Decompwlj and the Great Conjectures

Where Legendre, Andrica, Cramér, the Riemann Hypothesis, twin primes, Goldbach and Collatz actually touch the coordinates $a(n)=k(n)L(n)+d(n)$ — and where they only appear to.

Report of 24 October 2026 · framework: R. Eismann, arXiv:0711.0865 · all numbers recomputed this session; census, forced set, Legendre census and Collatz ramp checked in both Python/NumPy and PARI/GP 2.15.4

A coordinate change cannot lower the difficulty of a classical problem. The useful question is therefore narrower than "does decompwlj prove X": it is whether a conjecture enters the framework as an input, an output, an exact restatement, or not at all, and whether anything new can be said at the point of contact. The survey found one genuinely structural contact point — a single inequality, $\ell \le d^2$, that converts gap conjectures into statements about the level class — and several honest null results.

Two calibrations hold throughout. First, the framework's own native results — Theorem 1 proved* (level-classified density zero), Lemmas 1–5, Propositions 1–3, the column/lock structure — are unchanged by anything here. Second, Theorem 1's aggregate rarefaction is reproduced within about 1% by a Cramér random sequence of matching density (treatise §7), so it is not evidence of prime-specific structure and is not used below as such.

When the gap alone decides the class

Classification compares $k$ with $L=\ell /k$, so $a(n)$ is level-classified exactly when $k>\sqrt{\ell }$: when $\ell$ has no divisor in the window $(d,\sqrt{\ell }]$. If $d\ge \sqrt{\ell }$ the window is empty and arithmetic is never consulted.

On the primes, write $F=\{p_n:g_n<\ell (n)\le g_n^2\}$, the forced set. Since $\ell =p-g$, membership is the purely additive inequality $g^2+g\ge p$: a gap at least about $\sqrt{p}$.

These six primes already sit in the project's verification record as a closed exception set; what is added here is their reading as the complete list of primes whose class is decided by gap size alone. All six are level-classified, as Proposition B requires. Beyond 113 every level-classified prime is earned: $\ell$ had an open window and happened to have no divisor in it. Earlier atlas work found the same split across sequences — triangular numbers are 100% forced, the primes essentially 0%.

This is an Andrica-type statement with constant 1 in place of 2. Unconditionally only the trivial bound is available: gaps longer than $\sqrt{y/2}$ are disjoint, so $\mathrm{\#}F\cap [y,2y]\ll \sqrt{y}$ and $\mathrm{\#}F(x)\ll \sqrt{x}$. The best unconditional gap exponent, $g\ll p^{0.525}$ (Baker–Harman–Pintz), sits above $\sqrt{p}$ and decides nothing. The next page places every named gap conjecture against this threshold.

The gap ladder, read against $g^2+g=p$

The Firoozbakht line can be made exact. Under Kourbatov's bound $B(p)={\log }^{2}p-\log p-1$, a forced prime past $p_9=23$ needs $B(p{)}^2+B(p)\ge p$, which fails for every $p$ above roughly 3,200 (solved numerically here). The exhaustive list below that height is the six primes of page 2. So Firoozbakht's conjecture implies $F=\{5,13,19,23,31,113\}$ exactly — a clean conditional statement, but it adds nothing beyond the unconditional closure to $4\cdot {10}^{18}$, because Firoozbakht is itself verified only that far.

The instructive line is Andrica's. Its observed maximum is $0.6709$, at $p=7$ (recomputed to ${10}^7$), so data sit comfortably inside it — yet Andrica as stated tolerates gaps up to $2\sqrt{p}$, twice what $F$ can absorb. The forced set is thus a half-Andrica question: decompwlj supplies a natural reason to care about the constant in $g\le c\sqrt{p}$, and fixes the critical value at $c=1$. That is a reformulation, not progress; it is reported because the threshold is canonical rather than chosen.

Legendre's conjecture as a definability condition

The decomposition exists iff $a(n+1)<\frac{3}{2}a(n)$, and on the primes Lemma 2 is proved from Nagura's theorem (a prime in $(x,\frac{6}{5}x]$ for $x\ge 25$). Bertrand's postulate (ratio 2) would not suffice; Nagura's ratio $6/5$ does. Legendre's conjecture is the next rung down — an interval of length about $2\sqrt{x}$ instead of $x/5$ — and it has an exact home in the framework, as the domain of the map.

Corollary. Every term $q(m)$ with $5\le m<1.9\cdot {10}^9$ is level-classified and forced. For $2952\le m<1.9\cdot {10}^9$ this follows from L2, since ${\delta }_m$ is bounded by the gap straddling $m^2$, at most 1476 $\le m/2$ below $4\cdot {10}^{18}$. For $m<2952$ it is part of the direct computation below.

Verdict. Legendre's conjecture is equivalent to the decomposition being defined on $q$, and once defined the decomposition carries no arithmetic signal: $\ell /d^2\to 1/4$ exactly as for $a(m)=m^2$, whose terms are all forced ($\ell =m^2-2m-1<(2m+1{)}^2$). The framework restates Legendre cleanly and then, by its own geometry, has nothing to add.

A one-way street through gap moments

Nothing in the framework bears on the zeros of $\zeta$. The contact is indirect: RH controls prime gaps, and gaps control the forced set. Two classical consequences of RH matter.

Cramér (1920). RH implies $g_n\ll \sqrt{p_n}\log p_n$. Against the threshold $g\approx \sqrt{p}$ this misses by one logarithm, so RH does not, by this route, make $F$ finite.

Selberg (1943). RH implies

A second-moment bound is exactly what a set defined by large gaps needs.

So under RH the forced set is at most polylogarithmically sparse, against the unconditional $\ll \sqrt{x}$. The statement is true and it is the only implication from RH into the framework found in this survey — but it is about gaps, not about decompwlj: any set defined by $g\gtrsim \sqrt{p}$ inherits the same bound.

Computed from all 50,847,534 primes below ${10}^9$. The ratio to ${\log }^{2}x$ creeps upward and the ratio to ${\log }^{3}x$ falls, consistent with the conjectured order ${\log }^{2}x$ quoted in the literature on Selberg's theorem; seven decades cannot identify the order, and none is claimed.

What RH does not do here

Theorem 1 is unconditional: its proof is an upper-bound sieve plus gap telescoping, and RH supplies nothing it lacks. Problem 2 — is the level class infinite? — needs a lower bound for a pattern of consecutive primes, and contains the balanced-prime problem; I know of no route from RH to it. In the opposite direction, no statement about weights or levels implies anything about $\zeta$: the decomposition reads one consecutive pair at a time, while RH is a statement about all primes at once, and the two meet only through moments of the gap sequence.

Where the framework does carry arithmetic

Here the conjectures enter as exact restatements or as hypotheses, and the framework adds its own unconditional constants.

So the twin prime conjecture is literally "the weight-3 column is infinite" (founding Conjecture 1), and the Polignac family maps onto gap columns and lines (founding Conjectures 2, 3, 5, 6). These are dictionaries reformulation: no difficulty is removed.

Hardy–Littlewood as input. Theorem 2 conditional gives the asymptotic of the level-one stratum assuming a uniform Hardy–Littlewood $k$-tuple estimate and a Gallagher-type gap model; its density constant is $2C_2=1.3203236316\dots$, twice the twin-prime constant. Lemma 1 proved identifies that constant unconditionally, as the line average of the singular series $\mathfrak{S}(0,g,2g)$. This is the framework's strongest arithmetic content: a conjecture goes in, and a constant that does not depend on the conjecture comes out.

Mod-3 rigidity (Proposition 2 proved): for decomposable $p\ge 5$, exactly one of $3\mid g$, $3\mid \ell$ holds. It retires founding Conjectures 7–8 as elementary.

Below ${10}^7$: 21,837 balanced primes, every one on $L=1$ and level-classified (Figure 3, rose). Note the asymmetry with §2: the level class beyond 113 is entirely earned, which is why Problem 2 is arithmetic rather than a gap question.

What survives the control

The control experiment shows that the aggregate level share is generic for a sequence of prime density. The finer structure is not: admissibility-matched controls from the atlas work, the exact constant $2C_2$, the congruence lock $Pr[r\mid \ell |d=g]=1/(r-2)$ for odd primes $r>g$, and the weight columns are where the Hardy–Littlewood singular series becomes visible in decompwlj coordinates. Any future contact with a classical conjecture will come through this layer, not through the headline rarefaction.

Goldbach lives one step outside the prime sequence

Decomposing the primes reads one consecutive pair at a time, and its only additive identity is $\ell (n)+p_{n+1}=2p_n$. Applied to the primes, the framework therefore sees only the doubles $2p_n$, where Goldbach is trivial ($p_n+p_n$). That is the precise sense in which Goldbach is absent. But a Goldbach partition $2N=p_1+p_2$ with $p_1<N<p_2$ is itself a three-term progression $(p_1,N,p_2)$, and the decomposition can be applied to it directly: decompose $N$ against each prime $q_1<q_2<\cdots$ above it. With $d_j=q_j-N$,

So the 3/2 barrier of Lemma 2 becomes a natural Goldbach window: partitions with $N/2<p_1<N<p_2<\frac{3}{2}N$ are exactly the level-one decompositions of $N$ with a prime partner, and the case $N=p_n$, $j=1$ is Species I. Outside the window ($p_1\le N/2$) the framework has nothing to say.

Verdict. Goldbach is equivalent to: for every $N\ge 4$, some decomposition of $N$ against a prime above it has a prime partner ${\ell }_j$ — level one in the window, by (ii). This is the classical "primes equidistant from $N$" form (OEIS A047160) written in weight–level coordinates, and it is a dictionary: the level-one event is primality of $2N-q_j$, so the binary lower-bound problem is untouched. What the coordinates add is structural: the window $(N/2,\frac{3}{2}N)$ is the decomposition's own domain; Species I is its special case at prime centres; and the forced zone marks where "level one" and "prime partner" come apart. The window statement — every $2N\ge 88$ has such a partition — is a slightly stronger Goldbach variant, verified here to ${10}^7$; I have not found it stated in the literature and do not claim it is new.

\\ Goldbach depth of N (uses decomp_fact from page 8): [m*, j, first j with L_j = 1, inside window?]
gb(N) = { my(q = N, j = 0, r, l, f = 0);
  while (1, q = nextprime(q+1); j++; l = 2*N - q; if (l < 2, return([-1, j, f, 0]));
    r = decomp_fact(N, q); if (r[1] && r[2] == 1 && !f, f = j);
    if (isprime(l), return([q - N, j, f, r[1] > 0]))) };
\\ gb(43) = [24, 5, 3, 0]: 86 = 19 + 67 lies outside the window, and L_3 = 1 at a composite l_3
\\ gb(4895789) = [1842, 124, 124, 1]

The other three-halves

The constant $3/2$ appears in both theories for unrelated reasons. In decompwlj it comes from $\ell >d⟺a(n)>2d(n)⟺a(n+1)<\frac{3}{2}a(n)$. In Collatz it is the odd step $T(x)=(3x+1)/2$. One exact statement connects them, and it is a curiosity.

Both record sequences start as listed in the OEIS (1, 2, 3, 6, 7, 9, 18, 25, 27, … and 1, 2, 3, 7, 15, 27, 255, 447, …). They are sparse, so Proposition B makes most of them level by geometry alone — the same behaviour as any sequence growing at a fixed ratio. Nothing Collatz-specific is visible.

A Collatz-shaped iteration that is easy. The map $p\mapsto \ell (p)=2p-\mathrm{nextprime}(p)$, continued while the value stays prime, is the framework's own dynamics. Below ${10}^6$ its chain lengths are 0 (72,544 primes), 1 (5,453), 2 (475) and 3 (24); the longest is $6397\to 6373\to 6367\to 6361$. Every chain terminates because the map strictly decreases. Collatz is hard because $T$ is not monotone; the decomposition's maps are, and that difference is why no technique transfers in either direction.

What this report adds

One canonical threshold ($\ell \le d^2$) through which gap conjectures act on the classification; the forced set $F$ closed to $4\cdot {10}^{18}$ and Conjecture F; Proposition R under RH; Propositions L1–L2 placing Legendre at the domain of the map; Proposition G and the Goldbach window $(N/2,\frac{3}{2}N)$; Proposition C. All are restatements, one-way implications or elementary facts — the summary table on page 1 is the ledger. None changes the status of Theorem 1, Theorem 2 or Problem 2.

Compute it yourself (PARI/GP 2.15.4, as run)

\\ decompwlj x conjectures, PARI/GP 2.15.4. decomp_fact = project kernel (treatise §9), reflowed.
decomp_fact(n, n1) = { my(d = n1-n, l = n-d, D); if (l <= d, return([0,0,d]));
  D = divisors(l); for (i = 1, #D, if (D[i] > d, return([D[i], l/D[i], d]))) };
\\ forced-level primes (l <= g^2): exhaustive below 1476^2+1476; beyond, OeS-H-P gap tables close F
forced(X) = { my(F = List()); forprime(p = 5, X, my(g = nextprime(p+1)-p, l = p-g);
  if (l > g && l <= g^2, listput(F, p))); Vec(F) };
\\ Legendre sequence q(m) = nextprime(m^2): [M, increasing?, decomposable, level, forced, L=1]
legendre(M) = { my(q0 = 2, q1, r, s = [M,1,0,0,0,0]); for (m = 1, M, q1 = nextprime((m+1)^2);
  if (q1 <= q0, s[2] = 0); r = decomp_fact(q0, q1);
  if (r[1], s[3]++; if (r[1] > r[2], s[4]++); if (q0-r[3] <= r[3]^2, s[5]++); if (r[2] == 1, s[6]++));
  q0 = q1); s };
print(forced(1476^2 + 1476));   \\ [5, 13, 19, 23, 31, 113]
print(legendre(10^5));          \\ [100000, 1, 99996, 99996, 99996, 7477]

Expected output in the comments. The longer check file run this session also reproduces the census below ${10}^6$ ([78495, 18353, 5953, 12], 0 disagreements between decomp_fact and decomp_fordiv) and the 40/40 Collatz ramp; the Python engine reproduces the census rows to ${10}^7$ (664,576 / 138,049 / 44,011 / 28), the forced set to ${10}^9$ and the Legendre census.

References

1. R. Eismann, founding paper, arXiv:0711.0865; treatise, 10th ed. (project document, 19 Aug 2026).
2. T. Oliveira e Silva, S. Herzog, S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4·10¹⁸, Math. Comp. 83 (2014). Maximal gaps also from the table of known maximal gaps.
3. A. Selberg, On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943); exposition in arXiv:1802.07609; see also Erdős problem 233.
4. H. Cramér, Some theorems concerning prime numbers, Ark. Mat. Astr. Fys. 15 (1920); On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936).
5. A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Integer Seq. 18 (2015), 15.11.2.
6. R. C. Baker, G. Harman, J. Pintz, The difference between consecutive primes II, Proc. LMS 83 (2001). J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad. 28 (1952).
7. OEIS A117078, A117563, A118534, A001223, A002386, A006562, A006877, A006884.