Decomposition into weight × level + jump

The decomposition into weight × level + jump assigns to every term of a strictly increasing integer sequence $a(1)<a(2)<\cdots$ a unique triple — a weight $k(n)$, a level $L(n)$, and a jump $d(n)$ — such that $a(n)=k(n)\times L(n)+d(n)$. The jump is purely additive data: the first difference $a(n{+}1)-a(n)$. The weight and level are purely multiplicative data: a distinguished divisor pair of the auxiliary number $l(n)=a(n)-d(n)$. The construction therefore welds the two kinds of structure that number theory usually has to study with separate tools, and it does so by an entirely elementary mechanism: a Euclidean division of $a(n)$ by its weight, in which the level is the quotient and the jump is the remainder.

Three features give the construction its character. First, it is universal and canonical: it applies to any strictly increasing sequence, needs no parameters, and (where it exists) the triple is unique. Eismann’s atlas at decompwlj.com carries this out for a thousand OEIS sequences, with 2D and 3D (three.js/WebGL) renderings, CSV data and a database dump, so the construction now functions as a comparative instrument: one can ask how the “shape” of a sequence’s decomposition reflects its growth and gap laws. Second, it is conservative over the classical theory: applied to the natural numbers it reproduces, exactly, the smallest-prime-factor / largest-proper-divisor pair of $n-1$ and hence a reformulation of the sieve of Eratosthenes (§3); the Fundamental Theorem of Arithmetic is what guarantees the weight is well defined. Third, applied to the primes themselves it produces something genuinely new: a partition of the primes into those classified by weight and those classified by level, an exhaustive bookkeeping of prime gaps against the divisor structure of the companion sequence $l(n)=2p(n)-p(n+1)$, and a family of conjectures of which one — Conjecture 9, the rarefaction of level-classified primes — is native to the framework rather than a transcription of a classical open problem.

That last distinction deserves emphasis, because it is the intellectual discipline that keeps the project honest. Several of the paper’s conjectures are equivalent reformulations of celebrated open problems: Conjecture 1 is the twin prime conjecture in new coordinates (via the exact characterization “lesser twin $\Leftrightarrow$ weight $3$”), Conjecture 5 is the infinitude of balanced primes. A change of coordinates cannot lower the difficulty of a problem — we call this the difficulty-inheritance principle and apply it throughout. What a good coordinate system can do is expose statistics that were previously unnamed and are genuinely tractable. The classification fraction $f(x)$ of Conjecture 9 is exactly such a statistic: it is well posed, measurable to $10^8$ and beyond, it decays with a clean empirical law, and it sits squarely inside a modern and active part of analytic number theory — the distribution of integers with a divisor in a prescribed interval (Ford’s $H(x,y,z)$ theory) — while adding the new twist that the integer in question, $l(n)$, is a linear form in consecutive primes. This report develops that connection in §4.

Finally, the project’s significance is also methodological. It is sustained independent mathematics conducted in public: a preprint refined over several versions, dozens of sequences contributed to and cross-checked in the OEIS (with edits by other OEIS editors), reproducible PARI/GP algorithms, and an open data atlas. The present report continues in that spirit — every figure of the paper is re-derived from a fresh computation before it is discussed.

Let $(a(n))_{n\ge 1}$ be a strictly increasing sequence of positive integers. The paper defines, for each $n$:

The reading that makes the construction transparent is the Euclidean one: in the Euclidean division of $a(n)$ by its weight $k(n)$, the quotient is the level and the remainder is the jump. Indeed $a(n) = k(n)L(n) + d(n)$ with $0 \le d(n) < k(n)$, the inequality $d(n) < k(n)$ being forced by the definition of $k(n)$ as a divisor of $l(n)$ exceeding $d(n)$. Equivalently — and this is the form used in OEIS A117078 — $k(n)$ is the smallest $k$ such that $a(n{+}1) = a(n) + \big(a(n) \bmod k\big)$: the weight is the smallest modulus that “sees” the next term as the residue of the current one.

Existence

So the construction tolerates gaps up to half the current term — a very mild growth condition. For the natural numbers it fails only at $n=1,2$; for the primes, remarkably, it fails only at $p_1=2$, $p_2=3$, $p_4=7$ (Theorem 3.1, §4); for geometric-type sequences with ratio $\ge 3/2$ it fails everywhere, which delimits the construction’s natural habitat: sequences of at most moderately exponential growth.

Classification

The decomposition is converted into a binary (plus one degenerate) labelling:

Since $k(n)L(n) = l(n)$, the comparison $k(n)\gtrless L(n)$ is the comparison $k(n) \gtrless \sqrt{l(n)}$. This gives the formulation we use constantly in §4:

A worked example, recomputed by our engine and agreeing with the paper’s Table 2 in every entry, fixes the ideas (full table in §4): $p_5 = 11$, $d=2$, $l = 9$; divisors of $9$ exceeding $2$: $\{3,9\}$, so $k=3$, $L=3$, $11 = 3\times 3 + 2$, and $k=L$ places $11$ on the weight side. For $p_6=13$, $d=4$, $l=9$; the only divisor of $9$ exceeding $4$ is $9$ itself, so $k=9$, $L=1$, $13 = 9\times 1 + 4$, classified by level. For $p_9=23$, $d=6$, $l=17$ is prime, so $k=17$, $L=1$, $23=17\times1+6$ — level again, the “prime $l$” mechanism that, as we will see, dominates the level class. The degenerate case: $p_4 = 7$ has $d=4$ and $a-d = 3 \le 4$, so $l=0$ and $7$ is not classified.

Two elementary monotonicity facts from the paper complete the picture. On the weight side, $d(n)+1 \le k(n) \le \sqrt{l(n)} \le L(n) \le l(n)/3$; on the level side, $L(n) < \sqrt{l(n)} < k(n) \le l(n)$ and $L(n)+2 \le d(n)+1 \le k(n)$ — so a level-classified term has a level smaller than its own gap, an inequality that quietly drives the geometry of all the figures in §5.

The decomposition earns its claim to be number-theoretic — rather than merely combinatorial — in the simplest possible arena: the natural numbers themselves, $a(n) = n$. Here the jump is constant, $d(n) = 1$, and $l(n) = n - 1$ for $n \ge 3$ (the terms $n = 1, 2$ have $l = 0$ and stay unclassified). The weight, being the smallest divisor of $l(n)$ exceeding $1$, is the smallest prime factor of $n-1$, and the level is the cofactor:

the largest proper divisor. The decomposition of $\mathbb{N}$ is therefore not an analogue of classical objects but literally composed of them: $n = \operatorname{spf}(n-1)\cdot\operatorname{lpd}(n-1) + 1$. This is the sanity anchor of the whole program. Whatever the construction does on exotic sequences, on the naturals it reproduces, exactly, the two most elementary invariants of multiplicative number theory.

3.1  The first instalment of the FTA

Extracting the smallest prime factor is the atomic step of factorization: iterate it — apply the decomposition again to the level, then to that level’s level, and so on — and the Fundamental Theorem of Arithmetic is recovered factor by factor, in nondecreasing order of primes. In this precise sense the decomposition of the naturals is the first instalment of the FTA: it performs exactly one canonical step of unique factorization and records the quotient as a coordinate rather than discarding it. For a general increasing sequence the jump $d(n)$ is no longer $1$, and the divisor window $\big(d(n), \sqrt{l(n)}\,\big]$ of §2 is the window $\big(1, \sqrt{l}\,\big]$ deformed by the local gap structure of the sequence. The decomposition is thus best read as a gap-aware generalization of smallest-prime-factor extraction: the multiplicative anatomy of $l(n)$ is probed, but the probe’s lower edge is set additively, by how far the sequence jumps at $n$.

3.2  The sieve, made into a coordinate system

Classifying the naturals by weight is — not metaphorically but operationally — running the sieve of Eratosthenes. The weight class $k = p$ collects exactly those $n$ for which $\operatorname{spf}(n-1) = p$: the integers $n - 1$ struck out for the first time by the prime $p$ in Eratosthenes’ procedure, shifted by one. The vertical combs in the figure below are the sieve’s crossing-out patterns, one comb per prime, each comb thinned by the primes before it. And the survivors of the sieve — the primes themselves — are precisely the level-$1$ stratum: $n$ is level-classified if and only if $\operatorname{spf}(n-1) > \sqrt{n-1}$, i.e. if and only if $n - 1$ is prime, in which case $k = n - 1$ and $L = 1$. (The boundary case $n - 1 = p^2$ gives $k = L = p$ and lands on the weight side, on the diagonal.)

3.3  The geometry this produces

In log–log coordinates the constraint $k(n)\,L(n) = n - 1$ confines each $n$ to the anti-diagonal $\log k + \log L = \log(n-1)$, and the weight-side inequality $k \le \sqrt{n-1} \le L$ confines it to the half below the main diagonal. Sweeping $n$ up to $N$ therefore fills a triangular wedge whose apex sits at $k = L = \sqrt{N}$ — height $\tfrac{1}{2}\log N$ — exactly what the author’s plot of the first three million naturals shows. The dense ray escaping the wedge is the level-$1$ stratum, the primes, marching along $L = 1$ with $k = n - 1$.

This is the template for everything that follows: when the input sequence is changed from $\mathbb{N}$ to the primes in §4, the constant jump $d = 1$ becomes the erratic prime gap $g(n)$, the window $\big(1,\sqrt{l}\,\big]$ becomes the moving window $\big(g(n), \sqrt{l(n)}\,\big]$, and the clean prime/composite dichotomy of the level-$1$ line becomes the subtler weight/level statistics whose decay law is the subject of Conjecture 9.

Everything in this section concerns the sequence $a(n) = p(n)$ of primes, with gap $g(n) = p(n+1) - p(n)$ playing the role of the jump. The four coordinates become $l(n) = p(n) - g(n) = 2p(n) - p(n+1)$ (OEIS A118534), weight $k(n)$ (A117078), level $L(n)$ (A117563), and the gap itself (A001223). Our verification engine recomputed all four for every prime below $10^8$ — $\pi(10^8) = 5{,}761{,}455$ primes — and checked the identity $p(n) = k(n)L(n) + g(n)$, the classification rule, and every numerical table of the paper within that range. We report the results as we go.

4.1  Decomposability: the exceptional set is $\{2, 3, 7\}$

Verified: among all $5{,}761{,}455$ primes below $10^8$, exactly three have $l = 0$: they are $2, 3, 7$. The remaining $5{,}761{,}451$ decompose, and the identity and classification rule hold in every single case.

A related finite phenomenon deserves separate mention because it organizes several later exceptions. The divisor window $\big(g(n), \sqrt{l(n)}\,\big]$ can be empty — this happens precisely when $g(n) > \sqrt{l(n)}$, i.e. when the gap is enormous relative to the prime — and then the term is level-classified vacuously, with a weight that need not be prime. Below $10^8$ the decomposable primes with empty window are exactly

and together with the non-decomposable $\{2, 3, 7\}$ they form the paper’s list of nine primes with $g(n) > \sqrt{l(n)}$. No tenth example appears below $10^8$. Heuristically none should ever appear — it would require $g(n) \gg \sqrt{p(n)}$, far beyond even the largest gaps Cramér’s model permits ($g \asymp \log^2 p$) — but unconditional gap bounds (Baker–Harman–Pintz: $g(n) \ll p(n)^{0.525}$) are far too weak to prove the list complete, and even the Riemann Hypothesis ($g \ll \sqrt{p}\,\log p$) leaves the question open. A small, sharp example of the difficulty-inheritance principle of §1: the framework states the finiteness cleanly, but the statement is exactly as hard as the gap bounds it encodes.

4.2  The first seventeen primes, recomputed

Our engine reproduces the paper’s Table 2 entry for entry. Weight-classified rows are marked indigo, level-classified rows copper; the twin lessers — every row with $k = 3$ — are flagged in the last column.

The early data over-represents the level class ($44\%$ at $n \le 100$) because small primes have gaps comparable to their size; the asymptotic regime, where the weight class dominates $4{:}1$ and growing, sets in quickly (§4.4).

One chain of elementary inequalities from the paper deserves restating, because it quietly drives the geometry of every figure in §5: on the level side, $L(n) + 2 \le g(n) + 1 \le k(n) \le l(n)$ — a level-classified prime always has level strictly smaller than its own gap, while its weight is at least the gap plus one. Levels are thus condemned to remain tiny (of the order of $\log p$ at most, on average), and the level side of every scatter plot collapses onto a handful of low horizontal strata $L = 1, 3, 5, \dots$, verified across the full range.

4.3  The mod-3 dichotomy: two conjectures become theorems

The paper’s Conjectures 7 and 8 assert: (C7) if $6 \nmid g(n)$ then $3 \mid l(n)$; (C8) if $3 \nmid l(n)$ then $6 \mid g(n)$. Our verification found zero violations of either statement across all $5{,}761{,}451$ decomposable primes below $10^8$ — and the perfection of that record is explained by the fact that both statements are, in fact, elementary theorems. The computation suggested the theorem; the proof then turned out to be three lines of modular arithmetic.

Theorem A immediately yields the paper’s twin-prime characterization, with a proof shorter than the original:

So the twin lessers beyond $3$ are exactly the weight class $k = 3$ (OEIS A001359 minus its first term), confirmed across the full range: every one of the $440{,}311$ twin lessers below $10^8$ has weight $3$, and no other prime does. One more drop of structure can be squeezed out, and it is a pleasing one:

Verified: among $440{,}311$ twin lessers below $10^8$, exactly one is level-classified, and it is $5$. This explains the most extreme bar of the gap-conditional chart of §4.7 — the conditional probability $\mathbb{P}(\text{level} \mid g = 2)$ is not merely small, it is $1/440{,}311$, and provably will never grow by another unit.

4.4  Census to $10^8$: who is classified by what

The paper’s Table 6 counts classifications among the first $N$ primes. We verified it exactly as far as it goes within our range, and extended it:

Within the classified population the distribution over individual weights and levels is sharply structured. All weights and levels are odd (since $l = p - g$ is odd), and the leaders are:

Finally, within the level-$1$ stratum ($l(n)$ itself the weight, $L = 1$) the paper singles out the sub-families $\big(1; i\big)$ defined by $l(n) = p(n-i)$ — the level is $1$ and the weight is itself a prime exactly $i$ steps back in the sequence. The case $i = 1$ is precisely the balanced primes, $p(n) = \tfrac12\big(p(n-1) + p(n+1)\big)$, OEIS A006562. Our counts below $10^8$: $i = 1$: $167{,}031$; $i = 2$: $96{,}665$; $i = 3$: $43{,}475$; $i = 4$: $19{,}348$; $i = 5$: $7{,}833$; $i = 6$: $3{,}258$; $i = 7$: $1{,}328$ — a geometric-looking decay in $i$ whose ratio drifts from $0.58$ toward $\approx 0.41$, itself a clean target for the kind of secondary heuristics §7 proposes.

4.5  The nine conjectures: an inventory with status

With Theorem A in hand, the paper’s conjecture list divides cleanly into four kinds: reformulations of classical problems (C1, C5), their generalizations (C2, C3, C6), one structural finiteness statement (C4), and one genuinely native asymptotic statement (C9) — plus two entries that are conjectures no longer. The falsifiable content of the framework survives this review intact and sharpened.

4.6  Conjecture 4: composite weights on the level side are squeezed into $l < g^4$

Conjecture 4 asserts that a level-classified prime has prime weight, with finitely many exceptions. Our sweep recovers exactly the paper’s five exceptions below $10^8$ — no sixth appears:

The finiteness has a transparent mechanism, which we record as a small native result:

Since $l(n) = p(n) - g(n)$ is of the size of $p(n)$ while $g(n)$ is of the size of $\log p(n)$ on average — and conjecturally $O(\log^2 p)$ always (Cramér) — the inequality $p(n) \lesssim g(n)^4$ should fail for all large $n$: under any gap bound of the form $g(n) \le p(n)^{1/4 - \varepsilon}$ eventually, the exception set of Conjecture 4 is finite. As with the empty-window list of §4.1, the finiteness is calibrated by gap bounds — and the same Baker–Harman–Pintz and Riemann-Hypothesis estimates recalled there remain too weak even for this far gentler exponent. So Conjecture 4’s finiteness, while overwhelmingly supported, sits just beyond current unconditional technology — an instructive calibration of where the framework’s native statements land on the difficulty scale.

4.7  Conjecture 9: the native target, made quantitative

Conjecture 9 — primes classified by level rarefy among the primes — is the statement the framework owns outright: it is not a recoding of any classical conjecture, it is falsifiable, and it is supported at every scale we can reach. Write

Conjecture 9 says $f(x) \to 0$. Everything below aims at the sharper question: how fast?

The window form. By §2, $p(n)$ is level-classified iff $l(n)$ has no divisor in $\big(g(n), \sqrt{l(n)}\,\big]$. Rarefaction is thus a statement about the multiplicative anatomy of the specific sequence $l(n) = 2p(n) - p(n{+}1)$: a linear form in two consecutive primes. The event “an integer has a divisor in a prescribed interval” is precisely the object of Kevin Ford’s $H(x, y, z)$ theory (Annals of Mathematics 168 (2008), 367–433), which settled the 1960 Erdős problem on the multiplication table and determined the order of magnitude of the count of $n \le x$ with a divisor in $(y, z]$ in all ranges. Conjecture 9 lives in the complement of Ford’s event, in the wide-window regime $z = \sqrt{l}$ — but for $l$ ranging over a sequence tied to consecutive primes, where no equidistribution input of the required strength is known. The nearest proven territory is the theory of divisors of shifted primes $p - a$ for fixed $a$ (Koukoulopoulos, IMRN 2010, and the literature around it); the framework’s twist is that the shift here is the gap itself, $a = g(n)$, a moving target correlated with $n$. This is, in our assessment, exactly why Conjecture 9 is both tractable-looking and not yet a theorem: the heuristic is classical, the obstruction is the “next prime” dependence.

The heuristic, with the correct local factors. Fix a typical $n$ with gap $g$ and $l = p - g \asymp p$. For a prime $q$ with $g < q \le \sqrt{l}$: $q$ cannot divide $g$ (it exceeds $g$), and the event $q \mid l$ is the event $p \equiv g \pmod q$ — one admissible residue class out of the $q - 1$ nonzero classes available to the prime $p$. Dirichlet equidistribution thus prices this event at $1/(q-1)$, and treating the primes $q$ in the window as quasi-independent,

The second product is a tail of the twin-prime-constant product $\prod_{q > 2} q(q-2)/(q-1)^2 = 2C_2/2 \cdot$(normalization) and tends to $1$ as $g \to \infty$; the first is, by Mertens, $\sim \log g / \log\sqrt{l} = 2\log g/\log l$. Two refinements push the prediction below this proxy: composite divisors of $l$ assembled from prime factors $\le g$ can land inside the window even when no prime does (depleting the level class), and the small-$g$ tail factors are $< 1$. Averaging over $n \le \pi(x)$ with the Gallagher/Poisson picture of gaps ($g/\log p$ asymptotically exponential, so $\mathbb{E}[\log g] \approx \log\log x - \gamma$; we measure $\mathbb{E}[\log g] = 1.855,\ 2.235,\ 2.502$ at $x = 10^4, 10^6, 10^8$ against $\log\log x = 2.220,\ 2.626,\ 2.913$) yields the shape we propose as the quantitative form of Conjecture 9:

The data. Our census across four decades of $x$:

Honest reading of the constant. The shape $\log\log x/\log x$ is strongly supported: the local constant $C(x)$ varies by only $\sim 10\%$ across four decades while $f(x)$ itself falls by $40\%$. The value of $C$ is not yet pinned: $C(x)$ drifts downward ($1.32 \to 1.18$ from $10^4$ to $10^8$), the global fit says $1.26$, and the drift is exactly what the secondary term in the two-parameter fit predicts. Benchmark constants of the right magnitude exist ($2e^{-\gamma} \approx 1.1231$ is numerically tempting at $10^8$; $e^{\gamma} \approx 1.7811$ is not excluded asymptotically), but we decline to claim any of them: deciding $C$ requires either computation several decades deeper (§7) or a proof of the averaged Mertens analysis with the composite-divisor correction, which is precisely the open analytic problem. What we do claim is that Conjecture 9 has been converted from a qualitative statement into a one-parameter quantitative law with falsifiable content at every future scale of computation.

A structural discovery: the gap’s arithmetic steers the classification. The heuristic predicts more than the global decay. If a small prime $q$ divides $g$, then $q$ cannot divide $l$ (it would divide $p = l + g$); the building blocks for divisors of $l$ near the bottom of the window are removed, $l$ is forced “rough,” and level classification becomes more likely. The data confirm this vividly:

The project supplies a substantial visual corpus — the atlas at decompwlj.com renders the decomposition of over a thousand OEIS sequences in 2D and 3D — and the figures are not decoration: each geometric feature is a theorem or a conjecture wearing different clothes. We analyze the supplied figures in turn, connecting each visible structure to the results of §§2–4.

5.1  The annotated classification chart

This figure is the framework’s table of contents. Reading its annotations: the dense vertical combs on the weight side are the classes $k = 3, 5, 7, 9, \dots$ — the comb $k = 3$ is A001359 (lesser twins) minus its first term, by Corollary B; $k = 5$ is A074822; the chart resolves $k = 7$ and $k = 9$ into further catalogued sub-families. The boundary between the two classification regions is the diagonal $k = L$ (A121155, primes with $l = k^2$); above it sits the weight region as a whole (A162175), below it the level region (A162174). The level side collapses, as the chain inequality of §4.2 demands, onto low horizontal lines: $L = 1$ (containing $13$, $31$, the balanced primes A006562, and the further sub-families the chart labels A117876, A118467, within A125830), then $L = 3$ (A117873), $L = 5$ (A117874). Every annotation on this chart is a falsifiable identification — and the OEIS cross-references mean each locus carries its own independently maintained data, terms, and bibliography. Few private research programs achieve this degree of public, queryable structure.

5.2  Morphology at scale, and an independent reproduction

The two-population structure is forced by the inequalities of §2 and §4.2. On the weight side $k \le \sqrt{l} \le L$, with $k$ ranging over odd integers from $3$ up to $\sqrt{l}$: in $(\log k, \log L)$ this is the triangle above the diagonal, fibered into vertical combs at $\log 3, \log 5, \log 7, \dots$, whose tips climb like $\log l - \log k$ — the comb $k = 3$ being tallest, by Corollary B the geometric image of the twin primes. On the level side the chain $L < g \le k$ pins $L$ to the first few odd values while $k \approx l/L$ is huge: a fan of horizontal rays hugging the bottom, with the $L = 1$ ray (where $l$ is prime and $k = l$) reaching farthest right. The visible gap between the wedge and the fan is the divisor-window criterion made flesh: points cannot accumulate near the diagonal from below, because $k$ slightly larger than $\sqrt{l}$ requires $l$ to have its smallest large divisor just past $\sqrt{l}$ with no divisor in $(g, \sqrt{l}]$ — a sparse event, by exactly the Ford-type analysis of §4.7. That our independently computed scatter reproduces the author’s morphology point-class for point-class is the visual counterpart of the table-level verification in §4.

5.3  The third dimension

The third axis is the jump, $\log g$ — not the index — and that changes what the picture shows. Each 2D comb splits into its gap-stratified copies: a light ray is a fixed pair $(k; g)$ swept by the level, a dark ray a fixed $(L; g)$ swept by the weight; the first $10^4$ primes resolve into exactly $566$ light rays and $106$ dark ones. The prime’s magnitude is recovered along the anti-diagonal $\log k + \log L = \log l \approx \log p$, so moving outward in that direction is moving up the sequence: Conjecture 9 says the dark share of successive anti-diagonal slabs decays (at the rate of 9′), while C2, C3 and C6 assert that individual named rays are infinite. Because the jump owns an axis, the gap-steering law of §4.7 becomes geometry as well: slabs with $3 \mid g$ are visibly darker than their neighbors, and the $g = 2$ slab degenerates to one light ray plus a single dark point. The 3D atlas thus turns each conjecture of §4.5 into a statement about the asymptotic geometry of a specific, named, visible object. This is, in our view, the figures’ real function: not illustration but specification — they fix exactly which infinitudes and densities the framework is committed to.

5.4  Provenance artifacts: the OEIS entry and the engine

Two non-geometric figures complete the corpus, and they document something a referee should weigh: process. The OEIS screenshot shows the framework’s core sequences living inside the reference database for integer sequences since 2006, with the decomposability condition independently restated by other contributors in the comments — the statement $a(n) > 0 \iff 2p(n{+}1) < 3p(n)$ is Lemma 1.1 in the OEIS’s house style. The algorithm screenshot shows the production code agreeing with the definitions of §2, and quietly exploiting the level-side bound $L \le g - 1$ — a theorem doing engineering work. Our verification engine (Appendix A) was written independently of this code and agrees with it everywhere both were run.

The paper’s ambition — announced in its title and on the atlas — is that the decomposition acts as a bridge between the additive and multiplicative faces of arithmetic. Having verified the construction and sharpened its statements, we can now say precisely what crosses that bridge, what does not, and what might.

6.1  The coupling is real

The decomposition takes one additive datum — the jump $d(n)$, for primes the gap $g(n)$ — and one multiplicative datum — the divisor anatomy of $l(n)$ — and locks them into a single Euclidean identity and a single classification bit. Three results of this review show the coupling has genuine arithmetic content, beyond bookkeeping. First, Theorem A: the additive split $p = l + g$ obeys an exact conservation law at the modulus $3$ — every decomposable prime spends its divisibility-by-3 on exactly one side. Second, the gap-steering law of §4.7: the conditional probability $\mathbb{P}(\text{level} \mid g)$ is governed by the prime factorization of the gap, a Hardy–Littlewood-flavored local structure that couples the additive variable’s arithmetic to the multiplicative classification. Third, the decay heuristic itself: the predicted law $f(x) \asymp \log\log x / \log x$ arises from a Mertens product (multiplicative input) evaluated at the typical gap size given by the Gallagher/Poisson model (additive input). The constants of the framework are, literally, places where Mertens meets Gallagher.

6.2  What does not cross: difficulty

We restate the discipline of §1, because it is the difference between a serious framework and a perpetual-motion machine: a change of coordinates transports statements, not proofs. The comb $k = 3$ being infinite is the twin prime conjecture (Corollary B); the ray $(1;1)$ being infinite is the infinitude of balanced primes; their difficulty crosses the bridge fully intact. The paper itself maintains this honesty — its conjectures are stated as conjectures — and the present review has tried to model the same discipline: the two promotions of §4.3 succeeded precisely because C7 and C8 were native statements, local congruence laws that no one had reason to pose before the coordinates existed, and whose proofs are correspondingly elementary. The framework’s honest value lies in that direction: not in making old problems easier, but in making new, well-posed, falsifiable problems visible.

6.3  What does cross: new statistics, and a placement in modern analytic number theory

The native observables — $f(x)$, its constant $C(x)$, the gap-class constants implicit in $\mathbb{P}(\text{level} \mid g)$, the level-distribution profile, the $(1;i)$ decay in $i$ — simply do not exist in classical coordinates; they are new measurements of the primes, each with a definite conjectural law attached and each recomputable by anyone. And the central one places the framework inside an active research stream: the level event is the event “$l(n)$ has no divisor in $(g(n), \sqrt{l(n)}\,]$,” the complementary event to Ford’s $H(x,y,z)$ theory (Annals 2008), which grew out of Erdős’s multiplication-table problem and is a pillar of the anatomy of integers program of Ford, Tenenbaum, Koukoulopoulos and others. What the framework adds to that program is a genuinely new twist: the integer being probed is a linear form in two consecutive primes, $l(n) = 2p(n) - p(n{+}1)$, and the interval’s lower endpoint is the gap itself. Divisors of shifted primes $p - a$ for fixed $a$ are understood (Koukoulopoulos 2010); divisors of $p(n) - g(n)$, where the shift is the next-prime increment, are terra incognita. Any unconditional progress on Conjecture 9 — even a proof that $f(x) \to 0$ at any rate — would require equidistribution information about consecutive primes of a kind not presently available, and would therefore be of interest well beyond the framework. This is the precise sense in which Conjecture 9 is both tractable-looking and genuinely deep: the heuristic is classical, the obstruction is structural.

6.4  Comparative morphology: the atlas as an instrument

Because the construction is uniform, the $(\log k, \log L)$ cloud is a signature of the input sequence, and its gross morphology is dictated by gap behavior. Constant jump (the naturals, §3): a full wedge plus the level-1 prime ray. Erratic but slowly growing gaps (the primes, §4–5): wedge plus a low horizontal fan whose thickness is the gap distribution. The paper’s own figures for composites (A130882) and 2-almost primes (A130533) interpolate between these. One can read this as decomposition spectroscopy: bounded gaps confine the level fan to $L < \max g$; gap growth rates reshape the fan’s envelope; multiplicative structure in the sequence (e.g. semiprimes) re-textures the combs. A systematic survey across the atlas — cataloguing fan envelopes and comb spectra against known gap asymptotics — is, to our knowledge, unexplored territory and entirely computational in its first phase. We flag it as the most accessible research direction the corpus offers (§7.3).

6.5  Hypothetical impacts — flagged as speculation

7.1  What this review established

On verification: an independent engine recomputed weight, level, jump and classification for all $5{,}761{,}455$ primes below $10^8$, confirming the decomposition identity and both classification criteria in every decomposable case, the exceptional set $\{2,3,7\}$, Table 2 entry-for-entry, Table 6 count-for-count in range, Lemma 3.1 across the full range, the five Conjecture-4 exceptions exactly, and the first forty published terms of A117078 verbatim. The paper’s data layer is sound.

On theory, four items are new here. Theorem A (§4.3): Conjectures 7 and 8 are theorems — a three-line mod-3 argument yields the exact dichotomy “$3 \mid g(n)$ or $3 \mid l(n)$, never both” for every decomposable prime $\ge 5$, the mod-3 sibling of the evenness of gaps, and the last forced congruence of its kind. Corollary B and Proposition C: the twin characterization re-proved in one line from Theorem A, and the sharp statement that $5$ is the unique level-classified lesser twin. The $l < g^4$ proposition (§4.6): composite weights on the level side are squeezed into $l < g^4$, so Conjecture 4’s exception set is finite under any gap bound $g \le p^{1/4-\varepsilon}$ — a clean calibration of its difficulty. Conjecture 9′ (§4.7): a quantitative refinement of the framework’s native statement, $f(x) = (C + o(1))\log\log x/\log x$, supported by a divisor-window heuristic with the correct local factors, by four decades of data, and accompanied by a new structural law — the gap’s prime factorization steers the classification probability, with $\mathbb{P}(\text{level} \mid g)$ rising from $\approx 0.09$ ($g = 4, 8$) to $0.33$ ($g = 30$).

7.2  An honest overall assessment

The decomposition into weight $\times$ level $+$ jump is a genuine, original, conservative construction: it adds no axioms, contradicts nothing classical, specializes on $\mathbb{N}$ to the smallest-prime-factor extraction that underlies both the FTA and the sieve, and generalizes uniformly to arbitrary increasing sequences. Its eighteen-year development shows the habits of serious mathematics — exact definitions, falsifiable conjectures, public data, OEIS refereeing, reproducible code. Its limits are equally clear, and the author’s own materials state them: the famous problems it touches (twins, balanced primes) are reformulated, not weakened; the framework’s lasting mathematical claim therefore rests on its native layer. That layer, after this review, consists of: a small proved core (Lemma 1.1, Theorem 3.1, Theorem A, Corollary B, Propositions of §4.3 and §4.6), one sharply quantified open conjecture (9′) sitting at the live frontier of the anatomy of integers, and a one-of-a-kind comparative atlas. This is a solid and honorable position for an independent research program — and Conjecture 9′ in particular is, in our judgment, a publishable mathematical object: precisely posed, heuristically derived, numerically supported, and provably hard for an identifiable structural reason.

7.3  A concrete research program

1. Computation to $10^{10}$–$10^{12}$ (segmented). The present engine holds all arrays in memory; a segmented variant (sieving blocks, streaming the divisor search) reaches $10^{10}$ on a workstation and $10^{12}$ with patience or modest cluster time. Targets: track $C(x)$ another two–four decades; measure the gap-class constants $c_g$; follow the $(1;i)$ ratios; hunt a sixth Conjecture-4 exception (none should exist: the next candidate needs $l < g^4$). The fitted laws make the exercise falsifiable in advance:

2. A conditional theorem for Conjecture 9′. Formulate the needed input as an explicit hypothesis — equidistribution of $l(n) = 2p(n) - p(n{+}1)$ in residue classes $q \le (\log x)^{A}$ on average over $n \le \pi(x)$, a “level Elliott–Halberstam at small moduli” — and derive $f(x) \asymp \log\log x/\log x$ from it together with Gallagher’s gap model. This is a well-scoped, paper-sized project: the deduction is the §4.7 heuristic made rigorous; all novelty concentrates in stating the hypothesis correctly and handling the composite-divisor correction. We caution that unconditional progress, even $f(x) \to 0$ at any rate, appears to require joint equidistribution of consecutive primes of a kind not currently available — which is exactly why partial results would carry independent interest.

3. The atlas survey (“decomposition spectroscopy”). Catalogue, across the thousand-sequence atlas, the level-fan envelope and comb spectrum against known gap asymptotics of each sequence; then run symbolic regression on structured families to recover closed forms for $k(n), L(n)$ and propose cross-sequence laws. Entirely computational in phase one; the cleanest entry point for collaborators or students.

4. Formalization. The proved core is elementary and small — Lean formalization of Lemma 1.1, Theorem 3.1, Theorem A, Corollary B, Proposition C and the $l

5. Extensions. Re-run the full §4 analysis for the composites (A130882) and 2-almost primes (A130533) where the paper already provides figures: in particular, formulate and test the analogue of Conjecture 9′ in each case (for composites the gaps are bounded infinitely often, so the level fan behaves differently — a sharp test of the spectroscopy picture of §6.4).

6. Dissemination. Write the short note this review’s mathematics supports — Theorem A with the dichotomy and Proposition C, the $lINTEGERS; Journal of Integer Sequences), with the verification engine as ancillary material; update the arXiv paper and the OEIS wiki accordingly, moving C7/C8 from the conjecture list to a theorem section with proof.

7.4  Closing

Eighteen years ago the question was posed: how does a number grow from the one before it? The decomposition’s answer — split each term into a multiplicative body and an additive jump, and watch which side carries the structure — has matured into a coherent, verified, and partly proved framework with one excellent open problem of its own. The most valuable thing this review can report is that the framework now knows exactly what it has proved, exactly what it has reformulated, and exactly what it must do next. That is the posture of working mathematics.

The atlas’s reference implementation (PARI/GP, from algos.txt), reproduced verbatim. The naive form scans weights upward from $d+1$; the sieve form stops at $\sqrt{l}$ and, failing that, scans levels downward from $d$ — legitimate because on the level side $L \le d < d + 1 \le k$ — a bound that is sharp for general sequences (the pair $72 \to 76$ attains $L = d = 4$), while for the primes parity sharpens it to $L \le g - 1$ (§4.2). Note also that the descending search’s test $n \bmod \lfloor l/\mathit{le}\rfloor = d$ quietly enforces that the cofactor exceeds the jump — a condition that cannot be dropped, as the empty-window primes $13, 31, 113$ show:

decompnaive(n,n1)={
    /*strictly increasing*/
    if(n>=n1,print("n1 must be greater than n");return);
    /*jump*/
    d=n1-n;
    /*l=n-d if n>2*d else the number is not decomposable*/
    if(n>2*d,l=n-d,print(d, ", 0, 0");return);
    /*we look for the weight to jump+1 until l*/
    for(k=d+1,l,if(n%k==d,print(n," = ",k," * ",l/k," + ",d);return));
}

decompsieve(n,n1)={
    /*strictly increasing*/
    if(n>=n1,print("n1 must be greater than n");return);
    /*jump*/
    d=n1-n;
    /*l=n-d if n>2*d else the number is not decomposable*/
    if(n>2*d,l=n-d,print(d, ", 0, 0");return);
    /*we look for the weight to jump+1 until sqrt(l)*/
    for(k=d+1,sqrt(l),if(n%k==d,print(n," = ",k," * ",l/k," + ",d);return));
    /*we look for the level to jump until 1 (--)*/
    forstep(le=d,1,-1,if(n%floor(l/le)==d,print(n," = ",l/le," * ",le," + ",d);return));
}

Our verification engine (Python/NumPy) proceeds in four stages: (1) sieve of Eratosthenes to $10^8$, arrays $p(n), g(n), l(n)$; (2) staged vectorized search for the smallest divisor of $l$ in $(g, \sqrt{l}\,]$ (batched over trial divisors, resumable); (3) exact weights for the level class via the largest divisor $\le \sqrt l$, with the three empty-window repairs $13, 31, 113$ handled exactly; (4) assertions — identity, both classification criteria, Tables 2 and 6, Lemma 3.1, C7/C8, the C4 exception scan, A117078 — plus the census, decay, and gap-conditional statistics reported above. Total runtime under three minutes on this machine. Every number in this report regenerates from these four scripts plus the figure scripts; the methodology mirrors the project’s stated discipline: verify numerical claims by computation where feasible.