Scope, notation, and standards of evidence

Throughout, $a(n)$ is a strictly increasing sequence of positive integers and

$$d(n)=a(n+1)-a(n),\qquad \ell(n)=a(n)-d(n)=2a(n)-a(n+1).$$

The report uses the notation requested for the project: $a(n),d(n),\ell(n),k(n),L(n)$. For the prime sequence, $a(n)=p_n$ and $d(n)=g_n=p_{n+1}-p_n$. All plotted logarithms are natural logarithms, written $\ln$. A cutoff “$p\le X$” always uses the true successor of every $p\le X$, even when that successor lies above $X$. This is different from “the first $N$ primes.”

The following labels are deliberately non-interchangeable.

Source hierarchy

1. Significance: what the construction is—and is not

1.1 A canonical local coordinate change

The essential move is not the identity $a=kL+d$ by itself: every Euclidean division supplies identities of that form. The distinctive feature is how the divisor $k$ is selected. The next term determines the additive displacement $d$; the displacement determines the reflected integer $\ell=a-d$; and the arithmetic of $\ell$ selects the least divisor strictly beyond $d$. Thus no factorization parameter is chosen by hand.

This is a genuine additive–multiplicative coupling at a local scale. The additive datum is one forward difference; the multiplicative datum is a distinguished factor pair of one gap-shifted integer. For primes, the reflected point $2p_n-p_{n+1}$ lies one prime gap to the left of $p_n$. The framework therefore asks a sharp question: does that additively determined integer possess a divisor in a specified interval?

1.2 The real contribution

The framework makes three useful contributions.

First, it offers a uniform vocabulary for any increasing sequence whose relative growth is not too fast. Natural numbers, primes, polynomial sequences, and many bounded-gap sequences are treated by the same rule. Second, it turns familiar prime patterns into geometric strata—vertical fixed-weight fibres, horizontal fixed-level fibres, and a diagonal classification boundary. Third, it creates native divisor-window questions that do not arise in quite this form before the coordinate change, especially the proposed rarefaction of the level class.

The strongest defensible significance claim is therefore:

WLJ is a canonical local coordinate system that transports next-gap information into the divisor structure of $2a(n)-a(n+1)$, organizes many integer sequences into comparable strata, and suggests analytic questions about divisors of gap-shifted values.

1.3 Boundaries of significance

Several larger claims would be misleading.

• Not a new Fundamental Theorem of Arithmetic. Unique prime factorization already makes the natural-number specialization work. The uniqueness of $k$ is also built into the “least divisor” definition.
• Not the first additive–multiplicative bridge. Euler products, Dirichlet series, the circle method, and sieve theory already connect additive and multiplicative structure. The classical identity $\sum n^{-s}=\prod_p(1-p^{-s})^{-1}$ is the most basic example; see DLMF §27.4.
• Not a solution method for classical prime conjectures. Rewriting twin, balanced, or Hardy–Littlewood-type families does not reduce their difficulty.
• Not a route to RH or Goldbach currently. WLJ works at the scale of one consecutive-prime gap and one reflected integer. RH concerns global prime-counting errors and zeta zeros; Goldbach couples two independently varying primes in a sum. No implication or equivalence is known.
• Not universal for all increasing sequences. The construction fails whenever the next term is at least $3/2$ of the current term. It is eventually available for polynomial sequences, but not, for example, for a geometric sequence of ratio at least $3/2$; Fibonacci ratios tend to $\varphi>3/2$.

2. Definitions and elementary structure

2.1 Definition

Let $a(n+1)>a(n)$ and put $d(n)=a(n+1)-a(n)$. Define

$$\ell(n)= \begin{cases} a(n)-d(n),&a(n)-d(n)>d(n),\\ 0,&\text{otherwise}. \end{cases}$$

When $\ell(n)>0$, define

$$k(n)=\min\{r\in\mathbb N:r>d(n),\ r\mid\ell(n)\}, \qquad L(n)=\frac{\ell(n)}{k(n)}.$$

When $\ell(n)=0$, set $k(n)=L(n)=0$. Whenever the decomposition exists,

$$a(n)=k(n)L(n)+d(n),\qquad 0<d(n)<k(n).$$

The classification convention is:

$$\begin{array}{rcl} k(n)>L(n)&\Longrightarrow&\text{level-classified},\\ k(n)\le L(n)&\Longrightarrow&\text{weight-classified},\\ k(n)=L(n)=0&\Longrightarrow&\text{unclassified}. \end{array}$$

Ties $k=L$ belong to the weight class.

2.2 Exact existence criterion

Proposition 2.1 (proved). The decomposition exists at index $n$ if and only if

$$a(n+1)<\frac32a(n).$$

Proof. Since $d=a(n+1)-a(n)$, the definition requires

$$a(n)-d>d \iff a(n)>2(a(n+1)-a(n)) \iff a(n+1)<\frac32a(n).$$

If the inequality holds, then $\ell>d$, and $\ell$ itself is a divisor of $\ell$ exceeding $d$; the set defining $k$ is therefore nonempty. Its least element is unique. Conversely, if the inequality fails, every positive divisor of $\ell\le d$ is at most $d$, so no admissible $k$ exists. □

This criterion makes the domain transparent. Any sequence with $d(n)=o(a(n))$ is eventually decomposable. A bounded-jump sequence is eventually decomposable; a rapidly growing recurrence may not be.

2.3 The divisor-window criterion

The classification has an exact multiplicative interpretation.

Proposition 2.2 (proved). For a decomposable term,

$$\begin{aligned} \text{weight-classified} &\iff d<k\le\sqrt\ell\le L,\\ \text{level-classified} &\iff L<\sqrt\ell<k \iff \ell\text{ has no divisor in }(d,\sqrt\ell]. \end{aligned}$$

Proof. Because $kL=\ell$, $k\le L$ is equivalent to $k\le\sqrt\ell$. If a divisor $r\in(d,\sqrt\ell]$ exists, minimality gives $k\le r\le\sqrt\ell$, so the term is weight-classified. If no such divisor exists, the least divisor beyond $d$ lies above $\sqrt\ell$, giving $k>L$. □

Corollary 2.3 (level bound). Every level-classified term satisfies $L\le d$. Otherwise $L\mid\ell$ and $d<L<k$, contradicting the minimality of $k$.

The classification is therefore not an arbitrary comparison of factor sizes. It records whether the divisor interval immediately after the additive gap, up to the square-root barrier, is occupied or empty.

2.4 Worked examples

The $p=13$ row is particularly instructive. The factor $3$ is ignored because it lies below the strict threshold $k>d=4$; the selected weight is therefore $9$, which may be composite. This is why prime-sequence weights are not generally prime.

2.5 Audited PARI/GP algorithms

The supplied fordiv kernel captures the definition cleanly, but it does not validate monotonicity. Two supplied factorization variants return [d,0,0] in the non-decomposable case even though their normal tuple order is [k,L,d]; the canonical return is [0,0,d]. The older decompsieve performs a two-sided divisor search on one integer—it is not an Eratosthenes range sieve in the standard algorithmic sense.

The following memory-light kernel makes the contract explicit. PARI/GP’s fordiv visits positive divisors in increasing order; it avoids materializing the full divisor vector, although factorization cost remains implicit.

/* Return [weight, level, jump] for integers 0 < a < b. */
wlj(a, b) =
{
  my(d, ell);
  if (a <= 0 || b <= a, error("wlj: require 0 < a < b"));
  d = b - a;
  if (a <= 2*d, return([0, 0, d]));
  ell = a - d;
  fordiv(ell, k,
    if (k > d, return([k, ell/k, d]))
  );
  error("wlj: unreachable divisor state");
}

wlj_class(t) =
{
  if (t[1] == 0, return("unclassified"));
  if (t[1] > t[2], "level", "weight")
}

For a full natural-number interval, a genuine smallest-prime-factor sieve is asymptotically preferable to factoring each $n-1$ separately:

/* Smallest-prime-factor table, Eratosthenes style. */
spf_table(N) =
{
  my(sp = vector(N, i, 0));
  if (N >= 1, sp[1] = 1);
  forprime(p = 2, N,
    sp[p] = p;
    if (p <= N\p,
      forstep(m = p*p, N, p,
        if (sp[m] == 0, sp[m] = p)
      )
    )
  );
  sp
}

/* Decompose 3,4,...,N in one pass. */
wlj_naturals(N) =
{
  if (N < 3, return([]));
  my(sp = spf_table(N-1));
  vector(N-2, i,
    my(n = i+2, ell = n-1, k = sp[ell]);
    [k, ell/k, 1]
  )
}

For isolated or irregular sequence terms, fordiv is the safest exact implementation. For dense bounded ranges, an SPF table or segmented smallest-factor sieve is preferable. For very large prime censuses, one must generate each true successor and factor $\ell=2p-p_{\mathrm{next}}$; reported timing alone is not a substitute for releasing the code, checkpoints, range convention, and reproducible hashes. The relevant language semantics are documented in the PARI/GP function index.

3. Fundamental Theorem of Arithmetic and the sieve of Eratosthenes

3.1 Natural numbers: exact specialization

Take $a(n)=n$. Then $d(n)=1$, $\ell(n)=n-1$, and for $n>2$

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

The weight is necessarily prime: the least divisor greater than $1$ is the smallest prime factor. The level is the largest proper divisor of $n-1$. Hence

$$\begin{aligned} n-1\text{ prime} &\iff k=n-1,\ L=1,\ k>L,\\ n-1\text{ composite} &\iff k\le\sqrt{n-1}\le L,\ k\le L. \end{aligned}$$

Thus level-classified naturals are precisely shifted primes $q+1$; weight-classified naturals are shifted composites. Ties $k=L$ occur exactly at $n-1=q^2$ with $q$ prime.

3.2 Relation to the Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic (FTA) states that every integer $m>1$ has a unique prime-power factorization; see NIST DLMF §27.2. WLJ’s natural-number rule exposes the first step of that factorization:

$$m=k_1m_1,\quad k_1=\operatorname{spf}(m),\qquad m_1=k_2m_2,\quad k_2=\operatorname{spf}(m_1),\quad\ldots$$

Iterating the level/cofactor eventually recovers the prime factorization, with repeated smallest factors appearing as repeated steps. But the logical dependence runs from the classical arithmetic to the coordinate representation, not the other way around. WLJ does not exclude an alternative prime factorization independently; FTA does. The “unique triple” is unique because a minimum was specified.

3.3 Relation to Eratosthenes

At the sieve stage for a prime $q$, the surviving composites first struck by $q$ are exactly those $m$ divisible by $q$ and by no smaller prime. With $m=n-1$, this is exactly the WLJ column $k(n)=q$:

$$k(n)=q \iff q\mid(n-1)\text{ and no prime }r<q\text{ divides }n-1.$$

The points on $L=1$ are the survivors $m=q$ that were never struck before their own value. In that precise sense, the natural-number WLJ plane is an exact coordinate rendering of the sieve’s outcome.

Computationally, however, the supplied per-term routines are trial-division/factorization procedures. A genuine Eratosthenes sieve marks ranges of multiples and shares work across many inputs; see Melissa O’Neill’s “The Genuine Sieve of Eratosthenes” for the algorithmic distinction.

3.4 The sieve figures

Write $x=\ln k$, $y=\ln L$, and let $X$ be the largest value of $\ell=n-1$. The composite support satisfies the two exact linear inequalities

$$y\ge x,\qquad x+y\le\ln X.$$

The apparent “triangle” is therefore deterministic. Its apex is

$$\left(\tfrac12\ln X,\tfrac12\ln X\right),$$

the logarithmic image of the square-root barrier. The prime values of $\ell$ lie on $y=0$; they represent shifted primes $n=\ell+1$, not generally prime values of $n$.

The vertical occupancy is more precise than “multiples of $q$.” A fixed fibre $k=q$ consists of $\ell=qm$ for which $m$ has no prime factor below $q$. These are rough-number fibres. Their qualitative frequencies are governed by sieve products, but the saturated bitmap does not prove a Mertens asymptotic. The anti-diagonal is straight in log coordinates; $kL=X$ would be hyperbolic only on raw $(k,L)$ axes.

4. Application to the prime sequence: proved structure

4.1 Prime coordinates and the three exceptions

For $a(n)=p_n$, write

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

Whenever the decomposition exists,

$$p_n=k(n)L(n)+g_n,qquad k(n)=\min\{r>g_n:r\mid\ell(n)\}.$$

Theorem 4.1 (proved). The only non-decomposable primes are $2,3,7$.

The 2010 preprint proves this using explicit prime-counting estimates, but its printed argument suppresses a monotonicity detail. A shorter independent repair uses Nagura’s theorem: for $x\ge25$, there is a prime in $(x,6x/5)$. Hence for every prime $p\ge29$,

$$p_{\mathrm{next}}<\frac65p<\frac32p.$$

Checking the primes below $29$ leaves exactly $2,3,7$. See Nagura’s original 1952 paper; the preprint’s alternative route cites explicit estimates of Dusart, whose later bounds are available as arXiv:1002.0442.

4.2 Parity, coprimality, and order

For decomposable $p_n>3$, both consecutive primes are odd, so $g_n$ is even while $\ell(n)$, $k(n)$, and $L(n)$ are odd. Moreover,

$$\gcd(g_n,\ell(n))=\gcd(g_n,p_n)=1,$$

and therefore $\gcd(g_n,k(n))=\gcd(g_n,L(n))=1$. The basic inequalities are

$$3\le k(n)\le\ell(n),\qquad 1\le L(n)\le\frac{\ell(n)}3,qquad 2\le g_n\le k(n)-1.$$

Combining these with Proposition 2.2 gives the complete order geometry:

$$\begin{array}{ll} \text{weight class:}& g_n<k(n)\le L(n),\\[2pt] \text{level class:}& L(n)<g_n<k(n). \end{array}$$

The strict first inequality in the level row uses the parity refinement $L\le g-1$. These orders will explain the two 3D cones without appealing to visual density.

4.3 Weight three is exactly the twin column

Theorem 4.2 (proved). For $p_n>3$,

$$k(n)=3\iff g_n=2\iff p_n\text{ is the lesser member of a twin-prime pair}.$$

If $g_n=2$, then $p_n\equiv-1\pmod6$, so $\ell=p_n-2$ is divisible by $3$; the least divisor exceeding $2$ is $3$. Conversely, $k=3$ and $2\le g\le k-1$ force $g=2$.

One bookkeeping subtlety matters. The prime $5$ has $(k,L,g)=(3,1,2)$, so it is level-classified. Thus “primes of weight $3$” can mean either all primes with coordinate $k=3$, or only weight-classified primes in that column. Through $10^{10}$, those two conventions differ by the single point $p=5$ (and a count based on lesser twin members also handles $p=3$ separately). Reports should state which convention they use.

4.4 Balanced primes and level generation

A level-one prime has $L(n)=1$, hence $k(n)=\ell(n)$. The original project refines level one by writing $(1;i)$ when

$$\ell(n)=p_{n-i}.$$

For $i=1$,

$$p_n-g_n=p_{n-1} \iff p_n-p_{n-1}=p_{n+1}-p_n.$$

Therefore generation $(1;1)$ is exactly the sequence of balanced primes. This is an exact reformulation, but infinitude of balanced primes remains open. Infinitely many level-one primes would not by itself imply infinitely many balanced primes, because other generations $(1;i)$ may contribute.

4.5 The mod-3 rigidity

The original paper listed two conjectures:

$$6\nmid g_n\Longrightarrow3\mid\ell(n), \qquad 3\nmid\ell(n)\Longrightarrow6\mid g_n.$$

They are the same elementary fact, the second being the contrapositive of the first.

Theorem 4.3 (proved). For consecutive primes $p_n,p_{n+1}>3$,

$$3\mid\ell(n)\iff6\nmid g_n.$$

Proof. If $3\mid g$, then $\ell=p-g\equiv p\not\equiv0\pmod3$. If $3\nmid g$, the two endpoints are nonzero residues modulo $3$; primality of $p+g$ forces $g\equiv p\pmod3$, so $p-g\equiv0\pmod3$. Since $g$ is even, $3\mid g$ is equivalent to $6\mid g$. □

The current OEIS project page appropriately strikes C7 and C8 from the conjecture list. They should be recorded as elementary structure, not as two independent research achievements.

4.6 The Cube Lemma

Theorem 4.4 (Cube Lemma, proved). If a level-classified prime has composite weight, then

$$\ell(n)\le(g_n-1)^3.$$

Proof. Write the composite weight as $k=ab$ with $1<a\le b<k$. The factors $a$ and $b$ divide $\ell$ and lie below the least divisor $k>g$; therefore minimality gives $a,b\le g$. They are odd while $g$ is even, hence $a,b\le g-1$. The level bound gives $L\le g-1$. Consequently

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

The supplied fifth report contains an overbroad sentence suggesting that every proper divisor of $\ell$ is at most $g$; that is false because $k$ itself may be a proper divisor greater than $g$. The repaired proof above needs only the two factors $a,b<k$. The bound is sharp at $p=131$: $g=6$, $\ell=125=5^3$, $k=25$, and $L=5$.

The observed composite-weight level primes are

$$13,\ 31,\ 113,\ 131,\ 887.$$

This list is independently reproduced below through $10^6$. The project combines a scan beyond $3.21\cdot10^9$ with the published prime-gap verification through $4\cdot10^{18}$, where the maximal gap is $1476$. Any further exception within that verified range would satisfy

$$p=\ell+g\le1475^3+1476=3{,}209{,}048{,}351,$$

already inside the reported scan. This closes the five-element list through $4\cdot10^{18}$. It does not prove the global conjecture. The underlying finite prime-gap computation is due to Oliveira e Silva, Herzog, and Pardi; see their Math. Comp. paper.

4.7 Independent verification through one million

A separate Python implementation was written from the definition, using integer arithmetic, an SPF sieve, exact divisor enumeration, and no reuse of the supplied PARI/GP kernels. This is a cross-check, not evidence for an asymptotic theorem.

The first seventeen prime rows agree exactly with the 2010 preprint. The only non-decomposable values are $2,3,7$; the composite-weight level values are exactly $13,31,113,131,887$; and the weight-three equivalence, level bound $L<g$, and mod-$3$ equivalence have zero violations. The level share among decomposable primes at this value cutoff is

$$\frac{18{,}353}{78{,}495}=0.233811\ldots.$$

This must not be confused with the 2010 paper’s counts for the first $10^6$ prime indices, whose upper prime is much larger.

5. Prime conjectures, proofs, and modern context

5.1 Audited conjecture ledger

5.2 Why fixed weights are not Polignac columns

For a fixed weight $k$, one necessarily has an even gap $g<k$, together with

$$k\mid(p-g)$$

and the minimality requirement that no divisor of $p-g$ lies in $(g,k)$. Thus a fixed weight mixes a finite set of possible gaps and applies additional congruence and divisor filters. Weight $7$, for example, occurs with both $g=4$ and $g=6$. Only $k=3$ collapses to one exact gap family; $k=5$ happens to refine gap $4$, but this does not generalize.

If infinitely many primes had a fixed weight $k$, finite pigeonhole would force at least one even gap $g<k$ to recur infinitely often. The converse fails: infinitely many occurrences of a particular gap do not force a fixed WLJ weight. C2 is therefore Polignac-type, not equivalent to Polignac’s conjecture.

Modern bounded-gap theorems do not settle C1 or C2. Zhang proved a finite bound on the liminf of consecutive prime gaps; Maynard developed a stronger multidimensional sieve; the Polymath8 project reached the unconditional record $H_1\le246$ (and $H_1\le6$ only under the generalized Elliott–Halberstam hypothesis). These results imply that at least one even gap at most $246$ occurs infinitely often, not that gap $2$, every even gap, or any particular WLJ weight occurs infinitely often. See Zhang, Maynard, and Polymath8b.

5.3 Precise formulation of C9

Let

$$N_{\mathrm{lev}}(x) =\#\{p_n\le x:p_n\text{ is decomposable and }k(n)>L(n)\}.$$

Since only $2,3,7$ are non-decomposable, the natural value-cutoff form is

$$\frac{N_{\mathrm{lev}}(x)}{\pi(x)-3}\longrightarrow0.$$

The 2010 preprint and current OEIS records state this as Conjecture 9. The supplied fifth report labels it PROVED* and proposes the bound

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

which would indeed imply C9 by the Prime Number Theorem.

The proposed proof architecture

The outline is coherent and worth recording accurately.

1. Choose $M=(\log x)^{3/2}$. Since the sum of consecutive gaps up to $x$ is $O(x)$, indices with $g>2M$ contribute $O(x/M)$.
2. For $g\le2M$, the Cube Lemma confines composite-weight level points to $p=O(M^3)$, a polylogarithmic contribution.
3. Outside that exceptional window the weight is prime, say $q=k$, and $\ell=Lq$ with $L\le g-1$. A level point with gap $g$ forces the simultaneous primality of $$q,\qquad Lq+g=p_n,\qquad Lq+2g=p_{n+1}.$$
4. For an upper bound, consecutiveness may be relaxed: every actual level point supplies one prime triple, while additional nonconsecutive triples only enlarge the count.
5. A uniform dimension-three upper-bound sieve, summed over $g\le2M$ and odd $L<g$, is intended to yield the displayed estimate.

Why the status remains unrefereed

The attached HTML gives only these five steps and refers to a separate “submission manuscript” that was not supplied and could not be located as a public preprint. The analytic heart is asserted rather than proved: one needs a uniform Selberg upper bound for varying $(L,g)$, treatment of inadmissible or degenerate triples, and an averaged singular-series estimate of the form

$$\sum_{g\le2M}\sum_{L<g} \frac{\mathfrak S(L,g)}{L} \ll M\log M$$

with all parameter ranges and small cases controlled. No contradiction was found in the strategy, and standard sieve technology makes it plausible. But a plausible architecture is not a complete proof. The defensible statement is therefore:

5.4 Finite evidence for C9

The independent calculation gives the following value-cutoff shares.

The fifth report claims a complete value-cutoff census through $10^{10}$: $455{,}052{,}511$ primes, $455{,}052{,}508$ decomposable, and $71{,}670{,}808$ level-classified, for a reported share $0.1575$. Those are valuable project measurements but were not independently reproduced here; the raw $10^{10}$ census and source code were not among the attachments.

5.5 Level one and a conditional constant

When $g<\sqrt\ell$, level one is equivalent to $\ell$ being prime. Outside the exceptional region, a level-one point is therefore a three-term arithmetic progression of primes

$$\ell=p_n-g_n,\qquad p_n,\qquad p_{n+1}=p_n+g_n,$$

with the right pair consecutive. The supplied project report proposes the conditional asymptotic

$$N_1(x)\sim2C_2\frac{x}{\log^2x},$$

where $C_2$ is the twin-prime constant. Its separate mean-singular-series identity is consistent with Gallagher-style averaging. However, the stated hypotheses give a marginal prime-triple model and a marginal consecutive-gap model; multiplying them requires a joint assumption controlling the probability of no intervening prime inside the reflected-prime triple population. Unless that joint uniformity is added explicitly, the asymptotic should be called a heuristic prediction rather than a proved conditional theorem.

5.6 What RH, prime gaps, and finite verification do not imply

The Prime Number Theorem gives $\pi(x)\sim x/\log x$ and global density zero for primes among integers. It does not imply density zero for the WLJ level class inside the primes. RH would sharpen global prime-counting errors—see the Clay Mathematics Institute—but no implication between RH and C9 is known.

Prime gaps are unbounded, with strong quantitative results due to Maynard and others; see Maynard’s large-gap theorem. Conversely, exhaustive gap tables through $4\cdot10^{18}$ remain finite evidence. Neither a maximal gap of $1476$ on that range nor a long numerical census proves a global WLJ statement.

6. Detailed analysis of the supplied prime graphs

6.1 Common two-dimensional geometry

For every decomposable term, set

$$x=\ln k(n),\qquad y=\ln L(n).$$

Then

$$x+y=\ln\ell(n).$$

At a finite value cutoff $a(n)\le X$, one has $\ell(n)<X$, so $x+y<\ln X$. The diagonal $y=x$ is the exact class boundary: points on or above it are weight-classified, and points below it are level-classified. These two straight lines—the diagonal and anti-diagonal cutoff—explain the gross shape before any distributional theorem is invoked.

For geometry to be visually faithful, “square” must mean more than a square raster. The plot panel must use identical $x$- and $y$-limits and equal data-unit scaling. Every source graph below is displayed in a square matte without distorting its pixels; the independent reproductions also enforce equal mathematical aspect.

6.2 The annotated first-10,000-prime plot

The upper-left wing obeys

$$y\ge x,qquad x+y\le\ln 104{,}729\approx11.559.$$

Its vertical combs are fixed odd weights $3,5,7,9,\ldots$. Unlike in the natural-number plot, these weights need not be prime because all divisors at or below the gap are deliberately ignored. The lower-right wing obeys $y<x$; its horizontal bands are fixed odd levels $1,3,5,\ldots$. The stronger prime bound $L\le g-1$ keeps that wing near small $y$.

The diagonal points $k=L$ are assigned to the weight class. The annotations identify OEIS strata, but arrows do not prove set infinitude or asymptotic density. In particular, only the $k=3$ comb is exactly a named gap family. Higher combs are bounded-gap refinements.

6.3 The $p\le1.5\cdot10^6$ plot

The corresponding level share is

$$\frac{26{,}117}{114{,}152}=0.228791\ldots,$$

or about $22.88\%$. The percentages $17.11\%/82.89\%$, quoted in some earlier captions, came from the much larger first-$5\cdot10^7$-prime census and do not describe this image.

At this cutoff, $\ln(1.5\cdot10^6)\approx14.221$, so the upper wing again terminates on an anti-diagonal with midpoint near $7.110$. The level wing is horizontally banded at odd $L$; the dense baseline is $L=1$. More than $10^5$ points are compressed into a 963-pixel square, so apparent solid lines and darkness are dominated by overplotting. This picture demonstrates finite support and discrete strata, not rarefaction.

6.4 Independent equal-scale prime reconstruction

This color separation makes three facts visible without asking point darkness to carry a statistical claim: the classification is exactly diagonal, fixed weights form vertical fibres, and fixed levels form horizontal fibres. The blank sectors are forced by inequalities, not mysterious repulsion. Quantitative density questions require counts across multiple cutoffs, confidence or error analysis where sampling is involved, and ultimately proof.

6.5 The two 3D WebGL views

The project’s current 3D page identifies the coordinates as

$$(x,y,z)=(\ln k,\ln L,\ln g)$$

for $9{,}997$ dots, again the first $10{,}000$ primes minus $2,3,7$. An older description calling the third coordinate the index $n$ is wrong.

The exact order cones are

$$\begin{array}{ll} \text{weight class:}&z<x\le y,\\ \text{level class:}&y<z<x. \end{array}$$

Fixed gap $g$ gives a plane $z=\ln g$; fixed weight gives $x=\ln k$; fixed level gives $y=\ln L$; and the class boundary is the plane $x=y$. Perspective makes parallel sheets appear to converge. Occlusion, depth-dependent point size, antialiasing, and the absence of a legend prevent quantitative interpretation of color or density.

A publishable 3D replacement should use a square viewport, labeled axes, a fixed orthographic $(x,y)$ view plus one fixed oblique view, equal $x/y$ scaling, documented $z$-normalization, and consistent opacity across cutoffs. One version could color by class and another by $g\bmod6$. Static views alone should not be used to infer an asymptotic density.

6.6 The conjecture ledger screenshot

The screenshot usefully records the project’s internal taxonomy, but two badges require correction. Its C4 PROVED* label conflicts with the fuller fifth report, which correctly limits completeness to $4\cdot10^{18}$. The Cube Lemma is global; the five-member exception list is not. Its C9 PROVED* label refers to the internal Theorem B; the screenshot cannot validate a proof, and the full supplied report omits the central sieve details. C7 and C8 are one implication and its contrapositive, not separate contributions.

6.7 What the graphs establish

The figures are strongest as exact portraits of algebraic constraints:

• $x+y=\ln\ell$ creates the anti-diagonal finite cutoff;
• $x=y$ is the classification boundary;
• natural-number vertical fibres are least-prime-factor strata;
• prime vertical and horizontal fibres are fixed $k$ and fixed $L$;
• the 3D wings are the order cones $z<x\le y$ and $y<z<x$.

They do not establish infinitude, independence, limiting density, Mertens asymptotics, C9, or any classical prime conjecture. Any such conclusion must be supported by a stated statistic across controlled ranges or by analysis.

7. Hypothetical impact on multiplicative and additive number theory

7.1 The bridge stated at the correct scale

Classical additive number theory studies sums and representations; multiplicative number theory studies primes, factors, and multiplicative functions. WLJ does not merge those subjects wholesale. Its precise bridge is narrower and potentially useful:

$$\boxed{\text{next additive gap }d(n)} \quad\longmapsto\quad \boxed{\text{divisor window of }\ell(n)=2a(n)-a(n+1)}.$$

For primes, typical gaps have size comparable to $\log p$, while $\ell\asymp p$. The class asks whether $\ell$ has a divisor in $(g,\sqrt\ell]$. This transports a local additive statistic into a multiplicative avoidance event at a point whose location is itself gap-dependent.

7.2 Potential impact if the analytic program succeeds

A thin-set theorem coupled to prime gaps

A complete proof of the proposed C9 bound would be a real theorem about the scarcity of gap-reflected integers with an empty divisor window. Its novelty would lie in the coupling: $\ell$ is not sampled independently of the primes but is determined by a consecutive pair. Such a result would belong naturally beside sieve bounds for linear forms and the theory of divisors of shifted primes.

New asymptotic constants

If one could establish an asymptotic for level one or the full level class, the constants would measure a three-way interaction among prime-gap frequencies, prime-tuple singular series, and divisor-window avoidance. A rigorously derived constant would be more significant than a visual classification because it would quantify a new local correlation.

Transfer principles for other sequences

For a sequence with controlled gaps, one can ask whether its reflected integers behave like random integers, shifted primes, polynomial values, or a biased multiplicative ensemble. Comparing primes with natural numbers, polynomial sequences, and synthetic controls could identify which features are caused by the WLJ rule itself and which depend on the source sequence.

Algorithms for structured factor queries

WLJ also suggests a computational problem distinct from full factorization: find the first divisor of $\ell$ beyond a moving threshold $d$. On large batches, segmented sieves, partial factorization, and divisor-generation algorithms may exploit the narrow threshold and shared ranges. The mathematics of the coordinate system and the efficiency of its census are separable research questions.

7.3 The right analytic neighbourhood

The relevant external literature is not primarily RH or Goldbach. It is the distribution of divisors in short multiplicative intervals and of divisors of shifted prime values. Kevin Ford’s work on integers with a divisor in $(y,z]$, especially “The distribution of integers with a divisor in a given interval”, supplies a natural baseline. Dimitris Koukoulopoulos’ “Divisors of shifted primes” is closer still to the arithmetic sampling issue.

Neither paper directly treats $2p_n-p_{n+1}$, because the shift is the actual next prime gap and therefore depends on consecutiveness. That dependence is precisely the analytic obstacle. It can be discarded for some upper bounds, as the C9 outline proposes, but not for matching lower bounds or asymptotics.

7.4 Impacts that should not be claimed

• FTA: WLJ packages one factor/cofactor split; it neither generalizes nor reproves unique factorization.
• Twin primes and Polignac: the coordinates expose strata but provide no new lower bound for their infinitude.
• Goldbach: the reflected point uses a consecutive pair, not two independently chosen primes summing to an even integer. No reduction is known.
• RH: WLJ has no known relation to zeta zeros or RH-equivalent prime-counting errors.
• Cryptography: no demonstrated algorithmic advantage for factoring general integers follows from selecting a divisor beyond a threshold.

The intellectually honest impact is therefore conditional: the framework is already a useful descriptive coordinate system; it becomes a substantive analytic theory only when its native counting problems are proved with complete, externally checkable arguments.

8. Detailed conclusion and future research

8.1 What is established

The decomposition is a mathematically coherent coordinate change on the indices for which

$$a(n+1)<\frac32a(n).$$

It converts the next-step additive increment into a factor-threshold problem for the reflected integer $\ell(n)=2a(n)-a(n+1)$. Existence and uniqueness are elementary. The class boundary is exactly the occupancy or vacancy of the divisor window $(d,\sqrt\ell]$.

On the natural numbers, the coordinate system reduces exactly to the smallest-prime-factor decomposition of $n-1$. This explains the visible Eratosthenes columns and the shifted-prime baseline, while also fixing the correct logical claim: WLJ encodes the sieve’s first-strike partition and can recursively read a factorization; it does not replace FTA or introduce a superior sieve algorithm.

On the primes, all but $2,3,7$ are decomposable. The parity, coprimality, level bound, twin-prime identity, balanced-prime equivalence, mod-$3$ rigidity, and Cube Lemma are proved. The prime plane and 3D cloud have exact order geometry. These are durable results independent of any density conjecture.

The five composite-weight level exceptions are independently visible below $10^6$ and, accepting the project’s census together with the published finite gap table, are complete through $4\cdot10^{18}$. That statement is finite. The global C4 conjecture remains open.

The evidence for rarefaction is substantial: the independently recomputed share falls to $23.38\%$ by $10^6$, and the project reports $15.75\%$ by $10^{10}$. The project’s C9 proof strategy is plausible, but the supplied document does not prove the required uniform sieve and average singular-series estimates. As of the date of this report, C9 should be described as a proposed unrefereed theorem and as conjectural in the public OEIS/arXiv record.

8.2 What is not established

No attached source or external publication proves any of the following:

• infinitely many twin primes, balanced primes, fixed WLJ weights, fixed WLJ levels, or generations $(1;i)$;
• a global five-element exception list for C4;
• an unconditional asymptotic or lower bound proving infinitely many level-classified primes;
• a connection to RH, Goldbach, or a new proof of FTA;
• independence between prime gaps and the factorization of $2p_n-p_{n+1}$;
• the internal $10^{10}$ census by independently reproducible public code and raw data.

This negative list is not a dismissal. It identifies the exact boundary between an attractive coordinate system and a mature analytic theory.

8.3 Research agenda, ordered by mathematical leverage

Priority 1 — complete and publish the C9 argument

The highest-value next step is not a longer census but a self-contained manuscript. It should state and prove a uniform dimension-three upper-bound sieve for

$$q,\qquad Lq+g,\qquad Lq+2g$$

over the complete parameter range $g\le2(\log x)^{3/2}$, odd $L<g$, including inadmissible triples and small $q$. It should then prove the required average of the singular factors rather than cite it schematically. The gap-tail and Cube-Lemma reductions must be ordered so that “polylogarithmic” claims are made only after $g$ has been truncated. A public preprint would let independent specialists check constants, uniformity, and logarithmic losses.

Two outcomes would both be useful. A complete proof of the advertised bound would establish C9. A weaker but rigorous bound $N_{\mathrm{lev}}(x)=o(\pi(x))$ would establish the conjecture without preserving the proposed exponent. The mathematical goal should take priority over matching a preannounced rate.

Priority 2 — make the computations independently reproducible

Release the prime generator, decomposition kernel, checkpoint logic, boundary convention, aggregate tables, and cryptographic hashes of the raw census. Every row must decompose $p\le X$ using its true successor, not require $p_{n+1}\le X$. Cross-check at least three independent implementations: fordiv, factor-and-sort, and a segmented factor/sieve pipeline. Publish anchors for the first 17 rows, powers-of-ten counts, level-one counts, the Cube exceptions, and random row samples.

The project atlas should distinguish “computed by project code,” “cross-checked against OEIS,” and “independently reproduced.” Its current disclaimer that data are unverified is appropriately candid; a machine-readable audit manifest would make that disclaimer progressively removable.

Priority 3 — separate level one from higher levels analytically

Level one is a reflected-prime triple problem; higher levels are a divisor-window problem. They should not be forced into one heuristic constant. For level one, formulate an explicit joint hypothesis controlling both the primality of $p-g,p,p+g$ and the absence of primes between $p$ and $p+g$. This would repair the logical gap in the proposed conditional constant $2C_2$.

For $L>1$, compare the actual count with Ford’s divisor-in-interval distribution and Koukoulopoulos’ shifted-prime machinery. A first realistic theorem may be an upper bound averaged over $g$ or $L$, rather than a pointwise asymptotic. Any proposed numerical constant should be accompanied by a stability analysis under moving windows, not just cumulative ratios.

Priority 4 — understand the global C4 barrier

The Cube Lemma reduces any global exception to

$$p\le(g-1)^3+g.$$

Therefore an eventual bound $g=o(p^{1/3})$ would make further exceptions impossible after a finite point. Present unconditional technology is far weaker for this purpose: the Baker–Harman–Pintz short-interval exponent $0.525$ gives only $g\ll p^{0.525}$, whose cube is too large. See Baker–Harman–Pintz. Even the usual RH-scale gap bound would not cross the $1/3$ exponent threshold. Cramér-type polylogarithmic gaps would suffice, but are conjectural.

This identifies a precise reason the finite $4\cdot10^{18}$ closure does not automatically become global: the needed prime-gap inequality is stronger than current unconditional theory.

Priority 5 — develop sequence-to-sequence comparison theorems

For a general sequence with $d(n)\le B$, level classification is a divisor-avoidance condition with a fixed lower endpoint. It should be possible to prove broad upper bounds under hypotheses on how $\ell(n)$ occupies residue classes. Polynomial sequences, Beatty sequences, and carefully randomized controls would make useful test beds. Rapid-growth sequences should be analyzed separately because they fail the existence criterion.

A robust comparison program would report:

• the proportion of indices that are decomposable;
• the empirical distribution of $d$;
• class shares conditional on $d$;
• fixed-$k$ and fixed-$L$ counts with cutoff conventions;
• deviation from a matched random-integer divisor model;
• sensitivity to replacing the true next term by a shuffled or synthetic gap.

This would clarify whether apparent atlas “universality” comes from arithmetic structure or from the generic geometry of the definition.

Priority 6 — improve the visual evidence

Every 2D plot should publish the cutoff, count convention, log base, equal limits, equal aspect, and class counts. Use transparency or hexagonal/log-count bins to control overplotting. For 3D, fix a square viewport and camera, supply orthographic projections, and provide downloadable coordinates. Animate cutoffs only if the same opacity and scale are maintained. The visual goal is not spectacle but falsifiability: a reader should be able to infer which features are algebraic and which are empirical.

8.4 Final assessment

WLJ deserves to be taken seriously as a compact local coordinate system, not as a revolution in prime theory. Its natural-number fibre is classical and exactly solved; its prime fibre is richer because the point being factored depends on the true next prime. That dependence produces both the framework’s interest and its difficulty.

The decomposition’s most promising mathematical question is not whether familiar conjectures look different on the plot. It is whether one can prove distribution theorems for divisors of the gap-reflected values

$$2p_n-p_{n+1}.$$

Success there could yield a legitimate meeting point between prime-gap statistics, upper-bound sieves, and divisor-distribution theory. Until a full proof is publicly available and checked, the correct scientific posture is constructive restraint: retain the valid elementary structure, preserve the impressive finite evidence with explicit range labels, and treat the proposed new theorems as a research program rather than completed mathematics.

Appendix A. Reproducibility record

A.1 First seventeen prime rows

The independent implementation gives the following tuples. They reproduce Table 2 of the 2010 preprint exactly.

A.2 Independent test inventory

The verifier generated primes and smallest prime factors by an ordinary Eratosthenes-style sieve, enumerated divisors from prime powers, and then applied the definition literally. Through $p\le10^6$, it checked:

• exact agreement with all first 17 published rows;
• non-decomposable set $\{2,3,7\}$;
• zero violations of $k=3\iff g=2$ for $p>3$;
• zero violations of $L<g$ in the prime level class;
• zero violations of $3\mid\ell\iff6\nmid g$;
• composite-weight level set $\{13,31,113,131,887\}$;
• the single level-classified point in the $k=3$ column, $p=5$.

The square charts were generated from those independently computed tuples. The source code and machine-readable summary were used for this report’s internal validation; the HTML embeds the rendered figures so it remains self-contained.

A.3 Figure conventions

For every regenerated 2D WLJ scatter:

Coordinates are $x=\ln k(n)$, $y=\ln L(n)$, on a square canvas with equal data-unit aspect and identical axis limits. Zeros are omitted because $\ln0$ is undefined. The diagonal $y=x$ is the exact class boundary. The anti-diagonal $x+y=\ln X$ is a finite-cutoff envelope, not an asymptotic law. For prime value cutoffs, each $p\le X$ is decomposed using its true successor.

References and source links

1. Rémi Eismann, “Decomposition into weight × level + jump and application to a new classification of primes,” arXiv:0711.0865v4, revised 2010. Core project preprint; not refereed.
2. OEIS community, “Decomposition into weight × level + jump”, with linked sequence records A117078 (weight), A117563 (level), A162174 (level-classified primes), and A162175 (weight-classified primes).
3. Rémi Eismann, WLJ sequence atlas, prime decomposition page, and prime 3D visualization. Project sources; atlas data carry the author’s unverified-data disclaimer.
4. Supplied project files: links.txt, algos.txt, decompwlj_fordiv.txt, the seven audited images, the arXiv PDF, and Fable5_decompwlj_deep_analysis_5th_edition.html (13 July 2026). The later online sixth edition was consulted for current project status.
5. NIST Digital Library of Mathematical Functions, §27.2 (prime factorization and prime-counting notation), §27.4 (Euler products), and §27.12 (prime-number asymptotics and RH-related estimates).
6. J. Nagura, “On the interval containing at least one prime number,” Proceedings of the Japan Academy 28 (1952), doi:10.3792/pja/1195570997.
7. Pierre Dusart, “Estimates of some functions over primes without R.H.,” arXiv:1002.0442, 2010.
8. Melissa E. O’Neill, “The Genuine Sieve of Eratosthenes,” Journal of Functional Programming 19 (2009), Cambridge Core.
9. Yitang Zhang, “Bounded gaps between primes,” Annals of Mathematics 179 (2014), journal page.
10. James Maynard, “Small gaps between primes,” Annals of Mathematics 181 (2015), journal page.
11. D. H. J. Polymath, “Variants of the Selberg sieve, and bounded intervals containing many primes,” arXiv:1407.4897.
12. James Maynard, “Large gaps between primes,” Annals of Mathematics 183 (2016), journal page.
13. T. Oliveira e Silva, S. Herzog, and S. Pardi, “Empirical verification of the even Goldbach conjecture and computation of prime gaps up to $4\cdot10^{18}$,” Mathematics of Computation 83 (2014), doi:10.1090/S0025-5718-2013-02787-1.
14. Kevin Ford, “The distribution of integers with a divisor in a given interval,” Annals of Mathematics 168 (2008), journal page.
15. Dimitris Koukoulopoulos, “Divisors of shifted primes,” International Mathematics Research Notices 2010, arXiv:0905.0163.
16. R. C. Baker, G. Harman, and J. Pintz, “The difference between consecutive primes, II,” Proceedings of the London Mathematical Society 83 (2001), doi:10.1112/plms/83.3.532.
17. PARI/GP, function index and reference manual, especially divisors, fordiv, factor, and forprime.
18. Clay Mathematics Institute, Riemann Hypothesis, current status and problem description.