Weight × Level + Jump · decompwlj1 / 8

A research primer in eight pages · 17 September 2026

Weight × Level + Jump

term

weight

level

jump

Divide each term of an increasing integer sequence by the least divisor of that exceeds its jump , and the jump comes back as the remainder. This one canonical division reproduces the sieve of Eratosthenes on , turns the twin primes into a column, and supports a density-zero theorem on the primes, proved but not yet refereed. Here are its definitions, complete elementary proofs, the two analytic results, a reading of the graphs, the open problems — and, on the last page, what it might change.

Framework: Rémi Eismann, arXiv:0711.0865 · atlas decompwlj.com · OEIS A117078 (weight), A117563 (level), A118534 (). Numbering follows the tenth-edition treatise (19 Aug 2026); lettered statements are added here.

Status of claims: proved complete proof here or in print · proved* audited across editions, not yet refereed · conditional hypotheses named · heuristic contains an unjustified step · open. Counts marked † come from the tenth-edition census to (two engines sharing no code); unmarked counts were recomputed for this report, and constants and prime-gap records are quoted from the literature.

§1The construction

Definition 1 weight × level + jump

Let be positive integers. Put and if , otherwise . If , the weight is the least divisor of exceeding and the level is , so that

If we set and call not decomposable. A decomposable term is level-classified when and weight-classified when ; ties count as weight.

The weight exists because is itself a divisor exceeding ; it is unique because it is a minimum. Nothing is chosen or fitted, and depends on and alone.

Proposition A the remainder form proved

If , then for every : if and only if and . Hence and .

Proof. says with , that is and , and conversely. If no divisor of exceeds , matching .

So the level is the Euclidean quotient and the jump the remainder of divided by its weight — the founding definition of A117078. The minimum is the only informative selection (the largest admissible modulus is itself, forcing ), and reading as a congruence would admit : the inequality , carried by the remainder, is the whole construction.

Lemma 2 existence proved

is decomposable if and only if . On the primes exactly , and are not decomposable.

Proof. means , that is . By Nagura's theorem every interval with contains a prime, so once ; below 25 only , and fail.

Remark 1 the window criterion proved

A decomposable term is level-classified if and only if has no divisor in the window .

Proof. , and is the least divisor above .

Classification is therefore a question about divisors in an interval — the territory of Erdős, Tenenbaum and Ford — evaluated at , with the window's left end set by the sequence's own jump.

Figure 1 · Divisor ladders. Divisors of

on a logarithmic axis. Grey: divisors

, which the weight must clear. Tint: the window

; the vertical tick is

. The ringed dot is the weight (indigo: weight class, copper: level class). 17 and 19 share

and the divisor pair

; the jump alone swaps weight and level. 2557 and 2113 are the generic cases: a divisor (39) lands in the window, or the window holds no divisor and the weight jumps to 233.

§2 The plane and the base case2 / 8

**Table 1 · The first decomposable primes.** is the gap ; the class row reads W (weight) or L (level). The same seventeen primes, with 2, 3 and 7 undecomposable, open the founding paper; recomputed here.
	5	11	13	17	19	23	29	31	37	41	43	47	53	59
	2	2	4	2	4	6	2	6	4	2	4	6	6	2
	3	9	9	15	15	17	27	25	33	39	39	41	47	57
	3	3	9	3	5	17	3	25	11	3	13	41	47	3
	1	3	1	5	3	1	9	1	3	13	3	1	1	19
class	L	W	L	W	L	L	W	L	L	W	L	L	L	W

§2The plane, and the base case

Every decomposable term gives a lattice point on the hyperbola . On a log–log canvas that is square with equal aspect, is read as distance from the diagonal: the classification is its sign, and the bound is the anti-diagonal. Every weight–level figure below is drawn that way.

Proposition 1 Eratosthenes collapse proved

For one has , and, for , , the least prime factor, and . The term is level-classified if and only if is prime (then ), and exactly when is the square of a prime.

Proof. The least divisor of exceeding 1 is its least prime factor. If is prime, . Otherwise every prime factor of is at least , so , with equality iff .

This is a dictionary, not an analogy: the column is the set of integers struck first at stage of the sieve (shifted by one), the level class is the shifted primes, and decomposing again recovers the factorisation of . It is the one place where "the weight is a generalised least prime factor, the level class a generalised set of primes" is a theorem. Verified for : level class , ties .

Proposition B forced level proved

If , then is level-classified. Hence a polynomial sequence of degree , or of degree 2 with leading coefficient , is level-classified at all but finitely many terms.

Proof. gives . For with , and , so if and if ; decomposability holds eventually since .

The level share of a sequence is thus steered by : () sits as far from forced level as possible, the squares are level throughout, and the primes, with , sit in between.

weight class,

level class,

shading: log count per cell

Figure 2 · One classifier, three sequences. Every decomposable term below

, log–log, square with equal aspect; diagonal

, anti-diagonal

. Left,

: an arrowhead of columns at prime

; strictly below the diagonal only the row

survives (Proposition 1). Centre, the primes: columns again, now at odd

and including composite ones (9, 15, 21, …), and a level class opened into horizontal lines

— the jump

is wide enough for

to hold no divisor. Right, Cramér's model (each

kept with probability

, seed 20260917): the same anatomy and nearly the same level share, but every

and

occurs — no parity lines, no twin column. What separates centre from right is fine structure, and that is where §3 and §6 live.

§3 Elementary theory on the primes3 / 8

§3The elementary theory on the primes

Write , and , so is an arithmetic progression with difference . For decomposable (so ) the gap is even and odd, hence and are odd — used silently throughout.

Lemma 3 localisation and the level bound proved

(i) No divisor of lies in . If is level-classified, then moreover (ii) and (iii) .

Proof. (i) is the minimality of . (ii) is Remark 1. (iii) and , so by (i); is odd and even, so .

Parity is essential: on general sequences occurs ( gives ). Largest level below : at , ; below , with at †.

Lemma 4 the twin column proved

For a prime : if and only if .

Proof. If , the three numbers meet every residue class mod 3, and are primes above 3, so ; as , . Conversely with even forces .

Verified: for , i.e. twin pairs with (3, 5) — OEIS A007508(7). At : † each.

Difficulty conservation. By Lemma 4 the twin prime conjecture is the statement that the column is infinite. That is a translation, not a reduction; the same holds for every fixed column and line. Recasting a hard problem in these coordinates does not make it easier.

Proposition 2 mod-3 rigidity proved

For decomposable : if and only if .

Proof. As , . If then . If then , and would give ; so .

This settles the founding paper's Conjectures 7–8. It is a two-line congruence and should never be presented as a contribution; its use is downstream (§5, Figure 3d).

Proposition 3 cube bound proved

If is level-classified with composite weight, then . Among such primes are exactly (weights ), with equality only at .

Proof. Write with . Both are divisors of below , hence by Lemma 3(i), and odd, hence ; with this gives . So . For every gap is (Oliveira e Silva–Herzog–Pardi). If then , inside the census; if then , but every prime below has gap . The census to † finds exactly the five primes.

Lemma 5 level-one reduction proved

(i) If , then if and only if is prime. (ii) For the decomposable primes with are exactly ; (iii) among them the level-one primes with composite are and .

Proof. (i) means no divisor of in . A composite has a divisor , and then exceeds . (ii) forces ; the census settles it. (iii) Inspect the six.

Consequently for , and with error for all (gaps sum to at most ). Level-one primes are, up to that error, the middles of prime progressions whose upper step is a prime gap. Verified below : prime .

Corollary 2 balanced primes proved

Every balanced prime is level-one.

Proof. Then is prime, and because ; so and .

Corollary 3 mod-6 checkerboard proved

If is level-classified with prime weight , then if and only if .

Proof. with a prime other than 3, so ; apply Proposition 2.

**Table 2 · The invariant panel.** Every statement of §3 tested on every decomposable prime below (C engine); the PARI/GP kernel and the Python audit reproduce all tuples below exactly. Witnesses are complete exceptional sets, not samples. The tenth-edition census ran the first eight tests to † with the same sets and no violation.
statement	test	violations	witnesses below
Lemma 2	not decomposable	—
Definition 1	on every term	0	—
Lemma 3(iii)	on the level class	0	largest , at
Lemma 4	,	0	on each side
Proposition 2		0	—
Proposition 3	when is composite	0
Lemma 5(ii)		—
Lemma 5(iii)	with composite	—
Corollary 2	balanced	0	all balanced primes
Corollary 3	on the level class	0 with prime	fails only at , ,

§4 Rarefaction — Theorem 14 / 8

**Table 3 · The classification of the primes, decade by decade.** : level-classified; : its level-one stratum (Species I); : the rest (Species II); ; ties: . The last column is the envelope that Theorem 1 predicts to be bounded. Decomposable always (Lemma 2). Rows to recomputed here; rows marked † carried.
	decomposable					ties
	165	75	24	51	0.4545	3	0.6181
	1,226	390	135	255	0.3181	6	0.4348
	9,589	2,658	880	1,778	0.2772	8	0.3849
	78,495	18,353	5,953	12,400	0.2338	12	0.3310
	664,576	138,049	44,011	94,038	0.2077	28	0.3000
†	5,761,452	1,078,707	339,870	738,837	0.1872	79	0.2758
†	50,847,531	8,692,339	2,708,031	5,984,308	0.1709	187	0.2567
†	455,052,508	71,670,808	22,083,608	49,587,200	0.1575	483	0.2410

§4Rarefaction: the native theorem

The founding paper observed that the level share falls in every decade it could reach and conjectured that level-classified primes rarefy (its Conjecture 9). It is a question only the coordinates can pose, and it now has an answer.

Theorem 1 rarefaction of the level class proved*

. In particular : level-classified primes have relative density zero.

Corollary 1 normal form proved

Let be level-classified with . Then with prime, , odd. The level-classified with and number at most .

Proof. Proposition 3 and Lemma 3; and each pair fixes , with at most pairs.

Proof of Theorem 1. Let and split by gap and size.

Step 1 (large gaps). counts . Gaps telescope, , so at most of them exceed : .

Step 2 (degenerate range). counts , ; by Corollary 1, .

Step 3 (normal form). Every remaining prime has as in Corollary 1 and , so and : the triple consists of primes, , and is injective (for : , , triple ). With the number of for which , and are all prime, Here consecutiveness has been discarded: nothing asks to be the next prime.

Step 4 (sieve and sum). The three forms are linear with coefficients, shifts and pairwise resultants (, , ) of size . The Selberg upper-bound sieve in dimension 3 (Halberstam–Richert, Thm 5.7) gives, uniformly, with , so ; inadmissible triples contribute , and . Since and , and balances against .

Reading the proof

Why only an upper bound. A level prime couples a divisor condition on to the compositeness of . No sieve can condition on "no primes in between", but an upper bound may simply drop that constraint. The same move gives nothing from below — hence Problem 2.

The exponent is an artefact. comes from balancing against the saving of a triple. Heuristically , i.e. ; in Table 3, drifts from 1.23 at to 1.16 at . Restricted to the argument gives ; Problem 1 asks for unconditionally.

The asterisk records a social fact, not a doubt: the argument has passed internal audit across editions, but no referee has read it. Its ingredients are standard — telescoping, Proposition 3, a uniform upper-bound sieve — and a referee's attention belongs on the uniformity of Step 4 in .

No gap technology is used beyond telescoping. That robustness is a feature and a warning: §6 shows that the decay itself is what a random sequence of the same density also does.

§5 The constant of the level-one stratum5 / 8

§5The constant of the level-one stratum

By Lemma 5, Species I is the set of primes for which are three primes whose upper step is a prime gap. Two conditions are in play: a triple condition, which is Hardy–Littlewood territory, and a consecutiveness condition, which is gap-distribution territory.

For a tuple let be the number of classes it meets mod and . Write and . For even , with the twin-prime constant

Lemma 1 line average of the triple series proved

unless , in which case with , which equals with the prime-triplet constant. Moreover

Proof. If is odd, fills both classes mod 2; if it fills all classes mod 3; either kills a local factor. For the factors at 2 and 3 are and ; at they are if and otherwise, whose ratio is . So with , supported on squarefree integers prime to 6, . Averaging over (density ; the interchange is justified by , and ): The per-prime identity collapses this to .

That single identity — a triple-series factor times a divisor-average factor equals a pair-series factor — is why the twin constant appears in a statement about triples.

Theorem 2 asymptotic for Species I conditional

Assume (H1) the Hardy–Littlewood conjecture for , uniformly for ; and (H2) the Gallagher-type gap model: the number of with is , where , uniformly in the same range with negligible tail. Then

Derivation. Given a consecutive pair , (H1) relative to the pair density gives " prime" the conditional density . Weighting by (H2), , and Lemma 1 with partial summation gives , whence .

Where it is not a proof. The derivation assumes that conditioning on "no primes between and " does not bias the singular-series ratio at first order. The identification of the constant is unconditional (Lemma 1); the decoupling is not, and the data contradict it by about one percent.

**Table 4 · Gap-by-gap test at †.** Observed level-one counts ( prime) against the parameter-free prediction .
	observed	predicted	ratio
6	4,884,475	4,890,786	0.9987
12	3,784,976	3,718,233	1.0180
30	3,123,019	3,063,093	1.0196
36	1,112,790	1,142,292	0.9742
60	663,796	648,695	1.0233
all	22,075,357	21,826,295	1.0114

Two readings follow. First, the constant cannot be seen in raw counts: the windowed Species-I density near is about 1.06†, climbing toward 1.32 only logarithmically, because at barely samples the large gaps whose factors lift the average. The evidence is Lemma 1 plus Table 4, not extrapolation. Second, the stable excess of (Problem 4) has an identified mechanism, seen most cleanly one prime modulus at a time.

The interior-credit law heuristic. Fix and an odd prime . Among gap- primes, avoids the classes and mod ; spread evenly over the other , the lock class — that is, — would have share . Consecutiveness breaks the symmetry: a class with makes the interior point composite for free, while the lock class earns no such credit. To first order the lock share falls short of 1 by which vanishes for ( is empty) and for ( is inadmissible) — both confirmed to within at †. Over 391 cells the predicted shape is right and its size about 17% low; a single fitted factor absorbs the second-order term that nobody has yet derived.

Species II () carries 69% of the level class at † and has no identified constant. Over five decade windows stays within –†, and — over the Mertens mass — within –†, while the cumulative values drift down; an earlier working value is retracted. Identifying either constant is Problem 3, a question about divisors of shifted primes in a moving window (Ford, Koukoulopoulos).

**Table 5 · Species II by decade window.** at each window's geometric midpoint (recomputed from the counts of Table 3); carried†.
window
	0.7690	0.7705	0.7669	0.7656	0.7648
†	0.4445	0.4458	0.4432	0.4420	0.4410

§6 Reading the graphs6 / 8

§6Reading the graphs

(a) Level share

primesCramér model Aparity model B

(b) Primes ÷ model A

all level termsSpecies I,

(c) Weight columns,

prime

composite

(d) Gap and level, level class,

level-classified primes (log shading)

Figure 3. (a)

for the primes (

–

) and for two random sequences run through the same classifier (

–

†, tenth-edition seed); model A keeps each

with probability

, model B each odd

with probability

. (b) Ratios of the primes' shares to model A's, from four-digit shares. (c) Decomposable primes below

by weight, odd

. (d) Level-classified primes below

by gap

and level

, square with equal aspect.

(a) The decay is generic. Model A tracks the primes to within 1.5% at and within 0.5% from on (0.1709 against 0.1708 at ). The rarefaction rate is what the construction does to any sequence of density , so Theorem 1 must not be sold as prime-specific structure. Model B, which copies the parity of the primes, is worse: 19% too many level terms at and nearly twice the Species-I share, because a random odd is prime with probability , whereas the real is prime with density , which vanishes unless . Imitating one arithmetic constraint without the others moves the statistic the wrong way.

(b) The composition is arithmetic. The aggregate ratio stays within 1.5% of 1, but the Species-I ratio climbs steadily — 0.956, 0.984, 1.017, 1.043, 1.064 — crossing 1 between and (the crossing moves with the seed), toward if Theorem 2 holds. The first moment is generic; the second is not. A census stopping below would have concluded, wrongly, that primes are poorer than random in level-one terms.

(c) Columns are finite unions of gap events. The profile is not monotone, and every irregularity is exact arithmetic. Since forces , and belongs to , the column is exactly : primes counted both ways. Likewise splits into and : . A composite weight is gated by the gap: divides and is below , so (no exception among the primes below with composite weight). Thus needs and falls to , while () outruns (): once , Proposition 2 makes automatic and reduces to , which holds for about one admissible residue class in five, against one in seventeen for .

(d) The wedge is Lemma 3; its stripes are Corollary 3. All level primes below lie in the wedge without exception, in 481 cells, only at even and odd . Rows with () avoid multiples of 6, and all other rows live only on them: the row has primes at and two elsewhere, () and (). Below the only other exception is (weight 51).

§7 Conjectures, problems, ledger7 / 8

§7Conjectures, problems and the ledger

The founding paper closed with nine conjectures drawn from the first primes. Nearly two decades on, they sort into three kinds: proved elementary facts, one native theorem, and classical problems in new coordinates. The last kind is where a reader should be most careful.

**Table 6 · Status ledger.** "Native" statements cannot be formulated without the weight–level coordinates; "classical" ones are known problems in disguise.
item	statement	kind	status
Conj. 1	infinitely many primes of weight 3	classical: twin primes (Lemma 4)	open
Conj. 2	infinitely many primes of weight , each odd	classical: Polignac-type unions of gap events	open
Conj. 3	infinitely many primes of level , each odd	classical in disguise	open
Conj. 4	level primes have prime weight except 13, 31, 113, 131, 887	native; Proposition 3	proved to
Conj. 5	infinitely many balanced primes (level )	classical: three consecutive primes in progression	open
Conj. 6	infinitely many primes of level , each	classical: consecutive-prime patterns	open
Conj. 7, 8	;	native, elementary; Proposition 2	proved
Conj. 9	level-classified primes rarefy	native; Theorem 1	proved*
Lem. 1	line average of equals	classical tools	proved
Thm. 2		native + Hardy–Littlewood	conditional H1, H2
Prob. 1	unconditionally	native	open
Prob. 2	is the level class infinite?	native; contains Conj. 5 (Corollary 2)	open
Prob. 3	identify the Species-II constants ,	native; divisors of shifted primes	open
Prob. 4	derive the consecutiveness excess and the factor	native; mechanism heuristic	open
Conj. A	, with the same for primes and for Cramér's model	native universality	open

Three things the ledger does not say

Conjecture 4 is not proved in full. Proposition 3 is proved for all primes, but it only confines exceptions to , i.e. to gaps . The exceptional set is closed up to by exhaustive gap tables; beyond that height it would follow from — a bound Cramér-type conjectures predict with enormous room, but which neither the Riemann Hypothesis () nor the unconditional of Baker–Harman–Pintz supplies. Lemma 5(ii)–(iii) have the same status with in place of .

Problem 2 is weaker than Conjecture 5, not known to be easier. Theorem 1 is compatible with a finite level class. Infinitely many balanced primes would settle it, but so would any infinite family of primes whose avoids — a native question that exists only in these coordinates.

Nothing here touches the great problems. Twin primes remain twin primes when called the column ; Polignac remains Polignac; Goldbach does not appear; nothing bears on the Riemann Hypothesis. The framework's yield is its native rows — Propositions 1–3, Lemmas 1–5, Theorem 1, the column identities and the lock structure — and it should be judged on those alone.

Compute it yourself

The project's reference kernel in PARI/GP, with a census loop; output checked on PARI/GP 2.15.4 for this report.

\\ weight × level + jump of a, given the next term b
decomp(a,b) = {my(d = b-a, l);
  if (a <= 2*d, return([0,0,d]));
  l = a-d;
  fordiv(l, k, if (k > d, return([k, l/k, d])))}

\\ primes up to N: [decomposable, level, level-one, ties]
census(N) = {my(c = vector(4), r);
  forprime(p = 2, N,
    r = decomp(p, nextprime(p+1));
    if (r[1], c[1]++;
      if (r[1] > r[2], c[2]++; if (r[2] == 1, c[3]++),
        if (r[1] == r[2], c[4]++))));
  c}

? decomp(2113, 2129)
%1 = [233, 9, 16]
? census(10^6)     \\ about 0.2 s
%2 = [78495, 18353, 5953, 12]

What the control teaches. Every curve a construction like this produces must be run against random sequences put through the same classifier, and the control must be specified with care: density alone reproduces the aggregate, parity alone overshoots it by 19%, and neither reproduces Species I. A residual is evidence of arithmetic only against a control that matches the local congruence structure.

§8 Hypothetical impacts8 / 8

§8Hypothetical impacts on the future of number theory

Everything on this page is speculation, labelled by kind — program (a route with an identifiable technique), conjecture, or speculation — and by a plausibility judgement. None of it is a result.

1 · A calibrated bench for prime heuristics program · plausibility high

A census to yields exact, parameter-free observations in about four minutes on one core†, with three observables of known character: an aggregate share that is generic, a Species-I share governed by singular series, and a residue bias governed by consecutiveness. Random models of the primes — Cramér's, Granville's refinement, the Banks–Ford–Tao models — can be scored against all three at once. The likely impact is methodological: heuristics tested against samples instead of a handful.

2 · Consecutiveness conditioning as a portable tool program · plausibility medium

Conditioning on " exactly" is assumed harmless throughout gap statistics, and it is not: its first-order effect is computable from singular series and was confirmed over 391 cells. Its natural relative is the Lemke Oliver–Soundararajan analysis of biases between consecutive primes, built on the same Hardy–Littlewood bookkeeping; any claim of novelty must be measured against it. Carried to second order — deriving the factor rather than fitting it — the device would serve users outside this framework.

3 · Divisors in a moving window program · plausibility medium

Species II asks how often a shifted prime has no divisor in , a window whose left end is correlated with the shift. The theory of divisors in intervals (Ford's , Koukoulopoulos on shifted primes) is mature; this is a new normalisation of it. Success would compute , settle Conjecture A for Species II, and add a gap-coupled family to the Erdős–Tenenbaum–Ford theory.

4 · The same questions over program · plausibility medium–high

The obstruction on every page has one name: the word next. Over , with irreducibles ordered by degree then lexicographically, Hardy–Littlewood-type statements are theorems in several regimes (Sawin–Shusterman). There, at least partially, one may hope for Theorem 1 with the true exponent, Theorem 2 with its constant, and — the prize — an exact computation of the analogue of the , which over nobody can compute.

5 · Fingerprints for integer sequences speculation → program · plausibility medium

Level share, the plane and the forced-level ratio are invariants of any increasing sequence. Across the OEIS, measured against admissibility-matched controls, they could flag sequences whose statistics depart from what their density and local congruences predict — anomaly detection, not prediction of terms.

6 · A teaching and formalisation object program · plausibility high

§1–§3 are short, elementary, exactly checkable and exercised by real data: a natural first target for a proof assistant, and a clean classroom bridge from Eratosthenes to singular series and to the discipline of controls. Theorem 1 needs a formal Selberg sieve and is a larger project.

Falsifiable, for the next decade of census

If the models above are right, at : falls again and the envelope of Table 3 keeps decreasing; the aggregate ratio to model A stays within 1%; the Species-I ratio exceeds 1.064; the gap-by-gap ratio stays near ; the fitted lock factor stays in –.

What would count as failure

Conjecture A proved with residuals exactly as bookkeeping predicts; the Species-II constant found to be a known Buchstab evaluation; the function-field analogue a routine transcription; Problem 2 as stuck as balanced primes. What survives even then: the rigidity of §3, a theorem that answers a question only these coordinates pose, the column identities, and a consecutiveness calculus that may outlive its origin — a modest, honourable body of work. The largest defensible claim is this: the decomposition uses an additive datum, the jump, to select a multiplicative one, a divisor, canonically and for every increasing sequence, and on it is the sieve of Eratosthenes. It will not knock down a wall; it gives the additive–multiplicative divide an explicit, exactly verifiable instance.

References

R. Eismann, Decomposition into weight × level + jump and application to a new classification of primes, arXiv:0711.0865 (v4, 2010); atlas decompwlj.com; OEIS A117078, A117563, A118534, A007508, A006562, A002386.
J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad. 28 (1952).
H. Halberstam, H.-E. Richert, Sieve Methods, Academic Press, 1974 (Theorem 5.7).
G. H. Hardy, J. E. Littlewood, Some problems of "Partitio numerorum" III, Acta Math. 44 (1923).
P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976).
K. Ford, The distribution of integers with a divisor in a given interval, Ann. of Math. 168 (2008); D. Koukoulopoulos, Divisors of shifted primes, IMRN (2010).
T. Oliveira e Silva, S. Herzog, S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to , Math. Comp. 83 (2014).
R. C. Baker, G. Harman, J. Pintz, The difference between consecutive primes II, Proc. LMS 83 (2001).
A. Granville, Harald Cramér and the distribution of prime numbers, Scand. Actuar. J. (1995); W. Banks, K. Ford, T. Tao, Large prime gaps and probabilistic models, Invent. Math. 233 (2023).
R. J. Lemke Oliver, K. Soundararajan, Unexpected biases in the distribution of consecutive primes, PNAS 113 (2016).
W. Sawin, M. Shusterman, On the Chowla and twin primes conjectures over , Ann. of Math. 196 (2022).

Verification: Table 1, the rows of Table 3 to , every "verified" note and the data of Figures 1–3 were recomputed on 17 Sep 2026 by a C engine (least-prime-factor table, complete factorisation of ), an independent Python audit (plain sieve, trial division) and the project's PARI/GP kernel: all three agree on every one of the tuples below , and C and PARI/GP on the counts of Table 3 to . Items marked † are carried from the tenth-edition census, where two engines sharing no code agreed. Written with Claude (Anthropic) from the decompwlj project materials.