record: challenge Tables 2–3, every printed row and window; the paper's Tables 2, 3 and 6 under their stated conventions; the fourth report's values digit-for-digit wherever ranges overlap — census rows, all exception sets, window constants, all twelve generation counts. (3) Engine vs PARI: the 113 733-row fordiv re-derivation with zero mismatches and zero BPSW failures; the kernel-level cubex census returning exactly the five. (4) Engine vs sympy: 45 000 sampled rows re-derived by independent divisor enumeration, zero mismatches; the naturals identity \(k(n)=\mathrm{spf}(n-1)\) to 10⁶ with zero violations and the level class landing exactly on π(10⁶). (5) Engine vs OEIS: term diffs on A117078 (64 terms), A117563 (33), A121155 (12), A162175 (21) — zero differences. Structural invariants were swept globally on the full range, not sampled: Theorem 5 (zero violations in 455 052 508), the level bound (max \(L-g=-1\)), and the mod-6 bookkeeping closing to the unit at every mark.

Failure discipline — this session, in full (corrections item 12). (a) The first 10¹⁰ attempt ran unbudgeted and was killed by the per-call ceiling before its end-of-run save; because saving happened only at completion, nothing survived. The design response — budget argument, four-segment checkpoints, atomic renames — is what the paragraph above describes; the failure produced the architecture. (b) The complete table to 10⁶ was silently truncated at 10⁴ rows by a generic log cap on first build; the row count against π(10⁶) = 78 498 caught it before any downstream use, the cap was raised, and the range re-run from scratch. The rule stands from the fourth report: every bulk artifact gets at least one pointwise or cardinality audit before anything is computed from it. (c) The PARI batch executed 113 733 of 113 734 generated checks — one line swallowed by the file reader. Rather than silently rounding the claim up, the count is disclosed as-is; the sympy layer independently covers the sampled space. None of the three reached print.

Decomposition into weight × level + jump — deep analysis, fifth report · July 13, 2026 · framework and data: Rémi Eismann (arXiv:0711.0865 · decompwlj.com · OEIS A117078, A117563, A001223, A118534) · verification, analysis and text: the project's review team, this session · census engine: C, 455 052 508 decompositions in ≈ 23 minutes · adjudicators: PARI/GP 2.15.4, sympy, OEIS, the published tables · every number recomputed; every claim tagged; corrections log cumulative.