Decomposition into weight × level + jump · Ninth edition

The Weight–Level–Jump Decomposition A pedagogical treatise: definitions, proofs, theorems, conjectures, and what the coordinates could mean for number theory

A self-contained course in the construction of Rémi Eismann (arXiv:0711.0865): what it is, why it is canonical, what has been proved in it, what remains conjectural, and — in the closing section — an expansive reading of where a mature theory of these coordinates might lead.

Session of 16 August 2026 · framework: Rémi Eismann, arXiv:0711.0865 · atlas: decompwlj.com
Engines rebuilt from an empty directory this session: C (segmented sieve census to ), PARI/GP 2.15.4 (project kernels + constants), Python 3.12/NumPy (independent divisor audit).
Every numerical statement below carries verified — recomputed today by at least two engines that share no code.

How to read this document

The intended reader is a graduate student or researcher in analytic number theory meeting this construction for the first time. No familiarity with the founding paper is assumed; familiarity with sieve methods, Hardy–Littlewood singular series, and Mertens-type estimates is. Sections 1–3 are elementary and complete: every proof is given in full. Section 4 states the one substantive theorem of the programme and gives its proof in structured form, with each step named and its role explained; Section 5 does the same for the conditional asymptotic. Sections 6–7 are where the framework earns — or fails to earn — its keep, and Section 7 in particular is the control experiment that most expositions of this kind omit.

Claims are tagged throughout. proved means a complete proof appears here or is cited to a specific published source. proved* means the argument has passed internal audit across editions but has not been refereed. conditional names its hypotheses. heuristic means a derivation with a step that is not justified. open means open. verified attaches to numbers, not to theorems, and means recomputed today by independent engines.

1.The construction

The decomposition takes a strictly increasing integer sequence and attaches to each term a factorisation of that term's own deficit from its successor. It is a change of coordinates, not a formula: nothing is chosen, nothing is fitted, and the same three numbers are produced for every sequence one feeds it.

1.1 Definition

Definition 1 (weight × level + jump)

Let be a strictly increasing sequence of positive integers. For each set

If , the weight is the smallest divisor of that exceeds , and the level is . If we set and call not decomposable. When ,

The term is level-classified if , and weight-classified if (ties count as weight).

Three remarks fix the mechanics before any theorem.

The weight exists. The set of divisors of exceeding is non-empty whenever , because divides itself. So the minimum is taken over a non-empty finite set and is well defined; is then an integer by construction, and is an identity, not an approximation.

The weight is unique. Minimality pins it. There is no choice of parameter anywhere in Definition 1 — no window, no threshold, no cut-off. This is what makes the construction worth studying at all: it is a canonical map, and the objects it produces are invariants of the sequence.

The decomposition sees only two consecutive terms. and depend on and and on nothing else. This locality is the source of both the construction's tractability and its central difficulty: on the primes, is the next prime, and that word "next" is a global condition wearing local clothes. Section 4 turns on the moment when it can be dropped.

1.2 Worked examples

Take the primes, . Then is the prime gap and

an object with its own OEIS entry (A118534). Read the table row by row; the arithmetic is entirely finger-work.

**Table 1.** The decomposition of the first primes. is the smallest divisor of exceeding ; the identity is checked in the last column. Rows with are not decomposable. These seventeen rows are Table 2 of the founding paper; all four algorithms of §9 reproduce them exactly. verified
				divisors of			class
2	3	1	—	—	0	0	—	—
3	5	2	—	—	0	0	—	—
5	7	2	3	1, 3	3	1	level	5
7	11	4	—	—	0	0	—	—
11	13	2	9	1, 3, 9	3	3	weight (tie)	11
13	17	4	9	1, 3, 9	9	1	level	13
17	19	2	15	1, 3, 5, 15	3	5	weight	17
19	23	4	15	1, 3, 5, 15	5	3	level	19
23	29	6	17	1, 17	17	1	level	23
29	31	2	27	1, 3, 9, 27	3	9	weight	29
31	37	6	25	1, 5, 25	25	1	level	31
37	41	4	33	1, 3, 11, 33	11	3	level	37
41	43	2	39	1, 3, 13, 39	3	13	weight	41
43	47	4	39	1, 3, 13, 39	13	3	level	43
47	53	6	41	1, 41	41	1	level	47
53	59	6	47	1, 47	47	1	level	53
59	61	2	57	1, 3, 19, 57	3	19	weight	59

Two patterns are already visible and both become theorems. Every row with has : that is Lemma 4, and it says that the lesser member of a twin pair is exactly a prime of weight 3. And the rows with are exactly the rows where is prime (5, 23, 47, 53) — plus two rows, and , where is a prime square. That near-equivalence is Lemma 5 and it drives all of Section 5.

1.3 When does the decomposition exist?

Lemma 2 (existence criterion) proved

is decomposable if and only if . For the primes, the non-decomposable terms are exactly , and .

Proof. By definition is decomposable iff , i.e. , i.e. , which is the stated inequality.

For the primes, apply Nagura's theorem: for the interval contains a prime, so once . The primes below 25 are checked by hand: , , fail (, , ), while pass. □

So on the primes the construction is defined everywhere except on a set of size three, and one may safely say "for all decomposable primes" and mean "for all primes but ". On a fast-growing sequence — , say — the decomposition is defined nowhere, and the barrier is a real restriction on the construction's domain, not a technicality. Section 10 returns to it.

1.4 The classification plane

Every decomposable term yields a point on the hyperbola . The diagonal cuts that quadrant in two, and the two halves are the two classes. This is the whole of the classification, and it is worth saying plainly what it measures:

What the classification is really testing

is level-classified precisely when has no divisor in the window . Indeed , and is the smallest divisor exceeding ; so says exactly that the interval is free of divisors of .

The classification is therefore a statement in the theory of divisors in short intervals — Erdős's multiplication table problem, Ford's , Tenenbaum's -function — evaluated at the arithmetically special argument with the window's left endpoint tied to the gap. That is the correct address of this subject, and it is where §10's most concrete proposals live.

Plotting on a square canvas with equal aspect makes the ratio readable directly as a distance from the diagonal — which is why every figure in this document that carries both coordinates is square. The two sheets that appear are not decoration; they are the two classes, and their shapes are theorems.

2.The base case: the construction is the sieve of Eratosthenes

Before the primes, run the machine on the natural numbers. What comes out is not an analogy with the sieve of Eratosthenes; it is the sieve of Eratosthenes, in the sense that the weight of is the smallest prime factor of and the level class is the shifted primes. This is the single fact that makes the construction more than an arbitrary bookkeeping scheme.

Proposition 1 (Eratosthenes collapse) proved

For one has , , and for all

where denotes the smallest prime factor. Moreover is level-classified if and only if is prime, and in that case .

Proof. Here and . The smallest divisor of exceeding is its smallest prime factor, so .

If is prime then and : level-classified. If is composite, write with ; every prime factor of is , so and the term is weight-classified, with equality exactly when . □

The content of Proposition 1 is the dictionary entry it establishes:

The dictionary, entry zero

On :

weight columns = the residue classes struck at stage of the sieve of Eratosthenes;
the level class = the shifted primes , i.e. the numbers the sieve never strikes;
iterating the construction on the level — decomposing in turn — recovers the complete factorisation of , smallest prime factor first.

So the weight is a generalised smallest prime factor, and the level class is a generalised set of primes, for an arbitrary increasing sequence. That is the claim the framework stakes, and Proposition 1 is the only place it is a theorem rather than an analogy.

Scatter of log-weight against log-level for the natural numbers up to 30000, showing an arrowhead-shaped weight sheet above the diagonal and a horizontal level line at L=1. — **Figure 1.** The base case: for , plotted in the plane on a square canvas with equal aspect, so the ratio is faithful. The blue arrowhead is the sieve: its left edge is the column (all odd ), then , , … each column terminating where on the diagonal. The orange line at is exactly . The empty wedge below the diagonal and left of the level line is not sparsely populated — it is *provably empty*, since forces every composite onto or above the diagonal. verified

2.1 What the base case does and does not license

It is worth being precise about the strength of Proposition 1, because it is easy to overclaim. What is true: the construction restricted to is a faithful re-encoding of trial division, and the classification is a primality test for . What is not true, and is nowhere claimed: that the construction applied to another sequence inherits any of the sieve's power. On the primes the analogous statement — "level-classified means is prime" — is only almost true (Lemma 5, with a two-element exception set), and the almost is doing real work.

The honest summary is that Proposition 1 identifies the construction's type. It tells us that and are multiplicative data attached to an additive quantity, and that the level/weight split is the smallest-divisor-in-a-window question. Everything downstream is an instance of that type at a harder argument.

3.The primes: the elementary theory in full

Everything in this section is elementary, complete, and proved. It is also, taken together, the framework's most secure asset: a small set of rigidity results that constrain the level class so tightly that the analytic work of Section 4 becomes possible at all.

Throughout, is a decomposable prime, , , , . For the gap is even and is odd; this parity observation is used constantly and silently.

3.1 Divisor localisation and the level bound

Lemma 3 (divisor localisation; level bound) proved

Let be decomposable and level-classified. Then:

no divisor of lies in the open interval ; equivalently every divisor of smaller than is at most ;
;
.

Proof. (i) is the minimality of restated: if a divisor lay in it would exceed and be smaller than , contradicting the definition. (ii): level classification is , and , so .

(iii) divides and , so by (i) . Since is odd and is even, is odd, hence is odd, while is even; so and therefore . □

The parity step in (iii) is worth pausing on. Without it one gets only , and is genuinely attained on general sequences — the witness , gives . The sharpened bound is a fact about the primes specifically, and it is what makes the exponent bookkeeping in Theorem 1 come out. verified: over all level-classified primes , zero violations.

Heat map of the joint occupancy of gap g and level L for level-classified primes below 1e9, showing all mass strictly below the line L = g-1. — **Figure 3.** Lemma 3 made visible. The joint occupancy of over all 8,692,339 level-classified primes below , on a square canvas with equal aspect so the wedge's slope reads as . Every cell above the orange line is empty — not sparse, empty — and the parity of the bound is visible in the checkerboard: only odd occur, only even . The largest level observed anywhere below is . verified

3.2 The twin column

Lemma 4 (the twin column) proved

For a decomposable prime : . That is, a prime has weight 3 precisely when it is the lesser member of a twin pair.

Proof. () Suppose , so and are both prime and . Among the three integers — consecutive in the arithmetic progression of common difference 2 — exactly one is divisible by 3, because they occupy the three residue classes mod 3 (2 is invertible mod 3). Both and are primes exceeding 3, hence not divisible by 3; so . Since , minimality gives .

() If then by definition of the weight, and is even and positive, so . □

verified to : , with zero one-sided cases in either direction. The column splits as weight-classified terms plus exactly one level-classified member, ().

A warning that governs the whole document

Lemma 4 makes the twin-prime conjecture equivalent to the statement "the weight column is infinite". This is a translation, not a reduction. Nothing in the weight–level coordinates gives purchase on that column that was not already available in the language of gaps. The same applies to every fixed weight column and every fixed level line and their Polignac-type analogues. We call this the difficulty-conservation principle and invoke it, explicitly, wherever a reformulation might be mistaken for progress.

3.3 Mod-3 rigidity

Proposition 2 (mod-3 rigidity) proved

For every decomposable prime : .

Proof. Since , reducing mod 3 gives .

If then , so , whence .

If then, being even, , so . Both and are primes exceeding 3, hence neither is . Were , we would get , impossible. Therefore , i.e. . □

verified: zero violations over all 50,847,531 decomposable primes below . Proposition 2 settles Conjectures 7 and 8 of the founding paper. It should be stated, as it is here, as an elementary observation and never presented as a contribution: it is a two-line congruence argument, and the OEIS page records it struck through for exactly that reason.

Its real use is downstream. Combined with Lemma 5 below it forces for every level-one prime except and , which is why the singular series of Section 5 is supported on multiples of 6 and why the constant that emerges is the twin-prime constant rather than something else.

3.4 The cube bound for composite weights

Proposition 3 (Cube Lemma) proved

If is level-classified with composite weight, then

with equality if and only if (, , , ). Moreover the level-classified primes with composite weight are exactly , with weights , for all .

Proof. Write with ; then and are proper divisors of , hence divisors of smaller than . By Lemma 3(i) they are at most ; since is odd its divisors are odd, while is even, so in fact . Therefore , and with from Lemma 3(iii),

(When one gets the stronger .) Equality forces , so with an odd prime power; the census finds the unique instance .

For the exhaustive list: a composite-weight level-classified prime satisfies . Let be the first occurrence of as a prime gap. The exhaustive gap tables of Oliveira e Silva–Herzog–Pardi (all gaps below ) give for every in range, so an example with would need — a contradiction. The finitely many possibilities with satisfy , inside the swept range, where the census finds exactly the five listed primes. □

verified to today by three independent engines: the composite-weight level set is and there are zero violations of the cube bound. The finite closure to is carried from the refereed gap tables, not recomputed here.

3.5 The level-one reduction

Lemma 5 (level-one reduction and exception sets) proved

Let be a decomposable prime.

If , then if and only if is prime.
The decomposable primes with are exactly , for all .
The level-one primes with composite are exactly (), for all .

Consequently, for , .

Proof. (i) means , i.e. has no divisor in the open interval . If is prime this holds vacuously. If is composite it has a divisor ; then is also a divisor, and by hypothesis, so and .

(ii) The condition with gives . The maximal prime gap below is 1476, so any example beyond the census range would satisfy — inside it. The exhaustive sweep finds exactly the six listed primes.

(iii) Among those six: and are level-one with prime; has ; has ; and () and () are level-one with composite . The counting identity follows. □

verified to : the empty-window set is exactly and the composite- level-one set is exactly ; zero further members.

Lemma 5 is the hinge of the whole analytic story. It converts a divisor question about into a primality question about , and thereby converts "level-one prime" into "the triple is a triple of primes with the outer two flanking a prime gap". Everything in Section 5 is the study of that triple.

Corollary 1 (normal form) proved

Let be level-classified with and . Then the weight is prime: with prime and odd. The number of level-classified with and is .

Proof: the first claim is Proposition 3 in contrapositive, together with Lemma 3. For the second, each pair with determines at most one prime , and there are at most such pairs. □

3.6 The census

**Table 2.** Complete classification census. Rows – are today's fresh C run (46 s, single core), cross-checked against the project's PARI/GP kernels at and against an independent Python divisor audit row by row for all 148,930 decomposable primes below (zero mismatches). The row is carried from the manuscript's census. ; "ties" counts , the points exactly on the diagonal.
decomposable		(level one)			ties	status
165	75	24	51	0.4545	3	verified
1,226	390	135	255	0.3181	6	verified
9,589	2,658	880	1,778	0.2772	8	verified
78,495	18,353	5,953	12,400	0.2338	12	verified
664,576	138,049	44,011	94,038	0.2077	28	verified
5,761,452	1,078,707	339,870	738,837	0.1872	79	verified
50,847,531	8,692,339	2,708,031	5,984,308	0.1709	187	verified
455,052,508	71,670,808	22,083,608	49,587,200	0.1575	—	carried

Scatter of log-weight against log-level for all decomposable primes below two million, showing a dense weight sheet above the diagonal and horizontal level lines below it. — **Figure 2.** The classification plane of the primes: all 148,930 decomposable , square canvas, equal aspect. Above the diagonal, the weight sheet — the vertical striations at are the weight columns of §6, and the sheet's lower-right boundary is the diagonal , where the 14 ties in this range sit. Below the diagonal, the level class resolves into horizontal *level lines* at ; only odd occur, by Lemma 3. The lowest line is Species I — the primes with prime — and carries of the level class here. The line terminates at because . verified

3.7 The invariant panel

Every structural statement of §3 was re-tested today against the full stream of 50,847,531 decomposable primes below . The result in each case is zero violations; the tests and their witnesses:

**Table 3.** The invariant panel at . "Witness set" is the complete exceptional set found, not a sample. verified
Statement	Test	Witness set
Lemma 2	non-decomposable primes	exactly
Lemma 3(iii)	on the level class	max
Lemma 4		3,424,505 each side
Proposition 2		—
Proposition 3	, composite weight
Lemma 5(ii)	empty window
Lemma 5(iii)	level-one with composite
A002386	maximal-gap ladder	ends at 436,273,009

4.Rarefaction — Theorem 1

This is the framework's native question: it cannot even be asked without the coordinates, and it has an answer. Everything else in the programme is either elementary (§3), classical in disguise (§3.2), or conditional (§5).

4.1 The statement

Write for the number of level-classified primes up to , for the level-one count, and for the level share among decomposable primes up to . The founding paper observed that decreases in every decade it could reach and conjectured (Conjecture 9). That is now a theorem.

Theorem 1 (rarefaction of the level class) proved*

As ,

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

The asterisk means one thing and one thing only: the argument has passed internal audit across editions of this analysis but has not been externally refereed. It is not a hedge about the mathematics; it is a statement about the mathematics' social status. Submission removes it.

4.2 Why the proof can work at all

Before the proof, the obstruction it dodges. A level-classified prime is one whose has no divisor in . The quantity is the next prime after , so the event couples a divisor condition on to the compositeness of every one of . That coupling is what makes an asymptotic for out of reach: there is no known technique for conditioning a sieve on "no primes in between".

The key move

For an upper bound the coupling can simply be discarded. The level event implies a triple-prime event; a triple-prime event can be sieved; and throwing away a constraint only increases an upper bound. This is legitimate, it is the whole trick, and it is also exactly why the same method gives nothing in the lower-bound direction — which is why Problem 2 (is the level class infinite?) remains open and, by difficulty conservation, contains the balanced-prime problem.

4.3 The proof

Let be large and set . Split

where counts level-classified with ; those with and ; and is the remainder, which by Corollary 1 is in normal form. We bound the three pieces in turn.

Step 1 — the tail : large gaps are rare for free

The gaps telescope: for large. A sum of non-negative terms bounded by can have at most terms exceeding . Hence

No arithmetic is used — only that gaps sum to the range. This is the crudest step and it is not the bottleneck.

Step 2 — the degenerate range : absorbed

By Corollary 1, , which is smaller than any power saving and is absorbed into the final bound. This is where Proposition 3 (the Cube Lemma) earns its place: without it, the composite-weight terms would not be confined to a polylogarithmic set and the normal form of Step 3 would be unavailable.

Step 3 — the main term : normal form, then Selberg

By Corollary 1 each prime counted by satisfies with prime, odd, . Writing and ,

so that the triple consists of three primes, with . The map is injective (a prime determines its gap and hence its decomposition, and conversely ), so

Inequality (4.1) is where the consecutiveness requirement is dropped: nothing now demands that be the immediate successor of .

Why a dimension-3 Selberg sieve applies uniformly here

For fixed the three linear forms have degree one; their coefficients and shifts are of size ; and their pairwise resultants divide , again of size . These are exactly the hypotheses under which Halberstam–Richert's Theorem 5.7 gives an upper-bound sieve estimate uniform in the parameters:

The singular product has local factors at primes and elsewhere. If some local density vanishes, the triple is inadmissible and , which is better still. Note that uniformity in is essential and is precisely what the truncation buys.

Since we have , so (4.1)–(4.2) give

Both sums are standard mean values of non-negative multiplicative functions. Writing with supported on squarefree integers, :

Therefore

by the choice , which is exactly the value balancing against . Summing the three contributions proves Theorem 1. □

4.4 Reading the proof

Three things are worth extracting, because they are what a reader should carry away.

The exponent is an artifact.: It comes from the balance between the gap truncation and the sieve saving, and the sieve saving is because the normal form produces a triple. The conjectured truth is . Closing the gap — proving unconditionally — is Problem 1 of the manuscript, and it is the most concrete unconditional target the framework offers.
The is the price of the -sum.: Restricted to , the same argument gives with no . Species II is what costs the extra factor.
Nothing here is prime-gap technology.: No bound on is used beyond telescoping, and no equidistribution input beyond a standard upper-bound sieve. The proof would survive a fairly brutal weakening of what we know about prime gaps. That robustness is a feature, and §7 shows it is also a warning.

4.5 The finite- content

A bound of the shape makes a checkable finite prediction: the envelope statistic must stay bounded, and if the shape is right it should decrease. Fresh values:

**Table 4.** The Theorem-1 envelope. Monotone decreasing with no plateau — consistent with the bound being true but not sharp, i.e. with the conjectured and the exponent being an artifact of the dimension-3 normal form. verified at –; carried.

	0.2772	0.2338	0.2077	0.1872	0.1709	0.1575
	0.3849	0.3310	0.3000	0.2758	0.2567	0.241

Problem 2 (the remaining native question) open

Is the level class infinite? Theorem 1 is compatible with finiteness. Infinitude contains the balanced-prime problem (primes with occupy the level-one, generation-one stratum), so by difficulty conservation this is not expected to be easy — but unlike the twin-prime restatement of §3.2, it is a question that only exists in these coordinates.

5.The constant of the level-one stratum

Theorem 1 gives an upper bound with no constant. For the dominant stratum one can do better and identify the density constant exactly — at the cost of two hypotheses. The constant turns out to be , twice the twin-prime constant, and that identification is unconditional.

5.1 The reduction, restated

By Lemma 5, for ,

So the level-one primes are (up to the two exceptions ) the primes for which is prime, where is 's own gap to the next prime. Equivalently: sits in the middle of a three-term arithmetic progression of primes whose upper step is a prime gap. Two conditions are in play and they must be handled separately:

a triple condition — all prime — which is Hardy–Littlewood territory;
a consecutiveness condition — no primes strictly between and — which is gap-distribution territory.

5.2 Singular series, in one paragraph

Notation

For an admissible tuple , the Hardy–Littlewood singular series is , where . Write for the pair series and for the triple series of the progression. The twin-prime constant is , and .

5.3 The line average — where the twin constant enters

Lemma 1 (line average of the triple singular series) proved

unless , in which case

and

Proof. Vanishing. If then meets both classes mod 2 with , killing the local factor. If then is invertible mod 3 and covers all three residues, so and the factor vanishes. So .

Evaluation for . Then , giving local factors at and at . For : if , contributing ; otherwise , contributing . Collecting the constants and dividing out,

The average. Write with supported on squarefree integers composed of primes and ; then and converges. Hence

the interchange justified by non-negativity together with . Therefore

The evaluation. This is where the twin constant appears, and it appears through an exact per-prime identity. Using ,

since the factor of is . □

The identity at the heart of it

Every step above is bookkeeping except one: the per-prime collapse

which turns a triple-series local factor times a divisor-average local factor into a pair-series local factor. Both sides equal ; the left side simplifies because the awkward factors and cancel. Verified today in PARI/GP symbolically (exact ) and numerically to 38 digits. verified

5.4 The constant , and a correction

The manuscript prints . That is wrong from the eighth significant digit. Two independent routes computed today agree:

Correction to the printed value of

where is the classical Hardy–Littlewood prime-triplet constant. The identification is not a coincidence: , and is the same product with the prefactor . It makes the value independently checkable against the literature rather than against one's own code.

Route one: PARI/GP's zeta-accelerated prodeulerrat at 38 digits — the same routine that reproduces the published exactly. Route two: the direct Euler product over primes to with a rigorous tail bound, bracketing the accelerated value. The misprint is numerically harmless downstream (relative in every prediction using ), but the printed digits should read before submission. verified

5.5 Theorem 2 and its hypotheses

Theorem 2 (conditional asymptotic for Species I) conditional on H1 + H2

Assume:

(H1) the Hardy–Littlewood conjecture for the admissible triples , with uniformity for : ;

(H2) the Gallagher-type model for consecutive gaps: with , uniformly for , with negligible tail.

Then

Proof. By Lemma 5, up to an additive constant counts consecutive pairs with for which is also prime. Given such a pair, (H1) relative to the pair density assigns the extra event " prime" the conditional density , uniformly in the stated range. The interior primality constraints that make consecutive affect the pair count — which is what (H2) supplies — and not, at first order, the singular-series ratio. Hence

By Lemma 1 and partial summation, — the exponential weight has total mass and averages the line against its mean. Therefore . □

Where the argument is not a proof

The step "the interior constraints do not affect the singular-series ratio at first order" is a decoupling assumption, not a deduction. It treats the left point as independent of the compositeness conditions on . The data say this decoupling is wrong by a stable — see §5.6 — and §6 identifies the mechanism (the same residue classes that make an interior point automatically composite also tilt the divisor structure of ). Theorem 2 should therefore be read as: the constant is and the first-order model is right to about one percent, with the percent itself an object of study rather than an error bar.

5.6 The gap-by-gap test — isolating H1

Removing the gap-distribution layer isolates (H1): for each , feed the observed per-gap logarithmic mass through the ratio and compare with the observed level-one count. No gap model, no free parameter.

**Table 5.** Gap-by-gap test at , -prime convention (the two composite- exceptions removed, since the prediction models the primality of ). Singular series computed with the corrected . verified
	observed	predicted	obs/pred
6	672,962	674,806	0.9973
12	504,424	494,452	1.0202
18	373,734	370,238	1.0094
24	267,409	261,157	1.0239
30	368,006	359,816	1.0228
36	125,060	128,860	0.9705
42	135,393	131,025	1.0333
48	63,483	63,326	1.0025
54	47,031	45,012	1.0448
60	62,467	61,042	1.0233
66	26,584	26,664	0.9970
72	14,501	13,829	1.0486
78	12,791	12,642	1.0118
84	11,184	10,638	1.0513
	2,707,697	2,675,408	1.0121

Observed over predicted level-one counts per gap at 1e9 with Poisson error bars, mostly inside a shaded band from 0.97 to 1.05. — **Figure 6.** The full gap-by-gap test with Poisson error bars. Blue: gaps with observations; orange: small-count gaps, where the scatter is Poisson noise and should not be over-read (the outlier at , ratio 1.1318, rests on 484 observations — a fluctuation). Shaded: the window claimed in the manuscript. verified

A claim that must be reworded

The manuscript states that per-gap agreement "lies in at every gap ". Today's run reproduces the seven gaps outside that window exactly — (1.0513), 102 (1.0560), 108 (0.9669), 114 (1.0515), 132 (1.1318), 138 (1.0602), 144 (0.9627) — so the sentence is false as written. What is true, and is what the data support: of the 14 gaps with at least observations, 13 lie inside and the fourteenth () sits at 1.0513, marginally outside; the remaining gaps carry too few observations to distinguish from Poisson scatter. verified

5.7 The constant, honestly

A reader who wants to see emerge from the raw data will be disappointed, and it is important to say so plainly. The windowed Species-I density rises through

0.9040 ( $Misplaced &10^5,10^6\])  →  0.9472  (\(10^6,10^7\])  →  0.9804  (\(10^7,10^8\])  →  1.0079  (\(10^8,10^9\]) <span class="tag t-ver">verified</span> </p> <p>— still some 24% below the conditional asymptote at \(10^9$ , and approaching it only logarithmically. This is exactly what the model predicts: at the weight has barely begun to sample the large gaps whose enhancements raise the line average to its limit. The identification of the constant therefore rests on Lemma 1's exact identity plus the gap-by-gap test above — not on the raw density at accessible heights. Any presentation that shows only the raw density is misleading in one direction and any that shows only the identity is misleading in the other; Figure 5 shows both.

Three normalised diagnostics against x on a log axis: a decreasing envelope, a slowly drifting Species-II normalisation, and a rising windowed density approaching but well below the 2C2 line. — **Figure 5.** Three bounded diagnostics on one square canvas. Blue: the Theorem-1 envelope, decreasing. Orange: the Species-II normalisation , drifting slowly downward through 0.8046. Green: the windowed Species-I density, rising toward the dashed asymptote but far from reaching it. The honest reading is *bounded decrease and bounded increase* — none of the three has plateaued, and no constant should be extracted from any of them by extrapolation. verified

5.8 Species II — the level-above-one stratum

The stratum carries about 69% of the level class at and has no identified constant. Two empirical normalisations are stable:

**Table 6.** Species-II diagnostics, decade windows and cumulative. normalises against the working law ; normalises against the Mertens mass , the heuristic probability that an integer of size has no prime factor in . verified
window
windowed	0.7690	0.7705	0.7669	0.7656
windowed	0.4445	0.4458	0.4432	0.4420
cumulative at the upper end	0.8312	0.8204	0.8108	0.8046
cumulative at the upper end	0.4469	0.4460	0.4436	0.4422

The windowed sits in a band around across four decades; the cumulative diagnostic nonetheless drifts steadily downward, which is what refuted the earlier working conjecture . That conjecture is retracted and the retraction stands. Identifying either constant is open, and the natural technology is the theory of divisors of shifted primes — Ford's and Koukoulopoulos's work — since is literally a Mertens-normalised count of integers with no divisor in a short interval.

6.Exact structure: weight columns and the congruence lock

This section contains the framework's cleanest results: statements that are exact, provable in a paragraph, and false in any model that ignores arithmetic. They are also the clearest instances of the dictionary the construction advertises — a multiplicative weight column turns out to be a finite union of additive, residue-conditioned gap events.

6.1 Column identities

Column identity for

proved

For every : .

Proof. () If then , and is even and positive, so . By Lemma 4, forces . Hence , and .

() Suppose and . The weight is the least divisor of exceeding 4; the interval contains no integer, so that divisor is . □

Column identity for

proved

Proof. forces even, so , and is claimed by (Lemma 4). For : the interval is empty, so . For : the integers in are and ; is odd so ; hence and . □

verified at , on the nose:

; .

These identities also yield a clean measurement of an independence question. Within , the joint divisibility is observed times against the product prediction — a deviation of , or with (binomial). The two congruence locks are statistically independent at to the available resolution. verified

Figure 7. The weight columns at

. The profile is strikingly non-monotone, and the reason is arithmetic rather than noise: by the identities above, the column

draws on a single gap value while

draws on two, so the two are not comparable a priori. The alternation between prime weights (blue) and composite weights (orange) reflects the same mechanism at higher

. The column

— Lemma 4, the twin column — carries 3,424,505 primes, of which exactly one (

) is level-classified. verified

Logarithmic column chart of level occupancy at 1e9, showing only odd levels with those divisible by 3 systematically higher. — **Figure 8.** The level lines at . Only odd occur (Lemma 3), and the levels divisible by 3 (orange) stand systematically above their neighbours — the mirror, on the level side, of the mod-3 rigidity that shapes the weight side. The line is Species I. An occupancy law for individual level lines is open and is not pursued here. verified

6.2 The congruence lock

The next observation is trivial to state and surprisingly productive.

The lock proved

For an odd prime and a decomposable prime with gap ,

immediately, since .

So a divisibility condition on — a multiplicative fact, and one of the inputs to the weight — is identical to a residue condition on . If one now asks how often the lock closes, a flat prediction suggests itself: among gap- primes, is confined to the residue classes mod that are neither nor , and if it were equidistributed among them the lock class would occupy a share whenever .

Measured at , that flat law is right to within a small but structured bias: exact when , and deficient by up to as . The bias has an explanation.

6.3 The interior-credit law

The mechanism

Fix and an odd prime , and condition on being consecutive primes. Consecutiveness means every interior even offset has composite. Now observe: if , then , and the compositeness of that interior point is guaranteed for free. That residue class needs one fewer accident to achieve gap exactly , so it is over-represented among gap- primes. The lock class earns no such credit — its offset is odd, hence never in — so it is under-represented relative to the credited classes. That is the observed deficit.

First-order bookkeeping makes this quantitative with no free parameter. Let

be the Hardy–Littlewood conditional probability that the interior point is prime given the prime pair — the "prime risk" that the credit removes — where is measured within the gap- population. Weight each admissible residue by

the factor renormalising the risk of the non-credited points once is known. Each lands in exactly one admissible class (as ), so expanding at the lock class , where , gives the closed form

The structural zeros, explained

: , so exactly. : and the pattern covers all residues mod 3, so , giving and again — the interior point of a pair is always divisible by 3, so no prime can earn credit there. Measured:

: · : · : verified

The parameter-free test

Over all 392 cells with even , prime , and at least 5,000 gap- primes and 200 lock hits, the law predicts the measured deficit with and zero fitted constants — reproducing the full shape of the bias surface (both its growth in and its per- fine structure), with magnitudes running about 19% low. A single global factor

(weighted least squares through the origin) absorbs the second-order enhancement and gives over 391 degrees of freedom. Flagship cells, measured against the -corrected prediction:

**Table 7.** The interior-credit law at representative cells, . "Free" is the parameter-free first-order prediction; "" applies the single global second-order factor. verified
	measured	free
(2, 5)	1.0005 ± 0.0009	1.0000	1.0000
(4, 5)	1.0000 ± 0.0009	1.0000	1.0000
(4, 7)	0.9996 ± 0.0012	1.0000	1.0000
(6, 7)	0.9683 ± 0.0009	0.9741	0.9681
(12, 13)	0.9537 ± 0.0015	0.9618	0.9530
(30, 31)	0.9466 ± 0.0036	0.9555	0.9452

Scatter of measured against predicted lock deficit over 392 cells, tightly clustered along a line slightly steeper than the identity. — **Figure 9.** The interior-credit law over all 392 cells at : measured deficit against the parameter-free prediction, on a square canvas with equal aspect so a slope of 1 reads as . Colour encodes . The structural zeros at cluster at the origin; the orange line is the fitted second-order factor . The scatter about that line is consistent with the quoted binomial errors. verified

Status of §6.3

The derivation is first-order local calculus with an independence step, so the law itself is heuristic; its predictive success at is verified as stated; the structural zeros at are proved given pattern inadmissibility. What is open is the second-order constant: deriving from the next order of the same calculus (pairwise interior patterns, i.e. , plus the tilt that conditioning on "gap exactly " induces on the environment). The same machinery summed over residues rather than resolved by them is the natural route to the consecutiveness excess of §5.6 — the two are the same phenomenon seen from two sides.

7.The deflationary control

Any framework that produces curves must be asked the following question, and most are not: would a random sequence of the same density produce the same curves? Here it is asked, and part of the answer is uncomfortable.

Two random models were run through the identical classifier this session, at , with a fresh seed:

Model A — the classic Cramér sequence: each is included independently with probability .
Model B — parity-matched: each odd is included with probability , so that gaps are even and is odd, exactly as for the primes.

**Table 8.** The primes against two random controls, same classifier, same run. is the level share, the level-one share, both among decomposable terms. verified
primes	model A	model B	primes	A	B
0.2338	0.2384	0.2908	0.0758	0.0793	0.1596
0.2077	0.2093	0.2543	0.0662	0.0665	0.1338
0.1872	0.1878	0.2265	0.0590	0.0579	0.1158
0.1709	0.1708	0.2049	0.0533	0.0511	0.1021

Level share against x on a log axis for primes and two random controls; the classic Cramér control tracks the primes closely while the parity-matched control sits well above. — **Figure 4.** Rarefaction against its controls. Upper curves: the level share . Lower curves: the level-one share . The classic Cramér model (green, dashed) tracks the primes to within 1% at every decade; the parity-matched model (pink, dotted) — which one might naively expect to be the *fairer* control — overshoots badly. verified

The verdict, and it cuts both ways

The aggregate level share of the primes is generic. Model A tracks within 1% at every decade and to 0.1% at (0.1708 against 0.1709 — the three-decimal coincidence at the last decade is luck; the percent-level tracking across six decades is not). It follows that the rarefaction rate of Theorem 1 must not be presented as prime-specific structure: the decay shape is what any -density sequence does under this construction. Theorem 1 is a theorem about the construction as much as about the primes, and §10 argues that this is a feature.

The composition is arithmetic through and through. The parity-matched model overshoots the level share by 20% and doubles Species I (0.1021 against 0.0533 at ), because the real is prime with the Hardy–Littlewood conditional density , not the bare of a random odd sequence. Every one of §6's exact statements is false in both models. The framework's genuine content at the level of counting is the species split, its constants, and the exact lock and column structure — not the aggregate decay.

8.The ledger

What is proved, what is conditional, what is open, and — the column most often missing from expositions of a private framework — what is a restatement of something already hard.

**Table 9.** The complete status ledger. "Native" marks statements that cannot be formulated without the weight–level coordinates; "classical" marks statements that are known problems in disguise.
Item	Statement	Kind	Status
Prop. 1	Eratosthenes collapse on	native	proved
Lem. 2	decomposable ; exceptions	native	proved
Lem. 3	divisor localisation; ;	native	proved
Lem. 4	(twin column)	native	proved
Prop. 2	(founding Conj. 7, 8)	native	proved, elementary — never a contribution
Prop. 3	cube bound , sharp at 131 (founding Conj. 4)	native	proved
Lem. 5	level-one reduction; exception sets to	native	proved
Lem. 1	line average of equals	classical tools	proved
§6.1	column identities for ,	native	proved
Thm. 1	rarefaction (founding Conj. 9)	native	proved* — awaiting refereeing
Thm. 2		native + HL	conditional on H1, H2
§6.3	interior-credit law for the lock bias	native	heuristic; fit verified
§6.3	second-order constant	native	open
Prob. 1	unconditionally	native	open
Prob. 2	is the level class infinite?	native	open — contains balanced primes
Prob. 3	identify or	native	open (Ford–Koukoulopoulos)
Prob. 4	the consecutiveness excess	native	open; mechanism identified §6.3
—	working claim	—	retracted
Conj. 1	infinitely many primes	classical	open — is twin primes
Conj. 2,3,5,6	infinitude/structure of fixed columns and lines	classical	open — Polignac-type

Difficulty conservation

Recasting a classical problem in weight–level coordinates does not lower its difficulty. Twin primes remain twin primes when called "the column ". Polignac remains Polignac when called "the column ". Goldbach is not visible here at all. Nothing in this document bears on the Riemann Hypothesis. The framework's yield is the native row set — Theorem 1, Propositions 1–3, Lemmas 1–5, the column identities, the lock structure — and it should be judged on that and on nothing else.

9.Algorithms

Four algorithms compute the same function and are used here as mutual checks. All are in PARI/GP, following the project's convention that the reference implementation is the readable one.

9.1 The reference kernel

\\ The definition, transcribed. O(tau(l)) after one factorisation.
decomp_fact(n, n1) =
{ my(d, l, D);
  d = n1 - n; l = n - d;
  if (l <= d, return([0, 0, d]));          \\ not decomposable
  D = divisors(l);                          \\ sorted ascending; D[#D] = l > d
  for (i = 1, #D, if (D[i] > d, return([D[i], l/D[i], d])));
}

\\ Trial division with a parity wheel and the descending level search.
\\ Theta(sqrt l) worst case, but no factorisation is required.
decomp(n, n1) =
{ my(d, l, s, k0, step);
  d = n1 - n; l = n - d;
  if (l <= d, return([0, 0, d]));
  s = sqrtint(l);                           \\ exact integer square root
  if (l % 2, k0 = d + 1 + !((d+1) % 2); step = 2, k0 = d + 1; step = 1);
  forstep (k = k0, s, step, if (l % k == 0, return([k, l/k, d])));
  \\ Level branch.  If l has no divisor in (d, sqrt l], then k > sqrt l and
  \\ L = l/k is the LARGEST divisor le of l with le <= d AND l/le > d.  The
  \\ cofactor clause is essential: divisor pairs lying entirely below d occur
  \\ exactly for the empty-window primes 13, 31, 113 (e.g. l = 9, d = 4: the
  \\ divisor 3 has cofactor 3 <= d and must be skipped; the answer is k = 9).
  forstep (le = if (l % 2, d - !(d % 2), d), 1, -if (l % 2, 2, 1),
    if (l % le == 0 && l/le > d, return([l/le, le, d])));
}

decomp_auto(n, n1) = if (n < 10^8, decomp(n, n1), decomp_fact(n, n1));
dclass(r) = if (!r[1], 0, if (r[1] > r[2], 2, 1));   \\ 0 none, 1 weight, 2 level

The complexity crossover is real and worth stating: trial division costs on every level-classified term, whereas factoring a random is almost always far cheaper. At the two are comparable; at trial division would need iterations while decomp_fact answers instantly. Deep witnesses computed today:

? decomp_fact(1000000000039, nextprime(1000000000040))
%1 = [461, 2169197397, 22]
? decomp_fact(10^18 + 3, nextprime(10^18 + 4))
%2 = [47, 21276595744680851, 6]
? decomp_fact(10^24 + 7, nextprime(10^24 + 8))
%3 = [11909, 83970106642035435385, 42]

9.2 The census classifier

For a full census to neither kernel is fast enough. The C engine used here avoids factorisation entirely by exploiting the structure of the question. Write , where is the -smooth part (primes ) and the rough part (every prime factor ); let and let be the smallest divisor of exceeding ( if none). Since and weight-classification means :

**Table 10.** The classification decision table. Each case is a two-line argument; together they classify without ever factoring . Only the last row requires a primality test, supplied by deterministic Miller–Rabin on bases (valid below ).
case	reason	outcome
	is -smooth; enumerate divisors of	level iff ; then
	itself is a divisor in	weight
composite	and	weight
prime,	all divisors involving exceed	weight iff ; else level with

In the last case one checks that automatically, since . The engine streams consecutive primes from a segmented odd-bitmap sieve and classifies 50,847,531 decomposable primes in 46 seconds on one core. Every decomposable prime below is additionally re-classified inside the same run by two independent reference paths — a wholly naive upward divisor scan for , and the project's orig_sieve logic for — with a zero-mismatch gate. The rows are then dumped and diffed row by row against an independent Python implementation that enumerates full divisor sets. All gates passed.

10.Horizon: what these coordinates could do to number theory

This section is deliberately expansive. It is also, deliberately, the only section of this document where speculation is permitted — and every claim in it is labelled. Nothing here is a result; several things here are, in my judgement, worth a decade of somebody's attention.

Reading key for this section

Program — a concrete research direction with an identifiable existing technique and a plausible route to a theorem. Conjecture — a precise statement I believe, with stated evidence. Speculation — an idea with no evidence beyond its own coherence. I will not blur these.

10.1 The reframing: rarefaction is a universality class, not a fact about primes

Section 7 delivered what looked like bad news: the Cramér model reproduces the level share of the primes to within a percent across six decades. The instinct is to read this as deflation. I think that reading is exactly backwards, and that the correct reading is the most consequential idea in this document.

What §7 actually shows is that the map

is insensitive to arithmetic and sensitive only to density. That is the signature of a universality class. And a universality class is not a nuisance; it is a baseline. Once the generic backdrop is known exactly, everything that deviates from it is signal, and the deviation is measurable to four decimals over samples.

Conjecture A (rarefaction universality) — conjecture

Let be a random increasing sequence in which is included independently with probability . Then almost surely the level share satisfies for an absolute constant , and the same asymptotic holds for the primes with the same .

Evidence. Six decades of agreement between the primes and Model A to within 1% (Table 8); the fact that Theorem 1's proof uses no arithmetic input beyond a standard upper-bound sieve and gap telescoping (§4.4); the fact that the conjectured true order is exactly what a naive Mertens computation gives for a random sequence.

Why this matters. Conjecture A is almost certainly provable — a random model has no consecutive-prime coupling, so the lower bound that is out of reach for the primes is a routine second-moment computation for . If the constant can be computed exactly for the random model, then the decompwlj census becomes a spectroscope: measure for a real sequence, subtract the universal , and what remains is a number that no random model produces. For the primes that residual is currently against a universal constant we have not computed. Determining is, in my view, the highest-value cheap computation in this whole programme, and it is a self-contained probability problem.

Program 1 — the universal constant program

Compute exactly for the Cramér model; prove Conjecture A for the model; then re-express every Species-I and Species-II diagnostic of §5 as a ratio to the universal prediction. Expected outcome: the Species-II constant is revealed as (universal constant) (an arithmetic factor built from and ), and the arithmetic factor is the thing worth naming.

10.2 The function-field analogue: where the theorems are actually provable

The obstruction throughout this document has one name: the word "next". The decomposition of depends on , and conditioning on "no primes in between" is a technique we do not have over .

Over we do have it, or something close enough. The analogues of the Hardy–Littlewood conjectures in the function-field setting are, in the large- limit and in several fixed- ranges, theorems — obtained by geometric methods (monodromy computations, Sawin–Shusterman-type results, Bary-Soroker's work on irreducible polynomial tuples). Where the number field forces a hypothesis, the function field supplies a proof.

Program 2 — decompwlj over

program

Fix odd. Order the monic irreducibles of by degree and then lexicographically (any total order compatible with degree will do; the choice is a real one and its consequences should be studied). Define , , the weight as the least monic divisor of exceeding in the order, and . The programme is then:

establish the analogue of Proposition 1 — the construction should collapse to the sieve on , giving a genuine base case;
prove the analogue of Theorem 1 unconditionally and with the right exponent, since the dimension-3 sieve input is available as a theorem rather than a hypothesis;
prove the analogue of Theorem 2 with its constant — that is, prove unconditionally in what is conditional over ;
and — the item I would care about most — compute the analogue of the consecutiveness excess exactly, since in the function field the conditioning on "no irreducibles in between" is a computation on a moduli space rather than a hypothesis.

Item 4 is the reason this direction is worth doing. Problem 4 over is a number nobody can compute; over it may be a monodromy calculation.

I rate this the most likely of everything in §10 to produce a publishable theorem within a few years, and the least glamorous-sounding.

10.3 A quantitative calculus of consecutiveness

The interior-credit law of §6.3 is presented there as a fix for a small bias. I think it is the most transferable idea in the programme, and it deserves to be stated independently of weight and level.

The general device

Standard Hardy–Littlewood heuristics predict the frequency of an event for prime. What they do not do is predict conditioned on " exactly". Yet that conditioning is ubiquitous — it appears in Gallagher's model, in Montgomery–Soundararajan, in the recent work of Gafni and Tao on rough numbers between consecutive primes, and in every empirical study of prime gaps. It is almost always handled by assuming the conditioning is harmless.

§6.3 shows it is not harmless and that its first-order effect is computable from singular series alone: conditioning on gap exactly tilts the residue distribution of modulo any , by an amount determined by which interior points the residue class kills for free. The tilt is a – effect at accessible parameters, it has a closed form with no free constants, and it was confirmed here against 392 aggregate cells and 237 residue-resolved classes.

Program 3 — consecutiveness conditioning as a tool program

Develop the interior-credit calculus to second order (pairwise interior patterns, i.e. , plus the environmental tilt), derive the constant rather than fitting it, and publish the device on its own terms: a first-principles correction factor for any Hardy–Littlewood prediction conditioned on a prescribed prime gap. If that works, its natural consumers are not in this framework at all.

This is the item where I would place the highest probability of the decompwlj programme contributing something that outlives it.

10.4 Divisors of shifted primes, with a moving window

Recall from §1.4 that level classification is exactly " has no divisor in ". The theory of divisors of integers in short intervals is a mature subject — Erdős's multiplication table problem, Tenenbaum's -function, Ford's determination of , Koukoulopoulos on divisors of shifted primes. What decompwlj contributes to that subject is a new and slightly perverse normalisation: the integer is a shifted prime , and the shift is the prime's own gap — so the window's left endpoint is a random variable correlated with the shift.

Problem 3′ (sharpened) — program

Determine the asymptotic behaviour of

The denominator is the naive Mertens mass. Measured: at –, drifting downward by about per decade. The relevant analytic technology is Buchstab-type: the density of integers free of prime factors in a long interval is governed by the Buchstab function, and the correction factor here should be expressible through it, modified by the correlation between and the shift. I have not computed this constant and do not claim a value for it. Whether the drift terminates at a constant or continues logarithmically is, on the present data, undecidable — and saying so is the only defensible position.

The bold claim attached to this item is not about the constant. It is this: the decompwlj census is, incidentally, a high-precision dataset for the divisors-of-shifted-primes literature, and it is currently the only one indexed by the prime's own gap. That dataset has value independent of every theorem in this document.

10.5 The atlas as an instrument

About a thousand OEIS sequences have been decomposed and published at decompwlj.com. Today that is an archive. It could be an instrument.

Program 4 — regime certificates across the atlas program

For each decomposed sequence compute a fixed vector of invariants: the decomposable fraction (does the barrier bite?), the level share and its decay exponent, the species split , the weight-column profile, the tie density, and the residual against the Conjecture-A universal baseline. Then cluster.

The expected outcome is a taxonomy of integer sequences by a canonical additive–multiplicative invariant — not by growth rate, not by generating function, but by how the sequence's own increments factor. To my knowledge no such classification exists. Sequences that land in the same cluster as the primes despite having no evident relation to the primes would be, at minimum, interesting; and a sequence whose residual against the universal baseline is anomalously large is a sequence with hidden multiplicative structure.

This is also the natural home for machine-assisted conjecture generation. The invariants are cheap, the sample is a thousand sequences, and the ground truth (the decomposition itself) is exactly computable — which is precisely the setting where automated pattern search is worth trusting.

10.6 Beyond the barrier

The construction is undefined whenever . That excludes every sequence of exponential growth, which is to say most sequences one might care about. Two responses.

Speculation 1 (generations). The founding framework already speaks of "generations" — balanced primes are the level-one, generation-one stratum. Formalise it: when , pass to , and generally for the least with , if any. One obtains a decomposition at generation . This is a natural extension, and it is not obviously canonical — the minimality that pins the weight in Definition 1 does not obviously pin . I do not know whether a canonical extension exists, and I would not assume one does because the notation permits it.

Speculation 2 (multiplicative jumps). For geometric-type sequences the right analogue of the jump may be the ratio rather than the difference, which would make the construction a statement about the -adic or logarithmic structure of the sequence. This is pure speculation; I have no evidence that the resulting object has any of Definition 1's rigidity.

10.7 A two-variable object nobody has looked at

Speculation 3 — the weight–level zeta function

For a decomposable sequence define, formally,

The classification is the geometry of this object: the level class is the region where the -variable dominates. On the diagonal it degenerates to , which for the primes is a shifted-prime zeta function; off the diagonal it separates the two coordinates. Does admit meromorphic continuation? Does the line carry a distinguished singularity whose residue is the species split? I have no evidence for any of this. I include it because the object is natural, the framework produces it for free, and nobody appears to have written it down. The honest expectation is that is analytically intractable for the same reason everything else here is: the coupling to .

10.8 What would count as failure

A section like this is worthless without its negation. Here is the scenario in which the decompwlj programme's mathematical yield turns out to be small, and it is not far-fetched:

Conjecture A is proved, the universal constant is computed, and the primes' residual against it turns out to be exactly what bookkeeping already predicts — no new invariant.
The Species-II constant is identified and is a Buchstab evaluation already in the literature under another name.
The function-field analogue works but is a routine transcription, interesting to nobody outside the transcription.
Problem 2 (infinitude of the level class) stays open forever, as balanced primes have stayed open.

In that scenario what survives is: the rigidity results of §3 (which are small, sharp, and permanent), Theorem 1 (a genuine theorem answering a genuine question that only these coordinates can pose), the column identities of §6.1, and the interior-credit calculus of §6.3 (which may well outlive its origin). That is a real but modest body of work, and it would be an honourable outcome. Any presentation of this framework that cannot say so is not to be trusted.

10.9 The largest claim I am willing to make

Strip away the specifics and one structural fact remains. The decomposition takes an additive datum — the gap to the next term — and uses it to select a multiplicative datum — a divisor of the term. It does so canonically, with no parameter, for every increasing sequence, and on the natural numbers it reproduces the sieve of Eratosthenes exactly. Constructions with that property are rare. The additive–multiplicative divide is the central structural fact of number theory, and the objects that sit astride it — the circle method, singular series, the Selberg sieve, Erdős's multiplication table, the parity barrier itself — are the field's load-bearing walls.

The weight–level–jump decomposition is not going to knock down a wall. What it does is provide an explicitly computable, exactly verifiable, parameter-free instance of the divide, on which every heuristic in the subject can be tested against exact samples in under a minute of compute. In a field where most heuristics are tested against a handful of small cases or not at all, that is worth having. Theorem 1 shows the instance is not vacuous — a real question was posed in the coordinates and answered in them. Section 7 shows the instance is not credulous — the framework's own control experiment falsified one of its natural readings, and the framework survived by narrowing its claim.

A construction that poses answerable questions and destroys its own overclaims is doing what a piece of mathematics is supposed to do. Whether it eventually earns a place in the subject depends on §10.2 and §10.3 — on whether the function-field analogue yields unconditional theorems and whether the consecutiveness calculus works outside its home. Both are testable. Neither requires anyone to believe anything.

A.Verification record

Everything numerical in this document was recomputed today from source. The directory was empty at the start of the session; no data, no binary and no intermediate file was reused from a previous edition.

A.1 Engines

C census engine (written this session): Segmented odd-bitmap sieve, span , streaming consecutive primes to ; classification by the smooth/rough decision table of §9.2; exact on the level side, exact up to 256 on the weight side; deterministic Miller–Rabin on bases . Accumulates the census, the invariant counters, the joint histogram, weight and level histograms, per-gap level-one counts and logarithmic masses, lock numerators for and , residue histograms for , the Mertens mass, the exception lists and the maximal-gap ladder. Runtime 46 s, single core.
Internal gate: Every decomposable prime is re-classified by a wholly naive upward divisor scan, and every decomposable prime by the project's orig_sieve logic, inside the same run. Any disagreement aborts the run. Zero mismatches.
Python audit (written this session): Shares no code with the C engine: primes from a plain Eratosthenes list, full divisor sets from a trial factorisation, classification re-derived from the divisor set. All 148,930 decomposable primes below compared tuple by tuple . Zero mismatches. The invariants of §3 were independently re-derived over the same range.
PARI/GP 2.15.4 (project kernels): Four algorithms — orig_naive, orig_sieve, decomp, decomp_fact — agree on the 17 rows of Table 1 and on every prime below . Independent census to : 78,495 decomposable, 18,353 level, 5,953 level-one, 12 ties, 8,168 with , 8,168 with , zero mod-3 violations — equal to the C stream digit for digit. Proposition 1 re-verified for . Constants computed at 38 digits by prodeulerrat; the Lemma-1 per-prime identity verified symbolically to exact zero.
Random controls: Models A and B of §7, each, same classifier, seed 20260816 (a different seed from previous editions — the agreement with the record is therefore not a seed artifact).

A.2 External anchors reproduced

; the three non-decomposable primes give 148,930 decomposable, which is the point count of Figure 2.
The maximal-gap ladder A002386, entry for entry, ending at 387,096,133 and at 436,273,009 below .
(published twin-prime constant).
(published prime-triplet constant), giving
Deep decomposition witnesses at , , (§9.1), well beyond the census range.

A.3 Corrections and notes arising this session

The constant . The manuscript's printed is wrong from the eighth significant digit; the correct value is . Confirmed independently here by two routes (§5.4). Numerically harmless downstream (relative ), but the digits must be corrected before submission.
The per-gap window sentence. The manuscript's claim that obs/pred lies in "at every gap " is false as written; seven gaps fall outside, six of them on fewer than observations. The defensible statement is the one given in §5.6.
Figure-2 point count. Prior editions of this analysis quote 148,932 decomposable primes below . The correct count is , confirmed by three engines. Prior editions were off by two.
Convention note for the gap-by-gap test. The prediction models the primality of , so the two composite- exceptions (at ) and (at ) must be excluded from the observed counts. With the convention stated, the counts of Table 5 are exact; without it, reads one higher. This is a convention, not an error.
Lock cell census. This session admits 392 cells under a slightly more permissive inclusion rule than the 341 of the eighth edition; the fitted second-order factor is against 1.2324, and every flagship cell agrees to the fourth decimal. The two runs are consistent.

A.4 What was not done here

The row of Table 2 is carried from the manuscript's census. It was not recomputed this session.
The closure of the exception sets of Lemma 5 and Proposition 3 to rests on the refereed gap tables of Oliveira e Silva, Herzog and Pardi. Only the closure to was recomputed here.
Theorem 1's proof was audited for internal consistency, not refereed. The asterisk stands.