Decomposition into weight × level + jump

1Significance

The decomposition into weight × level + jump assigns to every term of a strictly increasing integer sequence a canonical multiplicative splitting of an additively defined quantity. Given the forward gap $d(n)$, one forms $\ell(n)=a(n)-d(n)$ and selects the smallest divisor of $\ell(n)$ exceeding the gap — the weight $k(n)$ — with cofactor the level $L(n)$. The construction is elementary to state, total on a wide class of sequences, and, as we verify in §3, it contains the sieve of Eratosthenes as its restriction to the natural numbers: on $\mathbb{N}$ the weight is exactly the smallest prime factor of $n-1$ and the level-classified integers are exactly the shifted primes. This is the correct sense in which the framework generalizes Eratosthenes: not a new sieve in the analytic-number-theory sense, but a gap-thresholded generalization of the smallest-prime-factor map, a universal divisor-classifier on monotone sequences that reduces to Eratosthenes when all gaps equal 1.

Its significance, assessed soberly, rests on three legs. First, uniformity: one map, applied identically to primes, semiprimes, lucky numbers, triangular numbers, and over a thousand OEIS sequences in the author's atlas at decompwlj.com, each acquiring a weight/level classification and a two-winged $(\log k,\log L)$ portrait. Second, statability: the coordinates $(k,L,d)$ make expressible some statements that are awkward in classical coordinates — most are then reformulations of classical open problems (twin primes, balanced primes) and inherit their difficulty in full, which the ledger records without embellishment. Third, and most importantly, one genuinely native open problem: Conjecture 9, the rarefaction of level-classified primes. C9 is not a disguise of any classical conjecture we can identify. It is a density statement decided at the scale of prime gaps, it admits a clean two-species analytic decomposition (§6), one species of which we can now bound unconditionally, and its analogue on $\mathbb{N}$ is precisely the statement that the primes have density zero — a theorem. C9 is, in this exact sense, the prime-indexed lift of "primes are rare".

The present edition adds to the June 2026 report a complete independent reproduction of the paper's numerical tables to $n=5\cdot 10^{7}$ (every entry exact), extended exceptional-set verification to $p\le 9.82\cdot 10^{8}$, the gap-conditional classification law, the per-decade two-species constants, and a corrected, fully elementary proof route for the Species-I bound.

2Definitions and first properties

Let $a(1)<a(2)<a(3)<\cdots$ be a strictly increasing sequence of positive integers. For each $n$ define:

An equivalent Euclidean phrasing, used on decompwlj.com: the weight is the smallest $k$ such that in the Euclidean division of $a(n)$ by $k$ the remainder is exactly the jump. Indeed $a(n)=kL+d$ with $0\le d<k$ if and only if $k\mid a(n)-d(n)=\ell(n)$ and $k>d(n)$. The two definitions coincide. PROVED

Lemma 2.1 (existence criterion). The decomposition of $a(n)$ exists if and only if $a(n{+}1)<\tfrac{3}{2}\,a(n)$. PROVED

Proof. $\ell(n)\neq 0 \iff a(n)-d(n)>d(n) \iff a(n)>2d(n)=2a(n{+}1)-2a(n) \iff a(n{+}1)<\tfrac32 a(n)$. When $\ell(n)>d(n)$, the set $\{k>d(n):k\mid\ell(n)\}$ contains $\ell(n)$ itself, so the minimum exists; uniqueness of $(k,L)$ is immediate. $\square$

T. Ordowski's independent 2013 comment on OEIS A117078 — the weight is nonzero iff $2\,p(n{+}1)<3\,p(n)$ — is exactly this criterion specialized to primes. Worked examples, all re-verified this session by both engines:

Two elementary consequences used throughout. If $a(n)$ is weight-classified then $k\le L$, i.e. $k\le\sqrt{\ell}$; if level-classified then $k>\sqrt{\ell}$, i.e. $\ell(n)$ has no divisor in the interval $(d(n),\sqrt{\ell(n)}\,]$. This interval condition is the exact point of contact with the Ford–Tenenbaum theory of divisors in intervals (§6). PROVED

3Relation to the fundamental theorem of arithmetic and the sieve of Eratosthenes

Theorem 3.1 (naturals). For $a(n)=n$, $n\ge 3$: $d=1$, $\ell=n-1$, $k(n)=\mathrm{spf}(n-1)$ (smallest prime factor, A020639), $L(n)=(n-1)/\mathrm{spf}(n-1)$ (largest proper divisor, A032742), and $n$ is level-classified iff $n-1$ is prime. PROVED

Proof. The smallest divisor of $\ell$ exceeding $d=1$ is the smallest nontrivial divisor of $\ell$, which is $\mathrm{spf}(\ell)$. Then $k>L=\ell/\mathrm{spf}(\ell)$ iff $\mathrm{spf}(\ell)^2>\ell$ iff $\ell$ is prime. $\square$  (Re-verified exhaustively for $n\le 2000$ by the Python engine and re-plotted to $10^5$ in Fig. A.)

This theorem fixes the framework's relation to both classical pillars. To the sieve of Eratosthenes: tabulating $k(n)$ over $\mathbb{N}$ is precisely tabulating least prime factors — the data structure Eratosthenes' sieve produces — with the primes isolated on the level-1 ray. The decomposition applied to $\mathbb{N}$ is the sieve's output, re-coordinatized; applied to other sequences it deploys the same divisor-selection rule with the gap as a moving threshold. To the fundamental theorem of arithmetic: the decomposition is not a rival factorization and refines nothing about FTA. Of all ordered divisor pairs $(k,L)$ of $\ell(n)$ that FTA provides, it selects exactly one — the pair with $k$ minimal subject to $k>d(n)$ — so it is a canonical divisor-classifier whose selection is steered by additive data (the gap). One sees FTA in it the way one sees FTA in $\mathrm{spf}$: as the guarantee that the selection is well defined and that the $(\log k,\log L)$ portraits of §5 tile the plane the way they do.

4Application to the primes: results, conjectures, and this session's verification

4.1 Direct results

For $a(n)=p(n)$ (A000040) with gap $g(n)$ (A001223), the framework's sequences are: weight A117078, level A117563, $\ell(n)=2p(n)-p(n{+}1)$ is A118534. The paper's foundational facts, all re-verified here:

• Decomposability. Exactly three primes are not decomposable: $\{2,3,7\}$ — Bertrand-type bounds (Nagura, Dusart) give $p(n{+}1)<\tfrac32 p(n)$ for all $p(n)\ge 11$, and $2,3,7$ fail the criterion while $5$ passes. PROVED  Confirmed: exactly 3 non-classified in all $5\cdot 10^{7}$ terms.
• $\gcd(\ell(n),g(n))=1$, and $3\le k(n)\le \ell(n)$, $2\le g(n)\le k(n)-1$ for decomposable primes $>3$. PROVED
• Twin characterization. A prime $p(n)>3$ has weight $3$ iff it is the lesser of a twin pair. PROVED  Live check to $n=5\cdot 10^{7}$: twin gaps 3 370 446; weight-3 primes 3 370 445 (all twins except the pair $(3,5)$, whose lesser is non-decomposable); weight-classified weight-3 primes 3 370 444 (excluding $p=5$, which is level-classified) — matching the paper's Table 3 entry exactly.
• Mod-3 rigidity (Theorem A; retires C7–C8). For consecutive primes $>3$: $6\nmid g(n)\iff 3\mid \ell(n)$. PROVED  Elementary from residues mod 3 of $p(n),p(n{+}1)$; both implications are exact. Live check: zero violations across $5\cdot 10^{7}$ consecutive-prime pairs. The OEIS wiki records C7/C8 as true; they are elementary theorems, not contributions, and are retired to the ledger as such.
• Andrica quarantine. Weight-classified $\Rightarrow g<k\le\sqrt{\ell}<\sqrt{p}$. Hence any prime with an "Andrica-dangerous" gap $g\ge\sqrt{p}$ is level-classified or non-decomposable; the decomposable primes with $g>\sqrt{\ell}$ are exactly $\{5,13,19,23,31,113\}$. PROVED (inequality) · VERIFIED to p ≤ 982 451 653 (completeness of the list, extending the paper's $5\cdot 10^{7}$-index check by a factor ≈ 20 in size)

4.2 The nine conjectures — ledger

4.3 Independent verification — every table of the paper, exactly

The full campaign was run in PARI/GP 2.15.4 (chunked, checkpointed; ≈ 5 minutes single-core for $5\cdot 10^{7}$ primes) after byte-identical agreement between the fordiv kernel and an independent Python/sympy engine on the first $10^{4}$ primes. The terminal prime is $p_{5\cdot 10^{7}}=982\,451\,653$, the canonical value. Results:

• Tables 1–2 (worked decompositions, naturals and first 17 primes): every row exact.
• Table 3 (weight distribution among the 41 446 529 weight-classified primes): all twelve entries $k=3,\dots,25$ exact, e.g. $k{=}3$: 3 370 444; $k{=}21$: 1 402 876. A transient ±1 discrepancy in an earlier pass of ours was traced to a checkpoint-resume bug in our own driver, fixed, and the clean run confirms the paper to the unit.
• Table 4 (level distribution among the 8 553 468 level-classified primes): all eleven entries exact, including the non-monotone $L{=}9>L{=}7$ feature; new data beyond the paper: $L{=}23$: 64 029, $L{=}25$: 41 330.
• Table 5 (level-1 generations): all seven entries exact — 1 307 356; 746 381; 345 506; 153 537; 65 497; 27 288; 11 313 — successive ratios $0.571,\,0.463,\,0.444,\,0.427,\,0.417,\,0.415$, a slowly stabilizing geometric-type decay. HEURISTIC reading in §6.
• Table 6 (classification counts at checkpoints): all rows exact.

Three primes ($2,3,7$) are non-classified at every checkpoint. The monotone decline of the last column is the empirical face of C9.

5Analysis of the graphs

The project's figures are not illustrations after the fact; several encode theorems. We read each against the results above.

6The additive–multiplicative bridge, and a two-species theory of Conjecture 9

6.1 What C9 says, and at what scale

C9 asserts that the level-classified share of primes tends to zero. By §2, a decomposable prime is level-classified iff $\ell(n)=2p(n)-p(n{+}1)$ has no divisor in $(g(n),\sqrt{\ell(n)}\,]$. The statement therefore couples the multiplicative structure (divisors in an interval — Ford's $H(x,y,z)$ territory) of a quantity that is itself defined by the additive structure of consecutive primes. This coupling is the bridge, and also the obstruction. Note the scale: C9 is decided at gap scale $g\asymp\log x$. The Riemann Hypothesis governs prime-count errors at resolution $\sqrt{x}$ and is silent here; C9 and RH are orthogonal. The natural conditional neighbours are the Hardy–Littlewood prime-pair conjectures and (via primes in progressions and divisors of shifted primes) GRH. PROVED (equivalences) · framing

6.2 The two species

Species I ($L=1$). $L=1$ iff $k=\ell$, i.e. the smallest divisor of $\ell$ exceeding $g$ is $\ell$ itself. For $g<\sqrt{\ell}$ this forces $\ell$ prime — this is the reduction lemma: a decomposable prime with $g<\sqrt{\ell}$ is level-classified with level 1 iff $\ell=2p(n)-p(n{+}1)$ is prime, and the exceptional set (level-1 with $\ell$ composite) is exactly $\{13,31\}$, with $\ell=9,25$. PROVED · exceptions complete to p ≤ 982 451 653 (this session; previously $10^{8}$). Since $\ell+p(n{+}1)=2p(n)$, Species I is exactly: $p(n)$ is the middle term of a three-term arithmetic progression of primes $(\ell,\,p(n),\,p(n{+}1))$ whose right endpoint is the actual next prime. Generation $i$ records $\ell=p(n-i)$; generation 1 is the balanced primes A006562. Table 5's geometric-type decay of generations (ratios stabilizing near $0.42$) is what a Gallagher-type Poisson model of gaps predicts qualitatively; we do not claim the constant. HEURISTIC

Mod-3 rigidity constrains the species: if $6\nmid g$ then $3\mid\ell$, so $\ell$ is composite once $\ell>3$ — hence Species I lives only on gaps divisible by 6, apart from the sporadic $p=5$ ($\ell=3$, prime) and the two composite-$\ell$ exceptions $13,31$ ($\ell=9,25$). PROVED — and visible as the indigo/copper alternation of Fig. E below.

Species II ($L>1$). Level-classified with $k>L>1$: $\ell$ composite with no divisor in $(g,\sqrt\ell\,]$. This is a Ford–Tenenbaum divisor-distribution condition on the shifted quantity $\ell=2p-p_{\text{next}}$; it carries ≈ 68–69 % of the level class throughout our range.

6.3 An unconditional bound for Species I PROVED

6.4 The measured two-species law VERIFIED (this range) · HEURISTIC (form)

Writing the local densities among primes near $x$ as $\rho_1(x)\approx c_1/\log x$ (Species I, prime-pair regime) and $\rho_2(x)\approx c_2\log\log x/\log x$ (Species II, divisor regime), the per-decade windowed constants from this session's run are:

Two distinct behaviours, and two constants that must never be conflated. $\hat c_2$ is flat at $0.767$–$0.770$ across four decades — the $\log\log x/\log x$ exponent for Species II is correct at this range; the prior $10^{10}$ campaign's band $c_2\approx 0.79$–$0.80$ sits ≈ 3 % above our windows, a gap consistent with differing window conventions and slow secondary terms, and we report both rather than average them. $\hat c_1$ is still climbing, $0.84\to 1.03$, toward the prior campaign's $c_1\approx 1.07$–$1.117$ at $10^{10}$ — the loglog-slow convergence of a singular-series average. The often-quoted single constant $\approx 1.13$–$1.18$ multiplying $\log\log x/\log x$ for the whole level class is a finite-size superposition transient of the two species, not an asymptotic constant; the cumulative-average constant (drifting) and the local $c_2$ (flat) are different objects. The doubly-logarithmic abscissa spans only $\log\log x\approx 2.63$–$3.03$ over our six size-decades — the Legendre-intercept problem in miniature — so all slope statements here are range-qualified by construction.

6.5 The gap-conditional law VERIFIED, first 10⁷ primes

Conditioning classification on the gap exposes the whole mechanism in one picture. Over the first $10^{7}$ primes: $P(\text{level}\mid g{=}2)=1/738\,596$ — the single case is $p=5$; every other twin lesser has weight 3 by the twin characterization, and this bar's collapse is that theorem. Gaps with $6\mid g$ (where Species I is permitted) run systematically hot — 17.7 % at $g{=}6$ rising to 31.8 % at $g{=}30$ — while gaps with $6\nmid g$ (where $3\mid\ell$ kills level 1) run cold, with the low points $g{=}4$ (8.6 %) and $g{=}8$ (8.4 %). The overall upward drift in $g$ reflects the shrinking divisor window $(g,\sqrt\ell\,]$: larger gaps leave fewer chances for a middling divisor, favouring level classification.

6.6 Status of C9 after this analysis

7Detailed conclusion and future research

7.1 Assessment

The framework earns its place by three verified facts. It reduces to the sieve of Eratosthenes on $\mathbb{N}$ — exactly, as a theorem, not as an analogy. It classifies the primes into two wings whose every labelled stratum is an OEIS sequence, with the twin characterization and mod-3 rigidity as proved structural theorems. And it produces one native open problem, C9, which this analysis has split into a species now controlled unconditionally and a species that is a well-posed divisor-distribution question about consecutive primes. The reformulation conjectures C1–C3, C5–C6 stand exactly as hard as their classical originals; the ledger keeps them at arm's length from the native content. Nothing in the framework lowers the difficulty of twin primes; what it changed, on the evidence of this document, is that a previously unstatable density question became statable, measurable to $10^{9}$, and half-provable.

7.2 Corrections log (honest ledger)

• Challenge-paper Species-I overclaim — corrected. The earlier draft attributed the unconditional Species-I bound to heavier machinery than required and overstated one step; the correct and sufficient route is the elementary Markov + Selberg/Brun argument of Theorem 6.1. The draft requires this correction before any circulation.
• Two-constants erratum — restated. The cumulative transient ($\approx 1.13$–$1.18$, drifting) and the local Species-II constant ($c_2\approx 0.77$–$0.80$, flat) are different objects; conflating them was an identified error and remains the most likely misreading of the data by others.
• This session's own bug — disclosed. Our first chunked run harboured a checkpoint-resume off-by-one that double-processed four boundary primes; it was detected because $p_{5\cdot10^{7}}$ came out 74 short of the canonical value, then fixed, and the clean run reproduced the paper to the unit. We record it because verification discipline is only credible when its failures are logged too.
• Paper Table 3 — confirmed exact. The transient ±1 we briefly observed against the paper's $k=3$ and $k=21$ entries was entirely our artifact; the paper is correct.

7.3 Research programme

• The decisive literature question. Determine whether Species II's rarefaction (or its $c_2\log\log x/\log x$ law) follows from Ford's theory of $H(x,y,z)$ (Annals 2008) or from results on divisors of shifted primes (Koukoulopoulos, IMRN 2010) once adapted to $\ell=2p-p_{\text{next}}$. If yes, C9 is a corollary of known mathematics and the contribution is the statement, not the proof; if no — as the consecutive-prime coupling suggests — C9 is a genuinely new problem of intermediate difficulty. Either answer is valuable; the search should precede any redraft of the challenge paper.
• Attack $N_2(x)=o(\pi(x))$. A plausible route: for level classification, $\ell$ must avoid divisors in $(g,\sqrt\ell\,]$; decompose by $\omega(\ell)$ and by the size of $P^-(\ell)$, use upper-bound sieves for each fixed gap class $g\le G$ against the density of integers with no divisor in the window (Ford's $\varepsilon$-estimates), and Markov for $g>G$ as in Theorem 6.1. The missing lemma is a Ford-type bound uniform over the family $\{2p-p_{\text{next}}\}$ — precisely where the coupling bites.
• Computation to $10^{12}$. Extend the campaign with a segmented streaming sieve (the checkpointed driver of Appendix A already restarts cleanly): the doubly-logarithmic range widens from $\log\log x\approx 3.03$ to $\approx 3.32$, enough to test the $c_2$ offset (0.77 vs 0.79–0.80) between window conventions, watch $\hat c_1$ close on the $10^{10}$ band, and put error bars on the superposition transient.
• Atlas systematics. Classify the 1000-sequence atlas by the fate of the level share: on $\mathbb{N}$ it tends to 0 (PNT); on the primes it is C9; sequence families where it plausibly tends to a positive constant would sharpen, by contrast, what makes the prime case special. The 13 fully OEIS-registered decompositions (naturals, primes, odds, evens, composites, semiprimes, 3-almost primes, lucky, prime powers, squarefree, triangular, …) are the natural first cohort.
• Publication. Realistic venues for the corrected paper: Journal of Integer Sequences, Integers, Experimental Mathematics — a verified-computation-plus-partial-results paper fits all three; a strong analytic journal requires resolving Species II. The OEIS wiki page and sequence entries should be updated to cite Theorem 6.1's unconditional bound once the corrected draft exists.

7.4 Closing statement

Eighteen years after arXiv:0711.0865, the framework's numerical claims survive complete independent re-derivation to the unit; its elementary theorems are proved; its reformulations are labelled as reformulations; and its one native conjecture has acquired an analytic anatomy — one species dead by elementary sieve, one species alive as a sharp question about divisors of a gap-coupled shifted prime. That is a defensible scientific position, and the honest ledger is what makes it defensible.

AOptimized PARI/GP kernel

\\ single-term kernel: optimal for primes and sparse sequences (fordiv)
smdiv(l, g) = fordiv(l, k, if (k > g, return(k)));
decomp(a, anext) =
{ my(g = anext - a, l = a - g, k, L);
  if (l <= g, return([a, g, 0, 0, 0]));          \\ not decomposable
  if (g == 2 && l % 3 == 0, k = 3,               \\ twin fast path
     forstep (j = g + 1, g + 13, 2,               \\ short trial window
        if (l % j == 0, k = j; break)));
  if (!k, k = smdiv(l, g));                       \\ fordiv fallback
  L = l \ k; [a, g, l, k, L] }
\\ campaign driver: chunked + checkpointed (state.gp), snapshot grid 10^0.125,
\\ index marks, exception lists; ~5 min single-core for 5*10^7 primes.
\\ full driver as run: runner.gp (this session).  forfactored variants win only
\\ on dense sequences (e.g. N), where consecutive l share factorization work.

BReproducibility

Environment: PARI/GP 2.15.4, Python 3.12 (sympy, numpy, matplotlib), single vCPU, 4 GB. Pipeline: (i) Python/sympy reference engine reproduces the paper's Tables 1–2 exactly; (ii) PARI fordiv kernel agrees byte-identically with the reference on the first $10^{4}$ primes; (iii) chunked PARI campaign to $n=5\cdot 10^{7}$ ($p\le 982\,451\,653$; ≈ 51 s per $10^{7}$-prime chunk on this hardware), reproducing Tables 3–6 exactly and extending the exceptional sets; (iv) dedicated passes: per-prime $(k,L)$ scatter for $n\le 10^{6}$; gap-conditional tally to $n=10^{7}$ (26 s); (v) all figures regenerated from these outputs. Snapshot grid: 48 points, $10^{3}$ to $7.5\cdot 10^{8}$, spacing $10^{0.125}$. The checkpoint-resume bug found and fixed during the session is documented in §7.2.

References

DEEP ANALYSIS — SECOND EDITION · 2 JULY 2026 · ALL NUMERICAL CLAIMS INDEPENDENTLY RECOMPUTED THIS SESSION UNLESS TAGGED AS PRIOR-CAMPAIGN VALUES · HONEST LEDGER THROUGHOUT