Decomposition into weight × level + jump — source-audited deep analysis

0. Scope, evidence and status

The construction is mathematically elementary to define, but its prime application mixes three distinct layers that must not be conflated: exact elementary identities, finite computation, and asymptotic number theory. This report keeps those layers separate.

proved Stable core

The definition, existence criterion, uniqueness by minimality, the divisor-window characterization, the exact natural-number reduction, the twin-column equivalence, mod-3 rigidity, and the elementary growth trichotomy admit short complete proofs.

verified Finite evidence

The uploaded reports give a complete census to 10¹⁰. This analysis does not rerun that 455-million-prime computation. It does independently recompute every prime through 10⁶ and verifies the fixed-window theorem through 2·10⁶.

conditional Asymptotics

The level-one constant $2C_2$ is derived under Hardy–Littlewood uniformity plus a consecutive-gap model. The singular-series average giving $2C_2$ is elementary/analytic; the passage to consecutive primes is conditional.

open Native questions

The infinitude and true asymptotic of the level class remain open. The project manuscript claims an unconditional density-zero upper bound; it is a serious internal proof but is not yet externally refereed.

Source hierarchy. The 2007–2010 arXiv paper and OEIS page define the framework and original conjectures. The July 2026 manuscript supplies the rarefaction proof and conditional level-one asymptotic. The fifth and sixth reports supply computation, taxonomy and later conjectural interpretation. The website itself warns that its thousand-sequence data have not been independently verified; this report therefore treats atlas-wide claims as a research resource, not as certified data.

1. Significance of the framework

The decomposition assigns multiplicative coordinates to a locally additive event. A strictly increasing sequence is observed through its next gap $d(n)$; that gap determines the shifted integer $\ell(n)=a(n)-d(n)$; the factorization of $\ell(n)$ then determines a distinguished divisor $k(n)$ and cofactor $L(n)$. The exact identity

$$a(n)=k(n)L(n)+d(n)$$

is therefore not merely a factorization of $a(n)$. It is a factorization of a number displaced from $a(n)$ by the next additive gap. This coupling is the genuine mathematical content.

1.1 What is genuinely new

The coordinate system is canonical: no parameter, optimization criterion or probabilistic model is chosen. It makes several familiar prime sets literal strata in one plane. Lesser twin primes form the weight-3 column; balanced primes form generation $(1;1)$ of the level-one stratum; fixed-gap and prime-tuple patterns become organized by weight and level.

The strongest conceptual advance in the later work is the divisor window. Level classification is equivalent to the absence of a divisor of $\ell$ in $(d,\sqrt\ell]$. Thus the construction transforms an additive statistic—how far the sequence jumps—into a moving multiplicative threshold.

1.2 What is not new

Rewriting twin primes as “infinitely many primes of weight 3” is an equivalence, not progress on the twin-prime conjecture. The same warning applies to fixed weight columns, fixed level strata and balanced primes. The framework earns mathematical value when it produces statements that do not exist before the coordinates are introduced: the classification density, the divisor-window taxonomy, exceptional-set bounds, and gap-conditioned divisor statistics.

Critical boundary. Nothing currently connects the framework directly to the zeros of $\zeta(s)$, the explicit formula, or the Riemann Hypothesis. The operative scale is the prime-gap scale, roughly logarithmic, not the square-root error scale of RH.

2. Definitions and elementary structure

Let \(a(1)

Jump: $d(n)=a(n+1)-a(n)$.

Auxiliary term: $\ell(n)=a(n)-d(n)$ when $a(n)-d(n)>d(n)$, and $0$ otherwise.

Weight: if $\ell(n)>0$, $k(n)=\min\{k>d(n):k\mid\ell(n)\}$; otherwise $k(n)=0$.

Level: $L(n)=\ell(n)/k(n)$ when $k(n)>0$, and $0$ otherwise.

Classification: level if $k(n)>L(n)$; weight if \(0

2.1 Existence and uniqueness

Existence criterion. The decomposition exists exactly when $$a(n+1)<\frac32a(n).$$ Indeed $\ell>d\iff a-d>d\iff a>2(a(n+1)-a)\iff 3a>2a(n+1)$. If it exists, $\ell$ itself is a divisor exceeding $d$, so a least such divisor exists; minimality makes $k$, hence $L$, unique.

2.2 The divisor-window form

Since $kL=\ell$, level classification satisfies $k>L\iff k>\sqrt\ell$. Because $k$ is the first divisor above $d$,

$$\boxed{\text{level-classified}\iff \ell\text{ has no divisor in }(d,\sqrt\ell].}$$

This formulation is more informative than the original inequality: classification asks whether a factorization appears inside a moving interval whose lower endpoint is additive and upper endpoint multiplicative.

2.3 Elementary level bound

If a term is level-classified, then \(L

2.4 Growth trichotomy

The sixth report introduces an elementary but useful taxonomy. Writing $a=a(n)$ and $d=d(n)$, exactly one regime holds:

N — nondecomposable

$a\le2d$. Then $\ell=0$.

Z — zero/empty window

\(2d

W — window open

$a>d(d+1)$. The interval $(d,\sqrt\ell]$ is nonempty; arithmetic decides whether it contains a divisor.

Interpretation

Much of the atlas is classified by growth before factorization matters. The interesting arithmetic resides in W.

Square graph of the growth trichotomy — **Figure 2.1.** The two absolute boundaries $a=2d$ and $a=d(d+1)$. The square canvas preserves the geometry of the $(d,a)$ plane.

Boundary correction. If $a(n)\sim c n^2$, then $4c>1$ forces eventual Z and $4c<1$ forces eventual W. At the exact boundary $4c=1$, the leading coefficient alone is insufficient; lower-order terms can decide. The latest sixth report states the equality case too categorically.

2.5 First prime examples

p	k	L	g	classification
2	0	0	1	not decomposable
3	0	0	2	not decomposable
5	3	1	2	level
7	0	0	4	not decomposable
11	3	3	2	weight (tie)
13	9	1	4	level
17	3	5	2	weight
19	5	3	4	level
23	17	1	6	level
29	3	9	2	weight
31	25	1	6	level
37	11	3	4	level
41	3	13	2	weight
43	13	3	4	level
47	41	1	6	level
53	47	1	6	level
59	3	19	2	weight

The mechanics are visible at $p=89$: the next prime is 97, so $g=8$ and $\ell=81$; the least divisor of 81 exceeding 8 is 9, hence $(k,L,g)=(9,9,8)$, a tie assigned to weight. At $p=887$, $g=20$, $\ell=867=3\cdot17^2$, and the first divisor above 20 is 51, giving $(51,17,20)$, a level classification.

3. Relation with the Fundamental Theorem of Arithmetic and the sieve of Eratosthenes

3.1 Exact collapse on the natural numbers

For $a(n)=n$, the jump is constant: $d(n)=1$, $\ell(n)=n-1$. The first divisor of $n-1$ exceeding 1 is its smallest prime factor:

$$k(n)=\operatorname{spf}(n-1),\qquad L(n)=\frac{n-1}{\operatorname{spf}(n-1)}.$$

If $n-1$ is prime, then $(k,L)=(n-1,1)$, so $n$ is classified by level. If $n-1$ is composite, $\operatorname{spf}(n-1)\le\sqrt{n-1}\le L$, so it is classified by weight. Therefore the level-classified naturals are exactly the shifted primes $p+1$, while the weight columns are the classes first struck by each prime in the sieve.

Independent audit. For every $3\le n\le10^6$, this report recomputed $k(n)$ and found zero violations of $k(n)=\operatorname{spf}(n-1)$. The number of level-classified terms is 78,498, exactly $\pi(10^6)$.

3.2 The FTA relation, stated precisely

The construction does not prove the Fundamental Theorem of Arithmetic. It selects the smallest prime factor in the natural-number case. Iterating on the cofactor recovers the prime factorization, but the uniqueness of the decomposition triple is only uniqueness of a selected minimum; FTA asserts uniqueness among all prime factorizations. The correct statement is that the decomposition exposes the first step of the FTA algorithm and renders the Eratosthenes sieve geometrically.

Square padded original sieve graph — **Figure 3.1 — supplied `sieveNb.jpg`.** Vertical columns correspond to smallest-prime-factor classes. The level-one ray is the shifted-prime set. Labels in the original image should be read as residue/sieving classes rather than as a new proof of primality.

Square graph of three million natural-number decompositions — **Figure 3.2 — supplied `naturaldecomp3M-2048.jpg`.** Three million naturals. The triangular/hyperbolic envelope reflects $kL=n-1$, while the vertical comb records the distribution of smallest prime factors.

3.3 Fixed-window counting and bounded jumps

The recent taxonomy isolates a general counting problem. For fixed $B$, let

$$W_B(x)=\#\{m\le x:m\text{ has no divisor in }(B,\sqrt m]\}.$$

A window-free $m>B^2$ has at most one prime factor above $\sqrt m$; apart from $B$-smooth exceptions it has the unique form $m=sP$, with $s\le B$ and $P$ prime. This yields

$$W_B(x)=\sum_{s\le B}\bigl(\pi(x/s)-\pi(s)\bigr)+E_B(x),\qquad E_B(x)=O_B\bigl((\log x)^{\pi(B)}\bigr),$$

and hence $W_B(x)\sim H_Bx/\log x$, where $H_B=\sum_{s\le B}1/s$. The statement is elementary for fixed $B$, and its counting identity was independently checked here.

B	direct $W_B(2\cdot10^6)$	prime-sum term	residual
1	148,934	148,933	1
2	227,433	227,430	3
3	281,504	281,497	7
4	323,043	323,033	10
5	356,907	356,890	17
6	385,573	385,552	21
8	432,526	432,488	38
12	502,029	501,953	76

Square chart comparing fixed-window counts and the exact prime sum — **Figure 3.3.** Direct divisor enumeration and the prime-sum formula are visually indistinguishable at this scale; the residual consists of the finite/smooth exceptional population.

A useful corollary follows: if a strictly increasing sequence has uniformly bounded jumps $d(n)\le B$, then its level-classified indices up to $N$ are $O_B(N/\log N)$. Thus rarefaction is automatic for bounded-gap sequences. Prime gaps are unbounded, so primes lie just outside this easy theorem.

4. Application to the sequence of prime numbers

For consecutive primes $p_n,p_{n+1}$, write $g_n=p_{n+1}-p_n$. Then

$$\ell(n)=p_n-g_n=2p_n-p_{n+1},\qquad p_n=k(n)L(n)+g_n.$$

The original paper and OEIS identify the corresponding sequences as A117078 (weight), A117563 (level), A001223 (gap) and A118534 ($\ell$). The only nondecomposable primes are 2, 3 and 7; for all sufficiently large primes the existence criterion follows from standard primes-in-short-interval results, and the remaining small cases are direct.

4.1 Weight 3 and twin primes

Lemma. For a decomposable prime $p>3$, $k(p)=3$ if and only if $p,p+2$ are twin primes. If the gap is 2, exactly one of $p-2,p,p+2$ is divisible by 3; the latter two are primes greater than 3, so $3\mid(p-2)=\ell$, and 3 is the first divisor above the gap. Conversely $k=3$ forces the even gap to be smaller than 3, hence 2.

This equivalence is exact, but it does not advance the twin-prime conjecture. It identifies that conjecture with the infinitude of the weight-3 column.

Counting convention. At $10^{10}$, the project reports $\pi_2(10^{10})=27,412,679$. The total number of decomposable primes with $k=3$ is 27,412,678: remove $p=3$, which is not decomposable. The number that is also weight-classified is 27,412,677: remove the exceptional level-classified prime $p=5$. The latest sixth report blurs these two counts in one sentence.

4.2 Mod-3 rigidity

For primes exceeding 3, both endpoints are nonzero modulo 3. Since $\ell=p-g$, elementary congruence bookkeeping gives

$$3\mid\ell\quad\Longleftrightarrow\quad 6\nmid g.$$

This proves the original Conjectures 7 and 8, which the OEIS page now strikes through as trivial. It also explains a strong comb structure in the level distribution.

Correct level-one corollary. It is not literally true that level one is empty whenever $6\nmid g$. There are exactly two small exceptions in the verified range and in the project’s closed exceptional analysis: $p=5$ with $(g,\ell)=(2,3)$, and $p=13$ with $(4,9)$. The correct statement is: every other level-one prime has $6\mid g$.

4.3 Conjecture ledger

#	Statement	Status	Assessment
C1	Infinitely many primes of weight 3	reformulation	Exactly twin primes.
C2	Every odd weight $k\ge3$ occurs infinitely often	reformulation	Polignac/prime-tuple type; contains C1.
C3	Every odd level occurs infinitely often	open	Fixed-level prime-tuple family; no simplification is known.
C4	Level-classified primes have prime weight except finitely many cases	project proof	Cube Lemma gives the reported set 13, 31, 113, 131, 887, closed computationally to $4\cdot10^{18}$.
C5	Infinitely many level $(1;1)$ primes	reformulation	Balanced-prime infinitude.
C6	Every generation $(1;i)$ is infinite	open	Graded Hardy–Littlewood family.
C7–C8	Mod-3 implications between gap and $\ell$	proved	Elementary congruence, already marked trivial on OEIS.
C9	Level-classified primes have relative density zero	proved*	Unconditional manuscript proof, awaiting external refereeing.

Square padded screenshot of the conjecture ledger — **Figure 4.1 — supplied July 2026 ledger screenshot.** The visual status table is useful, but “PROVED*” should be read as “complete project proof not yet externally refereed,” not as a journal-certified theorem.

4.4 The Cube Lemma and finite exceptional structure

If a level-classified prime has composite weight $k=ab$, then every proper divisor below $k$ is at most the even gap $g$, hence at most $g-1$ because $\ell$ is odd. Together with $L\le g-1$,

$$\ell=kL\le(g-1)^3.$$

Equality occurs at $p=131$: $g=6$, $\ell=125$, $k=25$, $L=5$. The project combines this cubic ceiling with exhaustive prime-gap tables through $4\cdot10^{18}$ to report the exact composite-weight level set $\{13,31,113,131,887\}$ over that range. The logic of the cubic bound is elementary; the global closure depends on the external gap computation and on correct first-occurrence comparisons.

4.5 Rarefaction theorem (project manuscript)

Let $N_{\mathrm{lev}(x)$ count level-classified primes up to $x$. The July 2026 manuscript claims

$$N_{\mathrm{lev}(x)\ll \frac{x\log\log x}{(\log x)^{3/2},\qquad \frac{N_{\mathrm{lev}(x)}{\pi(x)}\to0.$$

The proof architecture is coherent:

Use the Cube Lemma to reduce almost all level-classified primes to $\ell=LP$, where $P=k$ is prime and $L\le g-1$.
Truncate large gaps at $g\le2M$; the tail is bounded by the telescoping sum of gaps.
For fixed $(L,g)$, the level event implies simultaneous primality of $P,LP+g,LP+2g$. For an upper bound, consecutiveness can be discarded.
Apply a dimension-3 Selberg upper-bound sieve uniformly over $L,g\ll M$.
Choose $M=(\log x)^{3/2}$ to balance the gap tail and the sieve main term.

Referee checklist. The parts most deserving independent scrutiny are the uniform singular-factor bound in $(L,g)$, treatment of local degeneracies, the exact range of the Selberg sieve theorem invoked, and the summation of local factors. The argument is plausible and detailed, but this report cannot convert an unrefereed manuscript into an externally established theorem.

4.6 Two species

The level class naturally splits:

Species I: $L=1$. Outside the finite closed-window exceptions, this is equivalent to $\ell=2p_n-p_{n+1}$ being prime. It counts three-term prime configurations $(\ell,p_n,p_{n+1})$ in arithmetic progression with the right endpoint constrained to be the next prime.

Species II: $L>1$. This is governed by the absence of divisors in $(g,\sqrt\ell]$, and belongs to the analytic neighbourhood of Ford’s divisor-in-an-interval theory and Koukoulopoulos’s divisors of shifted primes.

4.7 Conditional level-one constant

For the triple $\{0,g,2g\}$, the singular series vanishes unless $6\mid g$. Averaging its local factors over $g$ gives exactly

$$\lim_{H\to\infty}\frac1H\sum_{g\le H}\mathfrak S_3(g)=2C_2=1.3203236316\ldots,$$

where $C_2$ is the twin-prime constant. Under a uniform Hardy–Littlewood conjecture for the triples and a Gallagher-type model for consecutive gaps, the manuscript derives

$$N_1(x)\sim 2C_2\frac{x}{\log^2x}.$$

The constant identity is robust; the asymptotic requires the unproved hypotheses and a first-order decoupling of the left prime from the interior compositeness constraints that make the right pair consecutive.

4.8 Species II and the fixed-window heuristic

The fixed-window theorem suggests a density $H_g/\log x\approx(\log g+\gamma)/\log x$. Substituting the typical prime gap $g\asymp\log x$ motivates a $\log\log x/\log x$ shape. This is a useful heuristic, not a direct consequence of the fixed-$B$ theorem: the gap varies, its distribution has a long tail, and $\ell$ is arithmetically correlated with both parent primes.

The project reports a stable windowed Species-II normalization near 0.773 and a Mertens-normalized ratio near 0.441. Neither constant has a proved closed form. An earlier working claim $c_2=1$ was correctly retracted when the cumulative estimator moved monotonically away from 1.

4.9 Census

The following table is reported by the fifth and sixth project reports under the complete-successor convention. The first row is independently reproduced here; larger rows are not independently recomputed in this analysis.

x	$\pi(x)$	decomposable	$N_1$	$N_{>1}$	$N_{\rm lev}$	level share
10⁶	78,498	78,495	5,953	12,400	18,353	0.2338
10⁷	664,579	664,576	44,011	94,038	138,049	0.2077
3·10⁷	1,857,859	1,857,856	116,139	250,384	366,523	0.1973
10⁸	5,761,455	5,761,452	339,870	738,837	1,078,707	0.1872
3·10⁸	16,252,325	16,252,322	912,172	1,999,786	2,911,958	0.1792
10⁹	50,847,534	50,847,531	2,708,031	5,984,308	8,692,339	0.1709
2.5·10⁹	121,443,371	121,443,368	6,227,807	13,848,468	20,076,275	0.1653
5·10⁹	234,954,223	234,954,220	11,719,154	26,181,298	37,900,452	0.1613
10¹⁰	455,052,511	455,052,508	22,083,608	49,587,200	71,670,808	0.1575

Square rarefaction chart through ten billion — **Figure 4.2.** The project census shows monotone decline of the total level share and of both species. The graph is evidence for rates and constants, not by itself a proof of a limit.

5. Analysis of the supplied graphs

Every image below is placed on a square canvas without stretching, so the weight–level aspect ratio is not distorted. Padding is visual only; the source pixels are unchanged.

Annotated prime classification in log weight and log level — **Figure 5.1 — `classification_primes.jpg`.** The diagonal $k=L$ separates the two classes. Vertical weight combs encode fixed weights, with $k=3$ the twin column. The sparse lower-right wing consists of small levels and large weights. The labels are a map of OEIS strata, not independent evidence for infinitude.

Square log level versus log weight plot — **Figure 5.2 — `Log_L-Log_k_1500000-R.jpg`.** The two wings are clearer at higher density. The upper-left wedge is weight-classified; the lower-right wedge is level-classified. Horizontal bands in the level wing reflect odd fixed levels. The empty diagonal neighbourhood is controlled by the minimal-divisor rule.

Independent square audit plot for primes to one million — **Figure 5.3 — independent recomputation to $10^6$.** All 78,495 decomposable primes: 60,142 weight-classified and 18,353 level-classified, with 12 ties. It reproduces the qualitative wings of the supplied graphs from a separate implementation.

Original graph explaining sieve classes — **Figure 5.4 — `sieveNb.jpg`.** For naturals, each vertical weight column is exactly a smallest-prime-factor class. The long level-one edge is the shifted primes. This graph has the strongest theorem-level interpretation of the supplied set.

Three million natural decompositions — **Figure 5.5 — `naturaldecomp3M-2048.jpg`.** The dense triangular wedge is the geometry of $kL=n-1$. The density differences among columns encode the probability that a given prime is the smallest prime factor.

Square padded 3D screenshot of prime decomposition — **Figure 5.6 — first 3D screenshot.** In three dimensions, the jump coordinate separates sheets that overlap in the two-dimensional $(k,L)$ projection. Apparent straight rays usually correspond to fixed gap, fixed weight or fixed level constraints.

Second square padded 3D screenshot of prime decomposition — **Figure 5.7 — second 3D screenshot.** Rotation reveals that the cloud is not a homogeneous surface: it is a union of arithmetic strata. The 3D display is exploratory; quantitative claims must be checked against arrays, because perspective and point density can create false visual regularity.

Conjecture ledger screenshot — **Figure 5.8 — conjecture/status ledger.** This is a provenance graph rather than a numerical graph. Its most important feature is the explicit separation of reformulations from native results. The report’s audit adds corrections to C7/C8 consequences and to weight-3 counting conventions.

5.1 Structural readings that are justified

The diagonal is exact: ties lie on $k=L$; the two classes lie on opposite sides. The vertical combs are exact fixed-weight strata. The level wing’s odd horizontal layers are exact because $\ell$ is odd for odd primes. The scarcity of large levels is constrained by $L\le g-1$, so it is not merely a plotting artifact.

5.2 Readings that remain heuristic

Visual thinning of the level wing does not prove rarefaction. Apparent power laws in log-log coordinates can be finite-range mixtures of Species I and II. The 3D rays do not by themselves imply algebraic relations beyond fixed-coordinate strata. Any claimed constant must be estimated locally by windows, not from a single cumulative cloud.

6. Hypothetical impacts on mathematics: an additive–multiplicative bridge

6.1 A precise bridge, not a metaphor

Additive number theory studies the arrangement of sequence elements and their gaps. Multiplicative number theory studies divisors and prime factors. Here the next gap $d$ selects the shifted integer $\ell=a-d$, and the divisor pattern of $\ell$ is interrogated above the same gap. The bridge is the map

$$\text{local gap }d\quad\longmapsto\quad\text{divisor window }(d,\sqrt{a-d}].$$

This creates a family of observables that are simultaneously gap-conditioned and factorization-sensitive.

6.2 Potential impacts

Unified taxonomy of prime patterns

Twin, fixed-gap, balanced and prime-triple phenomena can be compared in one coordinate system. Even when this does not solve them, it can reveal which phenomena share local congruence obstructions and which do not.

New divisor statistics of shifted primes

Species II asks about divisors of $2p_n-p_{n+1}$ in an interval whose endpoint is the actual prime gap. This moving-shift problem is adjacent to, but not identical with, established shifted-prime divisor theory.

Sequence-wide growth classification

The N/Z/W trichotomy can sort the thousand-sequence atlas into nondecomposable, growth-forced level, and genuinely arithmetic cases. That would turn a visual database into a theorem-guided research catalogue.

Computational falsifiability

The coordinates give exact arrays, exception sets and local distributions. Wrong constants can be retracted decisively, as happened for the proposed Species-II constant 1.

6.3 Connection with modern gap-scale work

Gafni and Tao study whether the interior of a prime gap contains a number whose least prime factor is at least the gap length. The present framework studies the exterior point $\ell=p_n-g_n$ and asks whether it has a divisor above the gap and below its square root. The two problems share a roughness threshold at gap scale, but constrain different integers and are not equivalent. Their coexistence suggests a broader programme of classifying “gap-scale roughness” at the left exterior, interior and right exterior of consecutive primes.

6.4 Limits on impact

No current theorem in the framework gives a lower bound sufficient to prove infinitely many level-classified primes. No fixed-weight or fixed-level infinitude becomes easier merely by renaming it. The approach is most credible as a source of new averaged questions, finite exceptional structure and conditional constants—not as a shortcut to classical conjectures.

Methodological caution. The fixed-window theorem is exact for fixed $B$, but the prime problem has random, unbounded and correlated $B=g_n$. Replacing $g_n$ by $\log p$ is a heuristic passage. A rigorous bridge requires averaging over the true gap distribution while retaining the arithmetic dependence of $\ell$.

7. Detailed conclusion and future research

7.1 Overall assessment

The decomposition is a legitimate and useful coordinate construction. Its best-established achievement is the exact Eratosthenes collapse on the natural numbers: weight becomes the smallest prime factor of $n-1$, level the complementary factor, and the class boundary the square-root stopping rule. Its best prime-level achievement is organizational: twin primes, balanced primes, divisor-window exceptions and the level/weight split become parts of a single structure.

The later project work raises the framework above visualization. The Cube Lemma, growth trichotomy, fixed-window count and bounded-jump rarefaction are elementary statements with independent interest. The rarefaction manuscript proposes a credible sieve proof for primes and a conditional explicit constant for Species I. The decisive next step is external refereeing, because the central asymptotic claim currently remains a project theorem rather than a community-validated result.

7.2 Research programme, ordered by leverage

Submit and referee the rarefaction manuscript. Verify the uniform Selberg-sieve step, local factors and exceptional ranges. This is more valuable than another decade of computation.
Publish the elementary taxonomy separately. State the N/Z/W trichotomy, fixed-window theorem and bounded-jump corollary with the equality case $4c=1$ formulated correctly.
Make the gap-conditioned model rigorous under one transparent hypothesis. Replace the informal pair-to-triple decoupling with a precisely stated uniform Hardy–Littlewood/Gallagher framework and derive all local constants from singular series.
Attack Species II with divisor-in-interval methods. Adapt Ford’s $H(x,y,z)$ and Koukoulopoulos’s shifted-prime results to a shift equal to the actual prime gap. The targets are the observed 0.773 and 0.441 normalizations.
Separate fixed-gap and varying-gap asymptotics. First prove or conjecture formulas conditional on $g_n=g$, then average over $g$. This avoids hiding the main correlation inside a fitted global constant.
Extend and publish reproducible census data. A 10¹¹ or 10¹² run should include code, hashes, checkpoints, exact conventions and independent spot audits. More data mainly reduces uncertainty in windowed constants; it does not replace proof.
Classify the atlas by regime. For each sequence, record whether N, Z or W dominates and which part follows from growth alone. Prioritize W sequences with unbounded but controlled gaps.
Formalize the elementary core. Lean or Isabelle verification of the definitions, existence criterion, natural-number collapse, twin column, mod-3 rigidity, trichotomy and fixed-window theorem is feasible.

7.3 Final judgment

The framework should be judged neither as a solution to classical prime problems nor as a mere plotting device. It is a canonical transformation that places an additive gap and a multiplicative divisor threshold in the same equation. Its durable contribution will depend on whether the recent native theorems survive refereeing and whether Species II can be connected rigorously to the modern theory of divisors of shifted primes.

Appendix A. Source audit, discrepancies and corrections

Item	Finding	Treatment in this report
Latest project state	The website lists a sixth report dated 25 July 2026, later than the uploaded fifth report.	Taxonomy results are included, but audited rather than copied verbatim.
Level one over non-multiples of 6	The sixth report states an empty level-one stratum for $6\nmid g$. Counterexamples are $p=5$ and $p=13$.	Corrected to “all level-one primes except 5 and 13 have $6\mid g$.”
Weight-3 census	The sixth report labels 27,412,677 as “primes of weight 3,” but this is the weight-classified count.	Total $k=3$: 27,412,678; weight-classified $k=3$: 27,412,677.
Quadratic growth boundary	The sixth report decides $a(n)\sim cn^2$ by $4c\ge1$. At equality, lower-order terms can matter.	Strict inequalities are stated; equality is left boundary-sensitive.
Prime density shape	The fixed-window theorem is sometimes described as deriving the prime $\log\log x/\log x$ shape.	Presented as a motivated heuristic after substituting a typical gap, not a theorem for correlated varying gaps.
Refereeing status	Theorem B is labeled PROVED* in project reports.	Described as a complete internal proof awaiting external refereeing.
Atlas verification	The main website explicitly says its downloadable sequence data have not been verified.	Atlas breadth is cited as scope; no global accuracy claim is made.

Independent small-range audit performed for this report

Using a separate Python/SymPy implementation and true prime successors, all primes through $10^6$ were decomposed. Results: $\pi(10^6)=78,498$; 78,495 decomposable; 5,953 level one; 12,400 level greater than one; 60,142 weight-classified; 12 ties; zero violations of mod-3 rigidity. The composite-weight level exceptions are exactly 13, 31, 113, 131 and 887; the empty-window primes are exactly 5, 13, 19, 23, 31 and 113. These match the project tables.

Appendix B. Optimized PARI/GP reference code

The minimal implementation should follow the definition directly and use fordiv, which enumerates divisors in increasing order. The first divisor exceeding the gap is therefore the weight.

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

decomp_class(a, b) = {
  my(v = decomp(a,b), k=v[1], L=v[2]);
  if (!k, return([v, "unclassified"]));
  [v, if(k > L, "level", "weight")]
}

regime(a, b) = {
  my(d);
  if (a >= b, error("successor must exceed term"));
  d = b - a;
  if (a <= 2*d, return("N"));
  if (d*d >= a-d, return("Z"));
  "W"
}

\\ Prime case, complete-successor convention:
decomp_prime(p) = decomp(p, nextprime(p+1));

For bulk prime ranges, repeatedly factoring every $\ell$ in GP is not competitive with a segmented sieve plus a specialized factorization/classification kernel. PARI/GP remains valuable as an independent adjudicator and for exceptional-set loops.

References and web sources

R. Eismann, Decomposition into weight × level + jump and application to a new classification of primes, arXiv:0711.0865 (2007–2010).
OEIS Wiki: Decomposition into weight × level + jump, definitions, algorithms, sequences and original conjectures.
decompwlj.com, atlas, graphs, downloadable data and links to the project’s report editions.
R. Eismann, uploaded manuscript, Rarefaction of level-classified primes in the weight × level + jump decomposition, with a conditional asymptotic for the level-one stratum, July 2026.
K. Ford, The distribution of integers with a divisor in a given interval, Annals of Mathematics 168 (2008), 367–433.
D. Koukoulopoulos, Divisors of shifted primes, IMRN 2010, no. 24, 4585–4627.
P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), 4–9.
H. L. Montgomery and K. Soundararajan, Primes in short intervals, Communications in Mathematical Physics 252 (2004), 589–617.
T. Oliveira e Silva, S. Herzog and S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4·10¹⁸, Mathematics of Computation 83 (2014), 2033–2060.
A. Gafni and T. Tao, Rough numbers between consecutive primes, arXiv:2508.06463 (2025).
OEIS sequences A117078, A117563, A001223, A118534, A121155, A162174, A162175 and A002386.

Social links listed in links.txt were treated as project communication channels, not mathematical evidence. All supplied images are reproduced for analysis; their square framing adds padding only and does not distort the original aspect ratio.