Decomposition into weight × level + jump
A deep analysis in number theory — significance, structure, proofs, censuses, and honest limits

Section 1Significance of the framework

The decomposition into weight × level + jump attaches to each term of a strictly increasing integer sequence a canonical triple: the jump \(d(n)=a(n{+}1)-a(n)\) is additive data — the local spacing of the sequence; the weight \(k(n)\), the smallest divisor of \(\ell(n)=a(n)-d(n)\) exceeding \(d(n)\), is multiplicative data — a divisor selected inside the reflected term \(\ell(n)\); and the level \(L(n)=\ell(n)/k(n)\) is the cofactor. The exact identity \(a(n)=k(n)L(n)+d(n)\) is the Euclidean division of \(a(n)\) by its weight, with the jump as remainder. Nothing is chosen: no parameter, no threshold, no normalization. The construction is canonical, and it is defined for the pair (term, successor) — the framework measures a sequence with the integers as a ruler, so its natural object is the pair \((a(n),a(n{+}1))\), never the term in isolation.

Three facts give the construction its significance. First, on the natural numbers it is the sieve of Eratosthenes, exactly (Section 3): the weight columns are the strike lists of the sieve, and the level-classified terms are the shifted primes. The construction therefore has a proved lineage — the sieve, not the fundamental theorem of arithmetic — and its uniqueness comes from a selection rule (“smallest”), which survives in monoids where unique factorization fails. Second, on the primes the coordinates make classical families visible as exact coordinate slices: the lesser twins are the single column \(k=3\) (a proved equivalence, not an analogy), balanced primes are the level-one generation-one stratum, and each fixed weight or level is an OEIS sequence. Third — and this is the part that did not exist before the coordinates — the classification into weight-classified (\(k\le L\)) and level-classified (\(k>L\)) primes poses a native question: does the level class thin out among the primes? That question (Conjecture 9 of the founding paper) is now settled unconditionally, with the quantitative bound of Theorem 1 below, by a proof whose every ingredient is visible in the coordinates: a cubic rigidity estimate, a reduction of the level-one stratum to the primality of \(2p_n-p_{n+1}\), and a three-dimensional Selberg sieve. Its density constant on the dominant stratum is identified conditionally as \(2C_2\), twice the twin-prime constant, through an unconditional singular-series average (Lemma 1 / Theorem 2).

The honest frame matters as much as the results, and this report keeps it explicit. Recasting a classical problem in these coordinates does not reduce its difficulty: the infinitude of the column \(k=3\) is the twin-prime conjecture; the infinitude of level (1;1) is the balanced-prime problem; nothing here bears on either, nor on the Riemann Hypothesis. The framework’s contribution is (i) the native rarefaction theorem and the sharp elementary structure around it, (ii) an identified constant with an unconditional identity behind it, and (iii) a working dictionary — developed further in Section 6 — between weight-column occupancy on the multiplicative side and Hardy–Littlewood local factors on the additive side. What it is not, the report says plainly, with the deflationary evidence attached (Sections 4.7, 6).

Section 2Definitions, existence, classification

PROVED Existence criterion. The decomposition exists iff \(\ell(n)>d(n)\), i.e. iff \(a(n{+}1)<\tfrac32 a(n)\) (equivalently \(2\,a(n{+}1)<3\,a(n)\), the form recorded by T. Ordowski on A117078). The weight is well defined because \(\ell\mid\ell\) with \(\ell>d\), and unique by minimality. In the Euclidean division of \(a(n)\) by \(k(n)\) the quotient is \(L(n)\) and the remainder is \(d(n)\): a remainder equal to \(d\) forces a modulus exceeding \(d\), which is exactly the constraint the minimality scans.

Worked rows (paper, Table 2; re-derived this session by all three engines). \(p=5,\,p'=7\): \(d=2,\ \ell=3,\ k=3,\ L=1\) — level-classified. \(p=11,\,p'=13\): \(\ell=9,\ k=3,\ L=3\) — weight (tie-free). \(p=13,\,p'=17\): \(d=4,\ \ell=9\); divisors of 9 exceeding 4: only 9, so \(k=9,\ L=1\) — level-classified with composite weight, the first of exactly five such primes (Section 4.3). \(p=23,\,p'=29\): \(\ell=17\) prime, \(k=17,\ L=1\).

A boundary witness worth keeping. For general sequences the level bound \(L\le d\) is sharp: the pair \(72\to76\) gives \(d=4,\ \ell=68,\ k=17,\ L=4=d\). For the primes, parity (\(\ell\) odd, \(g\) even) sharpens this to \(L\le g-1\) — one of the small rigidities the whole Section-4 architecture rests on. The self-test suite (Appendix A) checks this witness, the seventeen rows of Table 2, and the five composite-weight primes on every run; it passed on all engines today.

Two equivalent mod-form definitions (as on the OEIS wiki): \(k(n)\) is the smallest \(k\) with \(a(n{+}1)=a(n)+(a(n)\bmod k)\), and \(\ell(n)\) is the largest such modulus. These are the forms under which the sequences entered the OEIS in 2006 and were community-checked (entry A117078: edited and corrected by D. Reble and K. Brockhaus, April 2006).

Section 3Relation with the fundamental theorem of arithmetic and the sieve of Eratosthenes

PROVED Eratosthenes collapse. Take \(a(n)=n\). Then \(d\equiv1\) and \(\ell(n)=n-1\) for \(n\ge3\); the smallest divisor of \(\ell\) exceeding 1 is its smallest prime factor, so \[ k(n)=\operatorname{spf}(n-1)=\text{A020639}(n{-}1),\qquad L(n)=\text{largest proper divisor}=\text{A032742}(n{-}1), \] and \(n\) is level-classified iff \(n-1\) is prime, in which case \(L=1\) and \(k=\ell\). The weight columns are therefore precisely the strike lists of the sieve of Eratosthenes — the column \(k=2\) holds the numbers whose \(\ell\) is even, the column \(k=3\) those with \(\ell\) divisible by 3 and not 2, and so on — while the level-one line collects exactly the (shifted) primes the sieve leaves standing. The construction on \(\mathbb N\) does not resemble the sieve; it restates it. (Verified this session on all \(n\le 2000\) in PARI and re-drawn to \(n\le30\,000\) in Figure N2.)

Where the FTA enters — and where it does not. Since the jump is constant on \(\mathbb N\), the decomposition reduces to \(\ell=k\times L\) with \(k=\operatorname{spf}(\ell)\); iterating the construction on the level peels off smallest prime factors one at a time and reconstructs the full factorization — the fundamental theorem of arithmetic recovered as a fixed point of the decomposition, which is how the OEIS wiki presents it. But the logical dependence runs the other way and is weaker: the definition needs only the divisor lattice and a selection (“smallest divisor beyond the jump”). Selection uniqueness is not rigidity uniqueness. In the Hilbert monoid \(H=\{1,5,9,13,\dots\}\), where unique factorization fails (\(441=9\cdot49=21^2\)), the decomposition remains well defined and unique, because “smallest” still selects. The proved lineage of the construction is the sieve, not the FTA — a distinction that matters for the generalization program of Section 7, where the FTA is unavailable but sieves survive.

The picture that carries the theorem. In \((\log k,\log L)\) the naturals form an arrowhead (project figures sieveNb.jpg, naturaldecomp3M2048.jpg; new Figure N2): the weight sheet fills the wedge \(k\le\sqrt{\ell}\le L\) with its left wall at \(k=2\), the tip pinches on the diagonal at \((\tfrac12\log \ell_{\max},\tfrac12\log \ell_{\max})\) — the origin of the “C/2” annotation in the project schematic, half the logarithmic width of a cutoff at \(C\) — and the primes appear as the level-one baseline, growing thinner relative to the sheet exactly at the prime-number-theorem rate. The rarefaction of the level class of \(\mathbb N\) is the PNT read off a picture; the rarefaction of the level class of the primes is its one-floor-up analogue, and that is Theorem 1.

Section 4Application to the primes: proofs, conjectures, and the verified record

Throughout, \(p=p_n\) is a prime with gap \(g=g_n=p_{n+1}-p_n\) and \(\ell=\ell(n)=p-g=2p_n-p_{n+1}\) (A118534). The weight, level, jump sequences are A117078, A117563, A001223.

4.1 Existence and elementary rigidity

PROVED Non-decomposable primes. \(p_n\) is decomposable iff \(p_{n+1}<\tfrac32 p_n\); the failures are exactly \(2,3,7\). Beyond a finite check this is Nagura’s interval theorem (a prime in \((m,\tfrac65m]\) for \(m\ge25\)); the founding paper’s original argument via Dusart’s bounds on \(\pi\) gives the same. Consequently the decomposable count in every decade equals \(\pi(x)-3\); the census reproduces this identically at every mark (Table 4.1).

PROVED Direct relations (all decomposable \(p>3\)): \(g\) even, \(\ell\) odd; \(\gcd(\ell,g)=\gcd(k,g)=\gcd(L,g)=1\); \(3\le k\le\ell\); \(1\le L\le\ell/3\); \(2\le g\le k-1\); \(2g+1\le p\). For level-classified \(p\): no divisor of \(\ell\) in \((g,k)\), hence every divisor below \(k\) is \(\le g\); \(L\le g-1\) (parity-sharpened); \(k>\sqrt\ell\). For weight-classified \(p\): \(g+1\le k\le\sqrt\ell\le L\le \ell/3\) — whence, as the wiki notes, weight-classified primes satisfy the Legendre- and Andrica-type spacing bounds by definition; the content of those conjectures lives entirely in the level class.

4.2 The level-one stratum: reduction and exception sets

New this session — the closure frontier moves to \(2^{64}\). The exhaustive analysis of prime gaps has been completed to \(2^{64}\approx1.8447\cdot10^{19}\) by the Prime Gap Searches project (final push by LaGrou, Jacobsen, Smith and Gerbicz, per T. Nicely’s first-occurrence tables, mirrored at OEIS A000101); VERIFIED the maximal gap below \(2^{64}\) is 1550, following 18 361 375 334 787 046 697. Re-running the cap argument with 1550 bounds any new exception by \(p\le1550^2+1550=2\,404\,050\), still deep inside the swept range: the exception sets of Lemma 5(ii)–(iii) and of the Cube Lemma below are closed for all \(p\le 2^{64}\). The refereed anchor (Oliveira e Silva–Herzog–Pardi, Math. Comp. 2014) remains \(4\cdot10^{18}\); we recommend the manuscript keep the refereed bound in the statements and add the \(2^{64}\) extension as a remark with the distributed-computing citation. (Corrections log, item 16.)

Why only \(6\mid g\) matters at level one. By mod-3 rigidity, a level-one prime with \(6\nmid g\) has \(3\mid\ell\), so \(\ell\in\{3,9\}\) by (i)+(iii), i.e. \(p\in\{5,13\}\). The census confirms the count exactly: of the prior census’s 22 083 608 level-one primes to \(10^{10}\), all but the two primes 5 and 13 have \(6\mid g\) (22 083 606). This is the structural reason the singular series of Lemma 1 lives on multiples of 6.

4.3 The Cube Lemma and the normal form

PROVED Normal form. Outside the finite exceptional structure, a level-classified prime with \(g\le 2M\) and \(\ell>(g-1)^3\) has prime weight: \(\ell=L\cdot P\), \(P=k>\sqrt\ell\) prime, \(L\le g-1\) odd; and the count of level-classified \(p\le x\) inside the degenerate box \(\ell\le(g-1)^3,\ g\le2M\) is \(O(M^4)\). This turns Conjecture 4 of the founding paper into a theorem with an explicitly closed five-element exception set — its historical status as “conjecture” is retired below in the ledger. It is also the reason the original quartic-cap closure argument of an earlier report edition (log item 8) could be repaired: the cubic bound is what the record tables actually certify.

4.4 Theorem 1 — rarefaction of the level class (Conjecture 9 settled)

Architecture, in the coordinates. Set \(M=\lceil(\log x)^{3/2}\rceil\) and split \(N_{\mathrm{lev}}=E_0+E_1+S\): the gap tail \(E_0\) (\(g>2M\)) is \(\le x/M\) by the telescoping bound \(\sum_{p_n\le x}g_n\le 2x\); the degenerate box \(E_1\) is \(O(M^4)=O((\log x)^6)\) by the normal form; and the main term \(S\) counts, injectively, triples \((L,m,P)\) with \(g=2m\le2M\), \(L\le g-1\) odd, and \[ P,\quad LP+2m,\quad LP+4m \quad\text{all prime}. \] The decisive move is that the upper bound may discard consecutiveness: the level event implies the triple event, so the coupling \(\ell=2p_n-p_{n+1}\) that ties the gap to the divisor target — the precise obstruction to any unconditional asymptotic — is legitimately bypassed in this one direction. Selberg’s sieve in dimension 3 (Halberstam–Richert, Thm 5.7), uniform over the \(O((\log x)^3)\) pairs \((L,m)\), gives \(T\ll \mathfrak S\, y/(\log y)^3\) with \(\mathfrak S\ll h(m)h(L)\), \(h(n)=\prod_{q\mid n}(1+3/q)\); the two mean values contribute \(M\) and \(\log\log x\), and the choice of \(M\) balances \(S\) against \(E_0\).

What the exponent means — and does not. The level-one stratum alone yields \(N_1(x)\ll x/(\log x)^{3/2}\); the extra \(\log\log\) is the price of the \(L\)-sum. The conjectured truth is \(N_{\mathrm{lev}}(x)\asymp x\log\log x/\log x\) (Species-II dominated; Section 4.7), so \(3/2\) is an artifact of running a dimension-3 sieve on the prime-weight normal form. Closing the level-one exponent to the conditional truth \(2\) unconditionally is Open Problem 1. Shape check (this session): the Theorem-1 envelope statistic \(f(x)\sqrt{\log x}/\log\log x\) reads 0.331, 0.300, 0.276, 0.257, 0.241 at \(x=10^6,\dots,10^{10}\) — bounded and decreasing, exactly as an over-generous upper bound should behave against a smaller truth.

4.5 Theorem 2 and Lemma 1 — the level-one constant \(2C_2\)

The gap-by-gap test, isolating H1 (re-run in full this session at \(10^9\)). Feeding the observed per-gap mass \(\sum_{\text{gap }g}1/\log p\) through the ratio \(\mathfrak S_3/\mathfrak S_2\) predicts the level-one count at each gap with no gap model at all. Today’s complete recount at \(10^9\) reproduces the manuscript’s comparison column to every printed digit:

Why finite heights cannot see \(2C_2\) — the window ladder (recomputed today). The windowed estimator \(\hat c_1=\dfrac{\Delta N_1}{\Delta(\text{decomposable})}\cdot\log\sqrt{ab}\) over decade windows climbs 0.8431, 0.9230, 0.9324, 0.9719, 1.0024, 1.0280 for \((10^3,10^4]\) through \((10^8,10^9]\) — the last two matching the manuscript’s ladder exactly — and the prior-census endpoint gives 1.0486 aggregated over \((10^9,10^{10}]\), consistent with the manuscript’s finer intra-decade ladder rising 1.0561 → 1.0680 there. At \(\lambda=\log x\approx23\) the weight \(e^{-g/\lambda}\) has barely begun to sample the large \(6\mid g\) gaps whose \((q-1)/(q-3)\) enhancements lift the line average to its limit; the deficit from \(2C_2=1.3203\ldots\) is quantitatively the finite-height gap mix, converging at rate \(1/\log x\). This is precisely why extrapolation could never have identified \(c_1\), and an identity was required. (Figure N4.)

4.6 The census and the verification pipeline

The four-layer verification, executed today. (1) C engine: segmented sieve streaming consecutive primes; deterministic Miller–Rabin bases {2,3,5,7} (valid below 3.2·10⁹, comfortably above every \(\ell<10^9\)); full factorization with divisor scan for \(k\). (2) PARI/GP 2.15.4: the project’s fordiv kernel re-counted all of \(10^7\) from scratch — decomp=664 576, N₁=44 011, N_lev=138 049, zero mismatch — and the project self-test suite passed (Table-2 rows, the five C4 primes, naturals to 2000, the \(L=d\) witness); BPSW (ispseudoprime) agreed with isprime on all 78 495 values \(\ell\le10^6\). (3) Python: an independently coded sieve + divisor-scan re-derived every one of the 78 498 rows to \(10^6\) against the C output — zero mismatches. (4) OEIS: the full 10 000-term b-file of A117078 was pulled live and a 55-term stratified sample (endpoints, decades, and the large level-classified weights, e.g. \(k(9999)=104\,717\)) diffed programmatically — zero mismatches. External anchors reproduced today: the complete first-occurrence maximal-gap ladder to \(10^9\) (A002386: 1@2, 2@3, 4@7, 6@23, 8@89, 14@113, …, 250@387 096 133, 282@436 273 009) and \(\pi(x)\) at every decade; prior-census anchors (e.g. \(p_{5\cdot10^7}=982\,451\,653\), \(\pi_2(10^{10})=27\,412\,679\), ladder to 354@4 302 407 359) are cited, not re-run.

4.7 Species II: what is measured, what is open, what was withdrawn

The level class splits into two analytic species. Species I (\(L=1\), \(k=\ell\)) is Hardy–Littlewood territory: about 31% of the level class, governed by Theorem 2 with constant \(2C_2\). Every Species-I prime supplies a Goldbach witness for \(2p_n\), since \(2p_n=\ell+p_{n+1}\) with both parts prime — the implication runs one way only and yields nothing toward Goldbach, but as a census invariant it held with zero failures. Species II (\(L>1\): \(\ell=LP\), small odd \(L\), large prime \(P\)) is divisors-in-intervals territory — Ford’s \(H(x,y,z)\), Koukoulopoulos’ divisors of shifted primes — about 69% of the class, and its constant is genuinely unidentified:

• VERIFIED Windowed density: \(\hat c_2=\dfrac{\Delta N_{>1}}{\Delta(\text{decomposable})}\cdot\dfrac{\log\sqrt{ab}}{\log\log\sqrt{ab}}\) reads 0.7690, 0.7705, 0.7669, 0.7656 over the four decade windows \((10^5,10^6]\ldots(10^8,10^9]\) today, and 0.7648 on \((10^9,10^{10}]\) via the prior-census endpoint — squarely inside the documented band \(0.773\pm0.01\) with its window convention.
• VERIFIED Cumulative diagnostic: \(f_{>1}(x)\,\log x/\log\log x\) = 0.8312, 0.8204, 0.8108, 0.8046, 0.7999 at \(x=10^6,\dots,10^{10}\) — monotonically falling, the sequence that decided the retraction below.
• RETRACTED The interim claim \(c_2=1\) (log item 11) remains withdrawn: an estimator receding from the claimed value at every scale is not converging to it. The constant \(c_2\) is OPEN, band \(0.77\pm0.01\) windowed.
• OPEN Second manuscript flag from this session (log item 15). The manuscript’s Problem 3 also cites a “Mertens-normalized ratio \(R_2\approx0.4412\), flat to three decimals over \(10^6\)–\(10^{10}\)”, mass \(\approx 2\log g_n/\log\ell_n\) per decomposable prime. Today’s recount under that stated convention gives cumulative \(R_2=\) 0.4470, 0.4460, 0.4435, 0.4422 and windowed 0.4458, 0.4432, 0.4420 at \(10^6\!-\!10^9\): drifting in the third decimal, plausibly toward ≈0.441 but not flat over the stated range; the exact-product convention \(\prod_{g<q\le\sqrt\ell}(1-1/q)\) gives 0.41–0.40 and drifts more. Either the manuscript’s convention differs from both implementations or the flatness claim needs restriction to the top windows. Action: re-derive \(R_2\)’s exact pipeline convention before submission and reword; the identification question of Problem 3 is unaffected.

4.8 The congruence lock, and the consecutiveness bias as a measurable object

PROVED (trivially) For an odd prime \(r\nmid g\): \(r\mid\ell \iff p\equiv g \pmod r\). So weight-column occupancy at \(r\) is a pure residue event on \(p\) given the gap — the mechanism, verified across the whole atlas in earlier editions, by which the multiplicative side (which columns are occupied) reads off the additive side’s local data. For \(r>g\) the pair \((p,p+g)\) excludes two classes mod \(r\), leaving \(r-2\); equidistribution then predicts

\(\Pr[\,r\mid\ell \mid \text{gap }g\,]=\dfrac1{r-2}\)   HEURISTIC

Mechanism, and an honest failure of the first model. The deficit has a clean qualitative cause: among consecutive pairs of gap \(g\), residue classes \(p\equiv -j\pmod r\) for interior even offsets \(j\in\{2,\dots,g-2\}\) are enhanced, because \(r\mid p+j\) makes that interior point composite for free; the class \(p\equiv g\) (the one giving \(r\mid\ell\)) carries no such credit and is relatively depleted. This is the same interior-compositeness mechanism as the +1.2% aggregate excess of §4.5, now resolved by residue class. A one-parameter model — weight \(A\) per interior hit, so \(\Pr=1/\bigl((r-2-h)+hA\bigr)\) with \(h\) the number of hit classes — fits the large-\(g\) rows with \(A\approx1.08\!-\!1.11\) (e.g. \(r=7,g=6\): \(A=1.082\); \(r=11,g=10\): \(1.085\); \(r=13,g=12\): \(1.107\)) but fails at \(g=4\), where the single interior point yields \(A\approx1.00\) and no measurable bias. The exact local-factor law behind these numbers — presumably a Hardy–Littlewood computation over the interior pattern, in the spirit of Montgomery–Soundararajan — is a well-posed OPEN micro-problem this report contributes; deriving it would also predict the per-gap signature of §4.5 (deficit at \(g=36\), excesses at 12, 24, 30, 42) rather than merely measuring it.

4.9 The conjecture ledger of the founding paper

Two further statements deserve ledger lines. The comment-conjecture on A117078 — that 2, 3, 7 are the only primes with \(\log p<\sqrt g\), i.e. \(g>(\log p)^2\) — is Cramér-adjacent: OPEN in general, consistent with all exhaustive data to \(2^{64}\) (the record merit 34.94 there sits far below the \(\log p\approx44\) needed for a violation at that height). And the infinitude of the level class itself — compatible with Theorem 1, containing the balanced-prime problem — is the framework’s remaining native open question (Problem 2).

Section 5Analysis of the graphs

This section reads the project’s own graphics first — they are the historical record of how the structure was found — and then presents this session’s seven newly computed figures. All new figures are drawn on square canvases; the two \((\log k,\log L)\) portraits additionally use equal axis scaling, so that one decade of weight has the same length as one decade of level and the geometry of the two sheets is undistorted, as it must be for the diagonal \(k=L\) to mean what it says.

5.1 The provided project graphics, read closely

(a) Log_LLog_k_1500000R.jpg — the two sheets at 1.5 million primes. The canonical portrait. The dense upper-left wedge is the weight class: vertical striations are the discrete weight columns \(k=3,5,7,9,\dots\) (odd, coprime to the gap), its left wall the twin column \(k=3\), its roof the constraint \(L\le\ell/3\), its right frontier the diagonal, and its tip the pinch \(k\le\sqrt\ell\le L\) at \((\tfrac12\log\ell_{\max},\tfrac12\log\ell_{\max})\). The sparse lower-right wedge is the level class: horizontal striations are the level lines \(L=1,3,5,\dots\), its floor the level-one line running out to \(k=\ell\), its ceiling the parity-sharpened bound \(L\le g-1\) — which is why the level sheet stays low and flat while the weight sheet grows tall. The 83/17 density asymmetry between the sheets is Conjecture 9 made visible; that the lower sheet visibly thins rightward is Theorem 1 in pixels.

(b) sieveNb.jpg — the sieve schematic. The pedagogical key to Section 3. The three ellipsed columns are labelled exactly as strike lists: weight 2 = multiples of 2; weight 3 = multiples of 3 not 2; weight 5 = multiples of 5 not 2, 3 — the sieve of Eratosthenes column by column — while the long bottom ellipse collects the primes on the level-one line. The two ticks “C/2” and “C” encode the geometry precisely: with a cutoff at \(C\), the level-one line reaches \(\log(C-1)\approx\log C\) while the wedge tip sits at \(\tfrac12\log C\) — half the logarithmic width, the same pinch as in (a). One schematic, and the claim “this construction restates the sieve” needs no further argument.

(c) naturaldecomp3M2048.jpg — naturals to 3 000 000. The arrowhead at scale, axes in natural log (the tip at ≈7.45 matches \(\tfrac12\ln(3\cdot10^6)=7.46\)). Two features reward attention: the interior lattice mesh — horizontal and vertical alignments inside the wedge, which are the images of arithmetic progressions in \(\ell\) under the spf map — and the near-invisibility of the level-one baseline at this resolution. The second is the point: the primes thin at rate \(1/\log\), so the level class of \(\mathbb N\) rarefies against its weight class exactly as the PNT dictates. The picture is the base case of the rarefaction phenomenon whose prime-sequence analogue is Theorem 1.

(d) classification_primes.jpg — the annotated atlas map. Every labelled substructure is an OEIS object, and the annotations are exact: column \(k=3\) is A001359∖{3} (Lemma 4), \(k=5\) is A074822, \(k=7\) is A118741 = A119593 ∪ A118359 (split by gap), \(k=9\) is A119594 ∪ A118922; the diagonal is A121155 (\(k=L\), the square-\(\ell\) ties, weight by convention); level rows \(L=1,3,5\) are A125830, A117873, A117874, with the level-one row’s own decomposition into {13, 31}, the balanced primes A006562, and the generations A117876, A118467, …; the two sheets wholesale are A162175 / A162174. The figure documents the framework’s most concrete service: classical families become coordinate slices with stable identifiers, each independently maintained and checkable on the OEIS.

(e) lesser_3D.png and (f) Screenshot… three.js — the interactive atlas. The 3D embeddings (index or size along the depth axis, \(\log k\) and \(\log L\) transverse) show what the 2D projections compress: each weight column and level line is a full ray extruded along the sequence. In (e) the highlighted ray is the entire twin column — a classical family occupying exactly one structural fiber, the geometric content of Lemma 4. In (f), from decompwlj.com/3Dgraph, the two populations separate cleanly in depth as well: the boundary \(k\gtrless\sqrt\ell\) is a metric gap, not a fuzzy margin, which is why classification is stable under the eye long before it is counted.

(g) 3D_natural_numbers_2.JPG. The base case in 3D: the level-classified naturals (shifted primes) form the tight ray along the sequence axis; the weight-classified fan spreads into the \(k\)–\(L\) plane. The two-species geometry is present from the very first sequence the construction ever touched.

(h) A117078_20210214.png — the OEIS entry. Worth reading as provenance: definition in mod form; Wilson’s Mathematica implementation via Min@Select[Divisors[2p−np], #>np−p&] — independently equivalent to the divisor definition; Ordowski’s 2013 decomposability criterion \(2p_{n+1}<3p_n\); the 2006 edit-and-correction record. The comment-conjecture (“2, 3, 7 are the only primes with \(\log p<\sqrt g\)”) is Cramér-adjacent and remains OPEN — see the ledger note in §4.9.

(i) decompwlj_1_20231216.jpg — the decompsieve card. The published two-phase algorithm: ascending weight scan to \(\sqrt\ell\), then a descending level scan. The subtle point, documented in decompwlj_optimized.txt and preserved in this report’s Appendix A: the test n%floor(l/le)==d silently enforces \(\ell/le>d\) (a remainder \(d\) needs a modulus exceeding \(d\)), which is what protects the empty-window primes 13, 31, 113 — their divisor pairs lying entirely below the gap must be skipped. A naïve “largest divisor ≤ d” rewrite without the cofactor condition is wrong on exactly those three primes; the regression suite pins this.

(j) 150decompanimAnum.gif — the atlas in motion. Frame A000027 shown; across 150 sequences the arrowhead morphology recurs with sequence-specific tip position, wall spacing, and level-sheet thickness. The animation is the strongest argument that the atlas needs a quantitative morphology — the invariant vector proposed in Section 7 — rather than a catalogue of resemblances.

5.2 This session’s figures

Figure N1 — Primes, first 1 500 000, equal-aspect \((\log_{10}k,\log_{10}L)\). Recomputed from today’s engine (file kl_1500000.csv, 1 499 997 decomposable rows). Indigo: weight class; copper: level class; dashed rose: \(k=L\) (A121155). Matches the project reference Log_LLog_k_1500000R.jpg structure exactly; the level-one baseline is lifted just clear of the frame so its full extent to \(\log_{10}k\approx6.9\) is visible.

Figure N2 — Natural numbers to 30 000: the sieve, equal aspect. \(k=\operatorname{spf}(n-1)\) columns at \(\log_{10}2,\log_{10}3,\log_{10}5,\dots\); copper level-one line = the shifted primes. The base case of Section 3, drawn from an independent spf sieve.

Figure N3 — Rarefaction: \(f(x)\) per decade. Filled points: today’s complete counts to \(10^9\); hollow: prior census at \(10^{10}\). Teal: the conjectured order \(c\,\log\log x/\log x\) with \(c=1.163\) fitted at the top marks (the constant is descriptive, not identified). Dashed rose: the Theorem-1 envelope shape \(\propto\log\log x/\sqrt{\log x}\), pinned at \(10^6\) — an upper-bound silhouette, deliberately not a fit. \(f\) falls in every decade with no exception; Conjecture 9 is settled, and the plot shows how much daylight remains between theorem and truth (Open Problem 1).

Figure N4 — The level-one constant’s ladder. Windowed \(\hat c_1\) climbs 0.8431 → 1.0280 across six decade windows (today), with the \((10^9,10^{10}]\) aggregate 1.0486 from the prior-census endpoint; the copper line is the unconditional limit \(2C_2\) fixed by Lemma 1. Convergence of order \(1/\log x\): at accessible heights the exponential gap weight has hardly begun to sample the large \(6\mid g\) gaps whose \((q-1)/(q-3)\) factors carry the average to its limit. The figure is the visual argument for why an identity, not extrapolation, identifies \(c_1\).

Figure N5 — Gap-by-gap obs/pred at \(10^9\), isolating H1. Marker area ∝ observed count; grey band = the manuscript’s stated [0.97, 1.05]; teal line = today’s aggregate 1.0121 (matching the manuscript digit-for-digit). Rose labels: the seven small-count gaps found outside the band at \(10^9\) — ordinary Poisson noise, but a falsifier of the prose claim as written; see the erratum callout in §4.5 and corrections log item 14.

Figure N6 — Congruence lock and the consecutiveness bias. Measured \(\Pr[r\mid\ell\mid g]\times(r-2)\) at \(10^9\). Flat at 1.0000 to ~10⁻⁴ for \(g\ll r\) (with \(r=3,g=2\) exactly 1: that row is mod-3 rigidity); systematic dip as \(g\to r\), driven by interior-compositeness credit to competing residue classes. The exact local law is an open micro-problem (§4.8).

Figure N7 — The level sheet in \((g,\ell)\), all level-classified \(p\le10^7\) (138 049 points). Copper: the Cube-Lemma frontier \(\ell=(g-1)^3\), touched only at \(p=131\) (star). Teal dashed: \(\ell=g^2\); below it live exactly the six primes of Lemma 5(ii) — 13, 31 (stars, composite weight and composite \(\ell\)), 113 (star), and 5, 19, 23 (open circles). Everything else sits above both curves in the prime-weight normal form. The five stars are the entire composite-weight exception set, closed to \(4\cdot10^{18}\) and now to \(2^{64}\).

Section 6Hypothetical impacts: bringing multiplicative and additive number theory together

The claim to examine is that the decomposition is an additive–multiplicative bridge. This section states, with tags, what the bridge demonstrably carries, what it plausibly carries, and what it demonstrably does not.

6.1 What the bridge demonstrably carries

(i) A dictionary entry, proved and quantitative. The congruence lock \(r\mid\ell\iff p\equiv g\ (\mathrm{mod}\ r)\) converts weight-column occupancy — a statement about the divisor lattice of \(\ell\), the multiplicative side — into residue statistics of primes conditioned on their gap — the additive side’s local data, exactly the input of Hardy–Littlewood singular series. The first-order value \(1/(r-2)\) for \(r>g\) is the HL local factor’s footprint, verified today to ~10⁻⁴; the second-order consecutiveness bias (§4.8) is the same object at the next order, measured and awaiting its exact local law. This is why \(c_1=2C_2\): the level-one constant is a line average of triple singular series (Lemma 1), i.e. the dictionary applied and then averaged — an unconditional identity, not an analogy.

(ii) A native theorem whose proof crosses the bridge in one direction. Theorem 1 takes a multiplicative hypothesis on \(\ell\) (no divisor in \((g,\sqrt\ell\,]\)), rewrites it through the normal form as a simultaneous-primality event on three linear forms — additive combinatorics of the purest kind — and then discards consecutiveness to let a sieve through. The one-way traffic is the honest lesson: the coupling \(\ell=2p_n-p_{n+1}\), which ties the interval lower bound (gap) to the divisor target (shifted value), blocks the return trip; it is the precise obstruction to unconditional asymptotics in this framework, named as such and respected.

(iii) Independent confirmation that the frame is standard-compatible. Gafni–Tao (arXiv:2508.06463, Aug 2025) prove, for a cognate consecutive-gap event — gaps containing no \(g\)-rough number — an unconditional \(O(X/\log^2X)\) bound and a conditional asymptotic \(cX/\log^2X\) with \(c\) computed from Montgomery–Soundararajan singular-series averages (they estimate \(c\in(2.7,2.8)\)). The objects differ — they sieve inside the gap \((p_n,p_{n+1})\); the decomposition interrogates the reflected point \(p_n-g\) — but the analytic shape is the same: consecutive-gap event ⇒ density \(x/\log^2x\) ⇒ constant as an average of singular series. Theorem 2’s architecture, written independently, is a sibling of a 2025 Annals-track methodology; that is meaningful external calibration for the framework, and no more than that.

6.2 What it plausibly carries — the working program

OPEN Species II places the level class inside the theory of divisors in intervals: level-classified means \(\ell\) has no divisor in \((g,\sqrt\ell\,]\), i.e. \(H(x,y,z)\)-type conditions (Ford 2008) on the shifted prime \(\ell=p-g\) (Koukoulopoulos 2010). Whether Ford’s uniform estimates already pin the Species-II order — perhaps even the constant band \(0.77\pm0.01\) — is the focused literature question this program should answer next (Section 7). If yes, the bridge imports a mature multiplicative theory wholesale into a gap-conditioned setting; if no, the obstruction will again be the consecutive coupling, and identifying where Ford’s uniformity fails against it would itself be informative.

6.3 What it demonstrably does not carry — the deflationary record

Three findings, kept prominent because the framework’s credibility depends on them. First, reformulation is not progress: every fixed-column or fixed-level infinitude is a classical open problem verbatim (ledger, §4.9), and nothing here touches them or RH. Second, coordinate changes cannot lower difficulty: the level-one asymptotic needs HL-with-uniformity exactly as its classical formulations do; the coordinates re-pose, they do not re-price. Third, the Cramér deflation, established in the sixth-edition cycle and standing: a Cramér-random sequence reproduces the prime level-rate’s decay shape to within 0.8% in the top decade. The decay shape is generic to sequences of prime-like density; what is prime-specific is the arithmetic fine structure — the singular-series constants, the congruence lock, the mod-3 rigidity, the exact exception sets. Any impact claim must rest on the second list, and this report’s new measurements (the bias law of §4.8 above all) are chosen to enlarge exactly that list.

Section 7Detailed conclusion and future research

7.1 What this edition establishes

The seventh edition’s contribution is a full independent re-derivation and two pre-submission catches. Every headline number of the framework was recomputed from scratch today — the complete classification to \(10^9\) in 85 seconds on a fresh C engine, cross-checked by the project’s own PARI/GP kernel at \(10^7\) and a third-language pointwise audit at \(10^6\), with zero mismatches anywhere, and anchored externally by a live 55-term diff against the OEIS b-file of A117078 and the exact reproduction of the A002386 record ladder. The decade table matches the manuscript digit-for-digit; the gap-by-gap Theorem-2 test returns the manuscript’s 1.0121 aggregate and all seven printed per-gap ratios to four decimals; the twin column, mod-3 rigidity, exception sets, and window ladders all land exactly. The framework’s verified record is, as of today, re-verified.

The two catches, both logged and both fixable in minutes: the manuscript’s per-gap [0.97, 1.05] claim fails at \(10^9\) for seven small-count gaps (item 14 — reword, or restrict to \(g\le78\) at that height), and its \(R_2\)-flatness sentence does not survive either natural convention below \(10^9\) (item 15 — re-derive the pipeline’s exact convention and restrict the claim). One strict upgrade was also found: the exhaustive gap frontier now stands at \(2^{64}\) (maximal gap 1550), so the exception-set closures of Lemma 5 and the Cube Lemma extend beyond \(4\cdot10^{18}\) with a one-remark addition (item 16). None of the three touches a theorem.

7.2 The submission path (highest priority, unchanged)

The manuscript — “Rarefaction of level-classified primes …, with a conditional asymptotic for the level-one stratum”, targeting the Journal of Integer Sequences or Integers — is one administrative pass from submission: author email and affiliation lines (still TODO in the July compile), the two missing bibliography page numbers, plus now the item-14 and item-15 rewordings and, optionally, the \(2^{64}\) remark. Submission is what removes the asterisk from Theorem 1. The queued upstream errata remain queued: the challenge paper’s Table-1 lev1(10⁸) off-by-one (log item 6) and its Lemma-8 exception set, which should read the six-element \(\{5,13,19,23,31,113\}\) for the \(g\ge\sqrt\ell\) statement (with the level-one composite-\(\ell\) subset \(\{13,31\}\)), propagated also to the project website.

7.3 Open problems, ordered by tractability

• Problem A (manuscript Problem 1). Unconditionally \(N_1(x)\ll x/(\log x)^2\), closing 3/2 → 2 on the level-one stratum. Candidate route: replace the dimension-3 Selberg bound by a Montgomery–Soundararajan-style treatment of the triple with the gap variable averaged first — precisely the technology Gafni–Tao deploy; their Section-4 machinery is the natural first thing to try to transplant. OPEN
• Problem B. Derive the exact local law of the consecutiveness bias (§4.8): the residue-class weights among consecutive pairs of gap \(g\) modulo \(r\), reproducing the measured \(A\approx1.08\!-\!1.11\) for multi-point interiors and its collapse at \(g=4\), and through it the per-gap signature of §4.5. Self-contained, data-rich, and the cleanest new bridge entry on offer. OPEN
• Problem C (manuscript Problem 3). Identify \(c_2\) or \(R_2\) — after item 15’s convention repair — plausibly via Ford’s \(H(x,y,z)\) uniform estimates or Koukoulopoulos’ shifted-prime divisor theory; the focused question “does Ford 2008 already imply the Species-II order?” deserves a dedicated literature pass before any new computation. OPEN
• Problem D (manuscript Problem 2). Is the level class infinite? Rarefaction is compatible with finiteness; infinitude contains the balanced-prime problem. The framework’s last native open question. OPEN

7.4 The generalization line

The structural audit of earlier editions stands: four proved obstructions (characteristic-two collapse; the necessity of a monicity convention for units; the dimension obstruction in \(\mathbb Z[i]\), level rate 0.9951; no subtraction in free monoids), and two live targets. \(\mathbb F_q[t]\), \(q\) odd, remains the single most promising direction: gaps, divisors, and sieves all have clean function-field analogues, and the analogue of Conjecture 9 may be provable unconditionally with the true exponent where the integer statement needed Theorem 1’s compromise — a genuine test of whether the consecutive coupling is arithmetic or merely archimedean. Arithmetical congruence monoids \(M_{1,m}\) admit the construction with zero violations over 2.5M elements and a density exponent matching the Davenport constant of the class group — selection uniqueness at work where the FTA fails, per Section 3. Both belong on the post-submission agenda in that order.

7.5 The atlas program

The 1000-sequence atlas (decompwlj.com; full database, CSV and image dumps downloadable) is descriptive; the taxonomy discipline adopted in the sixth cycle should now be executed: attach to each sequence the invariant vector — decomposability fraction, level-share decay exponent and its Cramér-null comparison, occupied-column spectrum, congruence-lock pass/fail per small prime — and cluster. The congruence-lock screen is the uniform null model: any sequence whose column occupancy deviates from the residue prediction is flagged as carrying arithmetic beyond its density, and only D-class sequences with identified rates or constants warrant new conjectures. The 150-sequence animation (§5.1j) is the qualitative version of this program; the vector makes it falsifiable.

Appendix APARI/GP algorithms

The reference kernel (manuscript Appendix A), used verbatim in today’s \(10^7\) recount:

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

The optimized trial-division kernel from decompwlj_optimized.gp — parity wheel on odd \(\ell\), sqrtint, and the descending level branch whose cofactor condition \(\ell/le>d\) protects the empty-window primes 13, 31, 113:

decomp_td(n, n1) = {
  my(d, l, s, k0);
  if (n >= n1, error("need n1 > n"));
  d = n1 - n; l = n - d;
  if (l <= d, return([0, 0, d]));
  s = sqrtint(l);
  k0 = if (l % 2, d + 1 + !((d+1) % 2), d + 1);      \\ first admissible k
  forstep (k = k0, s, if (l % 2, 2, 1),
    if (l % k == 0, return([k, l/k, d])));
  forstep (le = if (l % 2, d - !(d % 2), d), 1, -if (l % 2, 2, 1),
    if (l % le == 0 && l/le > d, return([l/le, le, d])));
}

Dispatch and batch, per the project file: decomp_fact (via divisors()) changes the complexity class — trial division is \(\Theta(\sqrt\ell)\) on every level-classified term, while factorization is near-instant to \(\ell\sim10^{12}\) and beyond; decomp_auto switches at \(10^8\). Today’s runs of the project suite: selftest() — Table 2 (17 rows), the five C4 primes with weights 9, 25, 33, 25, 51, naturals to 2000 against spf, the \(L=d\) witness \([17,4,4]\) — all passed; the full \(10^7\) recount and the BPSW confirmation of all 78 495 primality verdicts \(\ell\le10^6\) — zero mismatches. For census-scale work the C engine of Appendix B is the production path; PARI is the independent adjudicator, which is the correct division of labor.

Appendix BReproducibility: the verification pipeline

Engines (this session). (1) C, gcc 13.3 -O3: segmented Eratosthenes (2²²-wide segments) streaming consecutive primes to \(10^9{+}4000\); per prime: deterministic MR {2,3,5,7} on \(\ell\) (valid < 3.215·10⁹; every \(\ell<10^9\)), else full trial-division factorization with MR early exit and a divisor scan for the least divisor \(>g\); accumulators for decades, per-gap masses, congruence locks, invariants, exception lists, record ladder; dumps for audits and figures. Wall time 85 s at \(10^9\). (2) PARI/GP 2.15.4, the project kernels above. (3) Python 3.12 / NumPy, independent sieve + divisor scan. (4) Live OEIS b-file diff. All four layers agree with zero mismatches on their overlaps; identity \(\ell=kL\) asserted engine-wide.

Anchors reproduced today: \(\pi(10^k)\) for \(k\le9\); decomposable \(=\pi-3\) at every mark; \(\pi_2\)-minus-one twin counts 58 979 / 440 311 / 3 424 505 at \(10^{7,8,9}\); A002386 through 282 @ 436 273 009; A117078 terms including \(k(9998)=104\,711\), \(k(9999)=104\,717\). Prior-census anchors cited, not re-run: the \(10^{10}\) row, \(p_{5\cdot10^7}=982\,451\,653\), \(\pi_2(10^{10})=27\,412\,679\), ladder to 354, and the intra-decade \((10^9,10^{10}]\) ladders. The distinction is maintained everywhere in this report.

This session’s own failures and near-misses (disclosed; none reached print). (a) The C engine’s first draft contained a garbled consecutive-prime streaming block and — more instructive — a composite-weight exception test conditioned on \(L>1\), which would have silently dropped 13 and 31 (both level-one with composite weight) from the C4 census. Both were caught in pre-run code review against the definitions; the engine never executed in the faulty state. The lesson is old but earns its keep: exception sets are where conventions bite, and the five-element list must be regression-pinned, never re-derived from memory of the predicate. (b) The first render of Figure N6 labelled \(1/(r-2)\) an “exact law”; it is the equidistribution heuristic (the residue reduction is exact, the value is not) — caught before embedding and reworded. (c) Provenance note: both project “PDFs” are page-image/text ZIP containers, not PDF streams; standard extractors fail on the magic bytes and the archive route was used. No numerical consequence.

Cumulative recordCorrections log (oldest first)

1. Species-I overclaim (first report; corrected in the second). Presenting level-one rarefaction as open overstated the case — a Selberg/Brun sieve already gives \(N_1\ll x/(\log x)^{3/2}\) unconditionally. Scope kept correct ever since.
2. Two-constants erratum (second report). Cumulative Species-II ratio ≈1.13–1.18 misread as a constant; it is a finite-size transient. Windowed \(\hat c_2\) is the reported quantity. (Item 11 is this erratum’s sequel.)
3. Checkpoint off-by-one (second report). \(p_{5\cdot10^7}\) 74 short of 982 451 653 from a driver checkpoint bug; caught by that anchor, all values regenerated.
4. Third report’s session failures (disclosed; none printed): memory-ceiling kill; time-budget overrun; unbounded naïve divisor search — each replaced by the staged, checkpointed design.
5. Population convention, Table-3 recount (third report). +9 over seven columns explained exactly by the nine level-classified primes with weight ≤ 23; the paper counts weight-classified only and is correct under its convention.
6. Challenge paper, Table 1, lev1(10⁸) (upstream, diagnosed third report). 339 869 vs the complete-convention 339 870: the boundary prime 99 999 989 excluded by a \(p_{n+1}\le x\) pipeline. Erratum queued upstream.
7. MathJax macro bug (third report, first build). Self-recursive \(\ell\) macro; rendering only.
8. Quartic-bound completeness overclaim (third report §4.5; repaired fourth). The \(g^4>p\) closure missed first-occurrence gaps 322–456; repaired by the Cube Lemma \(\ell\le(g-1)^3\), under which the record tables genuinely close the set.
9. Fourth report’s session failures (disclosed): segment-parity sieve bug passing four anchors silently, caught by the \(N_1(10^9)\) cross-check; an in-place patch without write-back; a cosmetic mathtext crash. Forced-odd segment bounds and asserted whole-file rewrites since.
10. Doubly-logarithmic span misquote (third report). Six-window span ≈0.89, not ≈0.62; prose-level only.
11. Species-II constant \(c_2=1\) — RETRACTED (interim; withdrawn in the fifth). Cumulative estimator 0.8312 → 0.7999 over \(10^6\!-\!10^{10}\), receding from 1 at every mark while windowed \(\hat c_2\) sits at 0.773 ± 0.004. \(c_2\) is OPEN; today’s recount reproduces the deciding sequence exactly (§4.7).
12. Fifth report’s session failures (disclosed): unbudgeted \(10^{10}\) sweep killed pre-checkpoint (replaced by budgeted chunks); a silently truncated \(10^6\) CSV caught by the \(\pi(10^6)\) row count; a 113 733/113 734 PARI batch (one line swallowed by the reader), covered by the independent audit.
13. Sixth-edition increment (summary of record): Lemma-8 upstream erratum queued — the \(g\ge\sqrt\ell\) exception set is the six-element \(\{5,13,19,23,31,113\}\), the two-element \(\{13,31\}\) being its level-one composite-\(\ell\) subset; the July 2026 manuscript already states both correctly (Lemma 5 (ii)/(iii)).
14. NEW — manuscript per-gap window at \(10^9\) (this session, §4.5). “[0.97, 1.05] at every \(g\le150\) at both heights” is false at \(10^9\) for \(g\in\{84,102,108,114,132,138,144\}\) (extremes 0.9627, 1.1318; all within Poisson noise). Reword before submission; \(10^{10}\) column retains prior-census status.
15. NEW — manuscript \(R_2\) flatness sentence (this session, §4.7). Under the stated Mertens mass, cumulative \(R_2\) runs 0.4470 → 0.4422 over \(10^6\!-\!10^9\) (windowed 0.4458 → 0.4420): trending toward ≈0.441 but not “flat to three decimals over \(10^6\!-\!10^{10}\)”; the exact-product convention is farther off. Re-derive the pipeline convention and restrict the claim. Problem 3 itself unaffected.
16. NEW — closure frontier upgrade (this session, §4.2–4.3; an improvement, not an error). Exhaustive gap analysis now complete to \(2^{64}\) (maximal gap 1550 @ 18 361 375 334 787 046 697; Nicely/PGS, mirrored at OEIS A000101). Lemma 5(ii)–(iii) and Cube-Lemma exception sets close to \(2^{64}\); keep the refereed \(4\cdot10^{18}\) in the statements, add the extension as a cited remark.
17. This session’s failures (disclosed; none reached print): the pre-run engine bugs, the “exact law” figure label, and the ZIP-container provenance note — Appendix B(a)–(c).

Seventh edition, compiled 5 August 2026. All “this session” computations executed today on the report pipeline (C / PARI-GP 2.15.4 / Python 3.12), figures generated from today’s dumps, fonts Fraunces · Spectral · IBM Plex Mono, palette per house style. Sources: arXiv:0711.0865 (v4, 18 Jan 2010, current); the July 2026 submission manuscript; the OEIS entries and wiki page (checked live today, C7/C8 struck through as trivial); decompwlj.com (live; database, CSV and image dumps available); Gafni–Tao arXiv:2508.06463; Nicely first-occurrence tables / OEIS A000101 attachment for the \(2^{64}\) frontier; Oliveira e Silva–Herzog–Pardi, Math. Comp. 83 (2014) for the refereed \(4\cdot10^{18}\) anchor. Nothing in this report claims progress on twin primes, balanced primes, Goldbach, or RH.