{"id":1257,"job_id":1432,"problem_id":1,"lane_id":null,"type":"audit","user_id":17,"model":"claude-fable-5-1","provider":"anthropic","report_md":"# Job #1432: audit of `paper/moire-primes.md`, built on the two rejected revisions (#9, #16) and their reviews (18, 39): the six required fixes and six inherited defects of review 39 applied, every disputed number recomputed\n\n**Caveat first.** No theorem of the paper changes and no number of the served text is altered except where a review showed it wrong; the mathematics of the spine (Redundancy, Copying, Seam, Exact Invariance, Zone Equivalence, pigeonhole) is untouched. The revision is the latest rejected text (#16) plus review 39's list, so the two earlier audits are carried, not redone.\n\n## 0. Custody\n\n- Served `paper/moire-primes.md` (snapshot main, 2026-09-19): sha256 56752f16…, the seed version (65 em dashes, no A144311 attribution corrections). Two revisions were rejected: #9 (report 18, @MichaelRobartes: the 181# attribution and two retained defects) and #16 (report 39, @nielsegberts: six bounded fixes F1–F6 and six inherited defects, \"most proposed edits can be retained\").\n- This revision: base = #16's exact bytes (eabc9ce8…), 20 hunks on top of it (`moire-primes.vs-return16.diff`), sha256 **bab9e4356999b16d8f4b3d1e457496bea9238bd8ecdba2ba2f8e15340402698a**. `patch` is the unified diff against the served file (34 hunks; re-applied with `patch`, output byte-identical). `patch1432.py` rebuilds the revision from #16's file with a count assertion on every edit.\n\n## 1. Review 39's required fixes, applied\n\n| # | where | what was wrong | change | evidence |\n|---|---|---|---|---|\n| F1 | §2 Crystallization Lemma, proof; §3 houses; abstract | \"every twin slot with r+2 < p²\" admits (1, 3) at x = 2; \"no later prime ever strikes below its own square\" contradicts the Redundancy Lemma (7 strikes 7 in T₅; the slot (11, 13) of T₅ is struck by 11); \"immortal\" slots | statement now has x ≥ 3, x < r and r + 2 < p²ₙₑₓₜ; \"apart from a prime's own position, no new strike below its square\"; permanent certification distinguished from tile membership; §3 says the certification is permanent and the slots die at their own folds, the house persists; \"three immortal houses\" → \"three houses\" | recomputed: new strikes of 7 in T₇'s period begin 7, 49; (47, 49) is a T₅ slot with 49 = 7² (`checks1432.out`) |\n| F2 | §2, §9 | Holt's [p², p²ₙₑₓₜ] called the upper part of the zone (p, p²) | short zone (p, p²) and expanded zone (p, p²ₙₑₓₜ) named; Holt's interval is the upper subinterval of the expanded zone, endpoint apart; \"one object\" → the shared frontier only | set comparison (36 ∈ [25, 49], 36 ∉ (5, 25)) |\n| F3 | §4 Unification Law | \"within about 1% at every grid point on both ranges\" | agreement stated per scale: 1.1% at x = 10⁸, 1.2% at 10⁷, 7.4% at u = 2 for x = 10⁶ (0.852 vs 0.793) | `attack2-05-07-integral-ladder.js` OUTPUT rows read; max discrepancies 7.44%, 1.13%, 1.09% |\n| F4 | §8 seam copy-law | \"exact to four decimals at depths 3 to 6, within 0.65% at depth 7\" | the five ratios printed (1.0000, 1.0000, 0.9999, 0.9998, 0.9935), 0.023% and 0.654%, 176/3,798 at depth 7, T₁₉ half-width 105, \"sampled, not the exact invariance\" | `attack2-04-10-hierarchy-oeis.js` table read |\n| F5 | abstract, §9 | G₂ \"the function as a studied object is not [published]\"; \"four sweeps\" | G₂ is a rediscovery: A144311's definition (m ≡ ±1 mod p, m = r + 1, is p ∣ r(r+2)) is the function; Carter 2008, Alekseyev terms 8–16 (2009), Wang 17–22 (2024); §9 bullet split into \"rediscovered\" and \"candidate contributions resting on dated searches\"; \"four audits, five query waves\" | OEIS A144311 text entry (job #1431 today: 22 terms); `PRIOR-ART.md` :89 and :495–508 |\n| F6 | §7 Scour | 181# attributed to `two-moire-argument.md`, which cuts at √width | 173# ≈ 10^68.2 and the 10^−63.7 sliver (30030/173# = 1.8 × 10⁻⁶⁴); the note's ~10⁷³/~10⁻⁶⁹ flagged as matching no stated cutoff | recomputed: log₁₀(173#) = 68.221649, log₁₀(30030/173#) = −63.744093; Scour primes 17 to 173 (34 of them) |\n\n## 2. Review 39's inherited defects, applied\n\n| # | where | change | evidence |\n|---|---|---|---|\n| I1 | §9 | Holt's machinery \"bounded by ∣s∣ < 2p₁ throughout\" → the driving-term transfer is stated under that bound in the papers read first, while 1408.6002 §6.1 Cor. 6.3 carries the transport for every gap size (PRIOR-ART's own retraction cited) | `PRIOR-ART.md` :658–663 read |\n| I2 | §4, §7 | \"exact proportion\" → over the full joint period, with finite-window fluctuation; \"misaligned in every window is the conjecture\" → the conjecture is infinitely many expanded zones (§6), since a periodic tile has slot-free windows | Zone Equivalence as stated in §6 |\n| I3 | §6 proof, §6 prose, abstract | the RS floor gives an upper bound on the quotient; PNT (K ~ p²/(2 ln p)) supplies the asymptotic equality; \"certified 2.00\" → 2.003 with K ≥ 73,411 and the full script path; \"we certify 2 ln p\" → the finite quotient, (2 + o(1)) ln p | recomputed: K = 79,661 at p = 1009, quotient 12.77 = 1.846 ln p; floor 73,411, certificate 2.003 (`attack-08-pigeonhole-theorem.js` OUTPUT) |\n| I4 | §8 (three places), §9 | \"no upper bound was published at any exponent\" → not found in the searches of `covering-dive.md` §2.2; \"first lower bound of any kind\" and \"first bounds of either kind\" → in those searches; \"No literature exists\" → none found; variance local factors given for odd p with the p = 2 case (one odd residue) | scoping only |\n| I5 | §2 Euclid proof | \"Suppose x ≥ 3 were the largest prime\" (at x = 2 the edge W − 1 = 1 is the excluded hole) | trivial |\n| I6 | §8 exponent paragraph | the one-class exponent 1 is Maier–Pomerance's conjecture, proven only in [1, 2]; the correction assumes it and the bias transfer | `exponent-control.md` \"The control\"; the same fix as beta2-note's #20 |\n\nAlso: a header note and a correction log at the end of §9 (house rule: refutations stay visible), listing every correction from #9, #16 and this pass; \"which are the first bounds of either kind\" scoped. Not changed: #16's other 30 hunks (review 39's table supports them; the Petersen author list, Pritchard 1981, Zhang 2014, KKL initials, HL supply factor 4, 437,987 vs 438,055, the 53-minute runtime, the 43# terms and A049296 scope were not re-verified beyond the arithmetic in `checks1432.out`, which reproduces the first-twin offset 84, the T₁₃ house ledger 155/152/149 under the self-strike convention and the pigeonhole coefficients 1.81, 1.79, 1.85, 1.87 at p = 17, 101, 1009, 4999); the coda as #16 trimmed it; `paper/wall-note.md`'s numbers (own audit stream).\n\n## 3. Calibration and limits\n\nEvery inserted number was recomputed here (`checks1432.out`, Python, seconds) or read from the cited script's embedded OUTPUT; the two OEIS entries were read in text format today (job #1431); Holt's pages, the DH book, FKMPT Remark 7 and the Rosser–Schoenfeld page were not opened (the paper's own readings, as in #9 and #16). Rungs: the corrected statements are proven where they are lemmas (F1, I3, I5), measured where they quote scripts (F3, F4, F6, I2's ledger), and scoped search claims elsewhere. Nothing here bears on the twin question. Transcript scrubbed as data (token, session id, account ids, paths, e-mail); this assignment's lines only. 70 of this handle's returns wait for a verdict.\n\nFiles: moire-primes.md (the revision; `revision` and `paper` file), moire-primes.patch (against the served file, also in `patch`), moire-primes.vs-return16.diff, patch1432.py, checks1432.out. Cites: returns #16, #9 (@Benjaminsen); reviews 18 (@MichaelRobartes) and 39 (@nielsegberts) on the paper record; OEIS A144311; `research/PRIOR-ART.md`, `two-moire-argument.md`, `attack2-05-07-integral-ladder.js`, `attack2-04-10-hierarchy-oeis.js`, `attack-08-pigeonhole-theorem.js`, `exponent-control.md`.\n","patch":"--- a/paper/moire-primes.md\n+++ b/paper/moire-primes.md\n@@ -1,6 +1,6 @@\n # Primes as Moiré Patterns: the Tile, the Family, and the Wall\n \n-*DRAFT v2, 2026-08-17. Sole author: Chris Benjaminsen (AI disclosure at\n+*DRAFT v2, 2026-08-17; audit revision 2026-09-19 (the corrections to this draft are listed at the end of §9). Sole author: Chris Benjaminsen (AI disclosure at\n the end). Flagship paper (Paper I) of the suite\n described in `paper/PAPERS.md`; technical detail deferred to Paper II (the\n twin-Jacobsthal bound), Paper III (the exact variance), and Paper IV (data and\n@@ -12,35 +12,40 @@\n ## Abstract\n \n This paper presents a new framework and vocabulary over classical\n-sieve-theoretic objects — a new lens. Stack the periodic multiples of the\n+sieve-theoretic objects, a new lens. Stack the periodic multiples of the\n primes you know on the number line, and the positions untouched by every wave\n form a repeating interference pattern: a moiré whose repeating unit we call\n the **tile**, whose period is the primorial, and whose holes contain all\n-further primes. Developed from scratch as a ten-year sequence of independent\n+further primes. Developed from scratch as a six-year sequence of independent\n rediscoveries (wheel factorization, Euler's and Schemmel's totients, the\n Hardy–Littlewood twin constants, Fermat's factorization, and most of the\n structural apparatus of Holt's cycle-of-gaps programme) and then audited\n against the literature, the lens earns its keep by what it makes newly\n speakable. We prove a spine of elementary structure theorems in its\n-vocabulary — among them an exact genealogy of twin candidates (every twin\n+vocabulary, among them an exact genealogy of twin candidates (every twin\n opportunity at every scale descends from a single ancestral slot, through\n-exactly three immortal houses fixed at the second fold, with birth cohorts\n+exactly three houses fixed at the second fold, with birth cohorts\n whose demographic shares freeze forever), an Exact Invariance Lemma for the\n-fossil record each prime leaves in the pattern, and one new unconditional\n-theorem: **for every prime p ≥ 17 the interval (p, p²) contains two primes\n-whose distance is at most (p² − p)/(π(p²) − π(p) − 1)**, a quantity that\n-Rosser–Schoenfeld makes (2 + o(1))·ln p and that is already 2.00·ln p at\n-p = 1009. We verify the framework's census against reality through a\n+fossil record each prime leaves in the pattern, and one unconditional\n+theorem, elementary in proof: **for every prime p ≥ 17 the interval (p, p²)\n+contains two primes whose distance is at most (p² − p)/(π(p²) − π(p) − 1)**,\n+a quantity that is (2 + o(1))·ln p by the prime number theorem, with an explicit Rosser–Schoenfeld certificate of 2.003·ln p at p = 1009, where the quantity itself is\n+1.85·ln p. We verify the framework's census against reality through a\n 7.42-trillion-position count, survey the parity wall through five doors and four\n faces, with the measured numbers behind each carried in the companion note, restate the Twin Prime Conjecture in the\n-framework's native form — *the Scour never achieves perfect local alignment\n-with the Grain* — and introduce the objects the lens found that our own\n-prior-art audit did not find in the literature: the twin Jacobsthal function\n-G₂, the exact two-class window variance, and the twin grain of the tile,\n-together with the difference map d ↦ G_d, whose closest published relative is\n-Ziller and Morack's paired Jacobsthal function h₂. A companion note (Paper\n-II) proves the first upper bound for G₂ at any exponent; a lower bound is\n-free and is likewise the first recorded, though it is nowhere near matching:\n+framework's native form, *the Scour never achieves perfect local alignment\n+with the Grain*, and introduce the objects the lens led us to, each with its standing after\n+our prior-art audit: the twin Jacobsthal function G₂, a rediscovery (its\n+values are OEIS A144311 shifted by one, Carter 2008, extended by Alekseyev\n+2009 and Wang 2024 to 22 terms, and that entry defines the function itself;\n+five search waves missed it by never shifting the ladder), the exact\n+two-class window variance and the twin grain of the tile, which the audit's\n+dated searches did not find in print, and the difference map d ↦ G_d, whose\n+closest published relative is Ziller and Morack's paired Jacobsthal function\n+h₂. A companion note (Paper\n+II) proves an upper bound for G₂, per our audit the first at any exponent; a\n+lower bound is free and is likewise, per the same audit, the first recorded,\n+though it is nowhere near matching:\n the band (2, 4.2665] between them is the problem this paper is about.\n \n ---\n@@ -50,30 +55,30 @@\n Take the number line and lay a wave of period 2 on it, striking every even\n position. Add a wave of period 3, then 5, then 7. Each wave deletes its\n multiples; what survives is the set of positions coprime to every stacked\n-prime — the *holes* of the combined pattern. Because the waves are periodic,\n+prime, the *holes* of the combined pattern. Because the waves are periodic,\n so is their superposition: the hole pattern repeats with period equal to the\n primorial 2·3·5···pₙ, and every prime larger than pₙ must land in a hole,\n-forever. This is the sieve of Eratosthenes seen as interference — a moiré, in\n+forever. This is the sieve of Eratosthenes seen as interference, a moiré, in\n the optical sense: simple periodic layers whose overlap produces structure far\n more intricate than any layer alone.\n \n The picture itself is not new, and we are precise about that at the outset.\n-Physicists have realized it literally: Petersen, Argüelles, Greenberg,\n-Kaminer, and Soljačić encoded the primes as intensity zeros of superposed\n-identical waves — \"mimicking the sieve of Eratosthenes,\" twin primes\n-included — in *Physical Review Letters* 122, 090201 (2019). Jason Davies'\n+Physicists have realized it literally: Petersen, Ceko, Svalbe, Morgan,\n+Bishop, and Paganin encoded the primes as intensity zeros of superposed\n+identical waves, \"mimicking the sieve of Eratosthenes,\" twin primes\n+included, in *Physical Review Letters* 122, 090201 (2019; arXiv:1812.04203). Jason Davies'\n interactive visualization \"El Patrón de los Números Primos\" (2012, after Omar\n E. Pol) made the same picture a small internet phenomenon; Jeffrey Ventrella's\n *Divisor Drips and Square Root Waves* develops primes as the negative space\n-behind overlapping periodic patterns at book length. The algorithmic core — a\n-wheel whose circumference multiplies by p as each new prime joins — is the\n-sieve of Pritchard (1979; \"Explaining the wheel sieve,\" *Acta Informatica* 17,\n-1982).\n+behind overlapping periodic patterns at book length. The algorithmic core, a\n+wheel whose circumference multiplies by p as each new prime joins, is the\n+sieve of Pritchard (*Comm. ACM* 24, 1981; \"Explaining the wheel sieve,\"\n+*Acta Informatica* 17, 1982).\n \n What is ours is the lens as a *system*: one metaphor family, one term per\n object, and the discipline of asking every question in its vocabulary. The\n repeating unit is the **tile** Tₚ (canonical alias: the primorial wheel mod\n-p#), of **width** |Tₚ| = p#. A new prime **folds** the tile — lays p copies\n+p#), of **width** |Tₚ| = p#. A new prime **folds** the tile, lays p copies\n end to end, then strikes its two twin-forbidden residues through every copy.\n Copies meet at **seams**; the tile is a palindrome (**the mirror**); each\n prime leaves a permanent **stratum**; the twin opportunities form a\n@@ -89,8 +94,8 @@\n either.\n \n One remark on provenance, because it shapes the paper's voice. The framework\n-was built between 2016 and 2026 in ignorance of the literature, as a private\n-research program (browser experiments from 2016, 26 folders of them from 2020). Nearly\n+was built between 2020 and 2026 in ignorance of the literature, as a private\n+research program (26 folders of self-contained browser experiments). Nearly\n everything in §§1–2 turned out to be known, some of it since 1857, and most of\n the structural apparatus of §§2–4 turned out to be Holt's, in print since 2007.\n We regard that as the strongest available evidence that the lens *works*:\n@@ -106,7 +111,7 @@\n \n **Redundancy Lemma (the kill image).** *When prime p folds the tile, the\n positions it strikes that were not already struck are exactly p × (the\n-previous hole set) — a p-times-magnified copy of the tile's own hole pattern.\n+previous hole set), a p-times-magnified copy of the tile's own hole pattern.\n In particular the first new strike is at p·1 = p (the prime striking itself\n as a candidate), the second at p², and one period of the new tile contains\n exactly φ(previous width) new strikes.*\n@@ -117,34 +122,45 @@\n strikes begin at p, fall silent, and resume at p². Per period there are\n φ(previous width) holes m in range, hence that many new strikes. ∎\n \n-The counts 1, 1, 2, 8 of new strikes per period — measured in this project's\n-`modifiercount.txt` years before the lemma was stated — are φ(1), φ(2), φ(6),\n+The counts 1, 1, 2, 8 of new strikes per period, measured in this project's\n+`modifiercount.txt` years before the lemma was stated, are φ(1), φ(2), φ(6),\n φ(30). The lemma says each fold deletes a scaled self-image: the moiré eats\n copies of itself. This is classical in effect (it is why Eratosthenes starts\n crossing out at p²), but the self-image phrasing is what later makes\n crystallization, the strata (§4), and the gap decoupling (§8) transparent.\n \n-**Crystallization Lemma (the frontier).** *In Tₓ, every hole in (x, p²ₙₑₓₜ)\n-is prime and every twin slot there is a real twin-prime pair; no later prime\n-ever strikes below its own square. Territory below the frontier p²ₙₑₓₜ is\n-final: possible = actual, permanently.*\n+**Crystallization Lemma (the frontier).** *In Tₓ with x ≥ 3, every hole in\n+(x, p²ₙₑₓₜ) is prime, every twin slot (r, r+2) with x < r and\n+r + 2 < p²ₙₑₓₜ is a real twin-prime pair, and, apart from a prime's own\n+position, no later prime strikes a new position below its own square.\n+Primality certified below the frontier p²ₙₑₓₜ is permanent; membership in\n+later tiles is not: the slot (11, 13) of T₅ is struck when 11 folds, at 11\n+itself.*\n \n *Proof.* A composite c below the frontier has a prime factor ≤ √c < pₙₑₓₜ,\n hence ≤ x, hence is already struck; by the Redundancy Lemma each later prime\n-q first strikes anew at q². ∎\n+q strikes anew first at q itself and then not again until q². ∎\n+\n+The endpoint clause is not decoration: (p²ₙₑₓₜ − 2, p²ₙₑₓₜ) is itself a twin\n+slot of Tₓ at x = 3, 5, 11 and 17 ((23, 25), (47, 49), (167, 169), (359, 361)),\n+and its upper member is the square that breaks it.\n \n We verified \"possible = actual below p²\" computationally at eight levels out\n to p = 9973 with windows to 10⁸: `equal=true`, 8 of 8\n (`research/04-crystallization-and-hl.js`). The moiré *crystallizes outward*;\n the quiet stretch (p, p²), where the newest prime has struck exactly once, at\n p itself, is **the zone**, and the entire Twin Prime Conjecture lives\n-there (§6). Holt names the same interval the *interval of survival* and its\n-upper edge the *horizon of survival* (arXiv:2603.25915); the two vocabularies\n-are describing one object.\n+there (§6). Two zones are in play in this paper: the short zone (p, p²),\n+where the pigeonhole theorem of §6 lives, and the expanded zone (p, p²ₙₑₓₜ)\n+of the Zone Equivalence Proposition. Holt's *interval of survival*\n+[p², p²ₙₑₓₜ] is the upper subinterval of the expanded zone, its upper endpoint\n+apart, and his *horizon of survival* p²ₙₑₓₜ is the frontier the two\n+vocabularies share (arXiv:2603.25915, as `research/PRIOR-ART.md` records it);\n+the intervals themselves are not one object.\n \n **Copying Theorem (the census).** *A twin slot is a position r with r and\n-r+2 both holes. Writing Dₓ for the census — the number of twin slots per\n-tile — each fold by p lifts every slot to p copies, of which exactly 2 die:*\n+r+2 both holes. Writing Dₓ for the census, the number of twin slots per\n+tile, each fold by p lifts every slot to p copies, of which exactly 2 die:*\n \n > *Dₓ = ∏₍₂<q≤x₎ (q−2), and per fold: (p−1)·D new copies created, exactly\n > 2·D destroyed.*\n@@ -152,7 +168,7 @@\n *Proof.* Folding tiles the old pattern p times, so each slot r lifts to\n r + j·(old width), j = 0,…,p−1. The old width is invertible mod p, so the p\n lifts occupy each residue class mod p exactly once. Exactly one lift lands in\n-r ≡ 0 and one in r ≡ −2 (mod p) — distinct classes for every odd p — and the\n+r ≡ 0 and one in r ≡ −2 (mod p), distinct classes for every odd p, and the\n fold kills those two copies and no others. ∎\n \n The count ∏(q−2) is Schemmel's totient (1869) at the primorial and OEIS\n@@ -167,13 +183,13 @@\n **Theorem (Euclid, ~300 BC, moiré form).** *There are infinitely many\n primes.*\n \n-*Proof.* Suppose x were the largest prime and build Tₓ. The tile has holes\n-besides 1 — the mirror guarantees it: if r is coprime to the width W, so is\n+*Proof.* Suppose x ≥ 3 were the largest prime and build Tₓ. The tile has holes\n+besides 1, the mirror guarantees it: if r is coprime to the width W, so is\n W − r, so the pattern is a palindrome whose edge W − 1 always survives. Any\n hole r > 1 is coprime to every prime ≤ x, so its smallest prime factor\n exceeds x: a prime larger than the largest prime. ∎\n \n-The subtlety is that the hole need not be prime — and usually isn't:\n+The subtlety is that the hole need not be prime, and usually isn't:\n 210 − 1 = **209 = 11 · 19**, and 30030 + 1 = 30031 = 59 · 509. Only its\n *factors* must be new, and that is all the contradiction needs. Euclid's own\n construction is literally p# + 1: the edge of the mirror, surviving by\n@@ -181,7 +197,7 @@\n edge *pair* (W−1, W+1) is a twin slot at every level forever, but the escape\n hatch was \"prime *or* has a new prime factor,\" and a pair needs both members\n actually prime. At p = 17 the edge pair factors as 510509 = 61·8369 and\n-510511 = 19·97·277 — the pair structure shatters. Singles have an escape\n+510511 = 19·97·277, the pair structure shatters. Singles have an escape\n hatch; pairs do not. That asymmetry is the difference between a\n 2300-year-old theorem and an open conjecture, visible in one line of\n factorizations.\n@@ -189,50 +205,53 @@\n ## 3. The family: a complete genealogy of twin opportunities\n \n The Copying Theorem says slots multiply; the lens asks *who begets whom*. The\n-answers turn out to be exact, verified, and — as far as the audit could\n-find — never before stated.\n+answers turn out to be exact, verified, and, as far as the audit could\n+find, never before stated.\n \n **No orphans.** If r and r+2 avoid all primes ≤ p, they avoid all primes\n < p: every twin slot of every tile reduces, mod any earlier width, to a twin\n slot of that earlier tile. No lineage is ever born after the beginning. We\n verified this at four levels (zero orphans among 1,638 slots checked;\n `research/genealogy.js`): the entire twin population of every tile, forever,\n-is **one family**, descending from the single ancestral slot (5,7) of T₃ —\n+is **one family**, descending from the single ancestral slot (5,7) of T₃,\n the wrap pair straddling the seam of the six-wide tile, created by the\n interference of 2 and 3 alone. (The 6k±1 template that every twin pair wears\n-is pure @2×@3 moiré.) The ancestor itself dies at the very next fold — 5\n-strikes position 5, the prime consuming itself as a candidate — the ancestor\n+is pure @2×@3 moiré.) The ancestor itself dies at the very next fold, 5\n+strikes position 5, the prime consuming itself as a candidate, the ancestor\n dies giving birth.\n \n **The Seam Lemma.** *After a fold, adjacent copies meet at seams k·(old\n-width), and every seam carries the pair (kP−1, kP+1) — twin slots by the\n+width), and every seam carries the pair (kP−1, kP+1), twin slots by the\n mirror. Each fold kills exactly 2 seam pairs and p−2 survive.*\n \n *Proof.* kP mod p sweeps every residue class exactly once as k does (P\n-invertible mod p); the pair dies iff kP ≡ +1 or −1 (mod p) — one k each. ∎\n+invertible mod p); the pair dies iff kP ≡ +1 or −1 (mod p), one k each. ∎\n \n Verified at six levels: survivors 3, 5, 9, 11, 15, …, 35 for folds 5 through\n-37, always exactly p−2 (`research/verify-ladder.js`). The Seam Lemma is the\n-Copying Theorem restricted to the **edge lineage** — the branch of the family\n+37, always exactly p−2 (`research/verify-ladder.js` through T₂₃,\n+`verify-ladder-big.js` for T₂₉ to T₃₇). The Seam Lemma is the\n+Copying Theorem restricted to the **edge lineage**, the branch of the family\n that keeps the seam address.\n \n **The three houses.** T₅ is the unique tile fully crystallized at birth: its\n-width (30) is smaller than its own frontier (49), so all three of its slots —\n-(11,13), (17,19), (29,31) — are certified real twin primes the moment they\n-exist, immortal by theorem. They are the family's complete and final\n-aristocracy. **House 11** and **House 17** are mirror images of each other\n-(the palindrome maps 11 ↔ 17 in T₅); **House 29** is self-mirror — it *is*\n+width (30) is smaller than its own frontier (49), so all three of its slots,\n+(11,13), (17,19), (29,31), are certified real twin primes the moment they\n+exist, and the certification is permanent by the Crystallization Lemma; the\n+slots themselves are struck when their own primes fold, and what persists is\n+the house, the residue class modulo 30 each founder names. They are the\n+family's complete and final aristocracy. **House 11** and **House 17** are mirror images of each other\n+(the palindrome maps 11 ↔ 17 in T₅); **House 29** is self-mirror, it *is*\n the edge, ≡ −1 mod 30, owner of every seam pair at every level forever. The\n Copying Theorem's uniformity makes the inheritance exact: **each house\n carries precisely one third of every census, forever.** Of T₃₁'s\n 6,226,553,025 slots, exactly 2,075,517,675 descend from each founder. For\n contrast, T₇ is the first tile containing *mortal* slots: (167,169) will be\n-executed by 13 — at 169 = 13², the first stratum kill in twin history — and\n+executed by 13, at 169 = 13², the first stratum kill in twin history, and\n (209,211) falls to 11.\n \n **Birth cohorts, with closed forms.** Each fold p, the edge slot bears its\n p−2 children: one remains the edge (the wrap), and p−3 *graduate* into the\n-interior as that fold's genuine newborns — the interior seam pairs.\n+interior as that fold's genuine newborns, the interior seam pairs.\n Everything else is copies. This yields an exact decomposition of the census\n by creation fold, telescoping with the Copying Theorem via 1 + (p−3) = p−2:\n \n@@ -243,8 +262,8 @@\n seam-address depth: born at fold p iff ≡ −1 mod the previous width but not\n mod the new one). We classified every slot of T₁₃ and T₁₇ by this rule:\n every cohort exact (990/396/88/10 + edge at T₁₃; 14850/5940/1320/150/14 +\n-edge at T₁₇; `research/birth-cohorts.js`). The demographic consequence is\n-striking — shares freeze at birth (a corollary of the Exact Invariance Lemma\n+edge at T₁₇; `research/birth-cohorts.js`). The demographic consequence:\n+shares freeze at birth (a corollary of the Exact Invariance Lemma\n of §4): share(@p) = cohort(p)/Dₚ forever. At T₃₁:\n \n | born at | count | share |\n@@ -254,11 +273,11 @@\n | @11 | 368,980,920 | 5.93% (= 8/135) |\n | @13 | 41,929,650 | 0.67% |\n | @17 → @31 | 4,192,964 | < 0.07% |\n-| eternal edge | 1 | — |\n+| eternal edge | 1 | (one slot) |\n \n Sums to 6,226,553,025 exactly. Two-thirds of every twin opportunity that\n will ever exist was born at the second fold; Houses 11 and 17 never absorb a\n-birth (they grow purely by copying); House 29 is the womb — every newborn\n+birth (they grow purely by copying); House 29 is the womb, every newborn\n from fold 7 onward arrives inside the edge house, and the births exactly\n compensate its graduations, holding it at one third.\n \n@@ -271,13 +290,14 @@\n \n **Strata and the Exact Invariance Lemma.** The Redundancy Lemma's kill image\n p × (holes) has a *head*: its densest part, landing exactly at p². Each\n-prime therefore digs a dent — a **stratum** — into the band [p², 2p²] of its\n+prime therefore digs a dent, a **stratum**, into the band [p², 2p²] of its\n own tile. The lens asks: does the dent heal, deepen, or persist under later\n folds? The answer is an identity:\n \n-**Exact Invariance Lemma.** *The in-period depth of any band of the tile —\n-its slot density relative to the tile average — is exactly invariant under\n-folding.*\n+**Exact Invariance Lemma.** *The in-period depth of any band of the tile,\n+its slot density relative to the tile average, is exactly invariant under\n+folding.* (A band is a set of offsets modulo the old width, and its count\n+after a fold is summed over the p copies the fold lays down.)\n \n *Proof.* Every slot's p lifts lose exactly 2 (Copying Theorem), so every\n band's total scales by exactly (p−2), the same factor as the census; all\n@@ -291,7 +311,7 @@\n `attack2-02-08-tomography.js`). The pattern is an archaeological record:\n every prime's ignition leaves a permanent, copied-forever stratum, its depth\n fixed at birth. (For small primes ≤ 13 the \"stratum\" is three or four\n-individual kills and layout luck dominates — 11's band is actually enriched;\n+individual kills and layout luck dominates, 11's band is actually enriched;\n the statistical law begins at p = 17. Consecutive strata overlap, since\n p²ₙ₊₁ < 2p²ₙ for close primes, so measured dents stack to 0.64–0.71.)\n \n@@ -303,28 +323,33 @@\n > ρ(u) = e^{2γ}/u² for 1 ≤ u ≤ 2, continuing as the pair-Buchstab square\n > (e^γ·ω(u))² on 2 ≤ u ≤ 3, pinned to 1 beyond,\n \n-derived independently by two routes in this project (Hardy–Littlewood +\n-Mertens on one side; the strata calculus on the other) and verified to ~1%\n-at every grid point tested (`research/attack2-05-07-integral-ladder.js`,\n-`attack2-03-09-depth-formula.js`; the derivation is Hardy–Littlewood-\n-conditional). One curve explains: the head cap e^{2γ} ≈ 3.17 (a fixed window\n+derived on 1 ≤ u ≤ 2 by two routes in this project (Hardy–Littlewood +\n+Mertens on one side; the strata calculus on the other), conjectured on\n+2 ≤ u ≤ 3 as the square of Buchstab's single-prime curve, and measured\n+against the curve on a twelve-point grid u = 1.2, …, 3.0 at three scales: at\n+x = 10⁸ (window 10⁶) the printed values agree to within 1.1%, at x = 10⁷ to\n+within 1.2%, while at x = 10⁶ the finite-size discrepancy reaches 7.4% at\n+u = 2 (0.852 measured against 0.793)\n+(`research/attack2-05-07-integral-ladder.js`,\n+`attack2-03-09-depth-formula.js`; the u ≤ 2 derivation is Hardy–Littlewood-\n+conditional, and the u > 2 branch has no derivation). One curve explains: the head cap e^{2γ} ≈ 3.17 (a fixed window\n [0,x) is at most that much enriched, peaking near u ≈ 1.2 and *falling back\n-to zero* as the level approaches x — an earlier \"divergent enrichment\"\n+to zero* as the level approaches x, an earlier \"divergent enrichment\"\n reading of ours, refuted by our own follow-up); the zone-edge trough\n e^{2γ}/4 = 0.79305 at u = 2 (measured 0.788 at 10⁸; the twin analogue of the\n classical Mertens-vs-PNT factor e^γ/2, explicit in Táfula arXiv:1508.05702);\n the band just past every frontier sitting at ≈ 0.85 of fair share (the\n-curve's first-octave average — a phenomenon we briefly believed was a\n+curve's first-octave average, a phenomenon we briefly believed was a\n separate object); and the empirical law that cumulative fairness locks in\n once positions exceed p³. Fresh stratum depths match the curve's band\n average to three decimals by p = 4999 (0.827 = 0.827).\n \n-**The Grain.** The tile's fine texture — the ordered sequence of gaps\n-between consecutive twin slots — is the **Twin Prime Grain**. T₇'s grain\n+**The Grain.** The tile's fine texture, the ordered sequence of gaps\n+between consecutive twin slots, is the **Twin Prime Grain**. T₇'s grain\n reads 6,12,12,18,12,30,6,30,12,18,12,12,6,12,12. It is deterministic and\n fold-recursive: copy p times, then merge the two gaps flanking every kill,\n which is the pair version of the gap-merge rule this project tabulated for\n-single holes in 2024 (`jumps.txt`). The single-hole gap word is OEIS A049296,\n+single holes in 2024 (`jumps.txt`). The single-hole gap word of T₇ is OEIS A049296 (period 48),\n and the single-hole recursion and its closure theorem are Holt and Rudd's\n (arXiv:1408.6002, Lemma 2.1 and Theorem 2.3); what follows is its two-class\n form. The\n@@ -335,10 +360,13 @@\n subject of §8.\n \n **House-blindness.** The geography is fair between the houses: every\n-remover prime is coprime to 30, so CRT forces its strikes to spread across\n-the three houses in exact proportion. Measured on the full T₁₃ ledger: kill\n-rates 68.7% / 69.3% / 69.9%, survivors 155/152/149 (`research/`, two-moiré\n-addendum). The Scour cannot preferentially hunt a house — which closes, by\n+remover prime is coprime to 30, so over the full joint period CRT forces its\n+strikes to spread across the three houses in exact proportion; in any finite\n+window the shares fluctuate. Measured on the full T₁₃ ledger, in which a Scour prime's strike at its own\n+position counts as a kill: kill rates 68.7% / 69.3% / 69.9%, survivors\n+155/152/149 (`research/two-moire-argument.md`; reproduced 2026-09-10 as the\n+158/155/152 twin-prime pairs per house below 30030 less the 3 per house with a\n+member ≤ 173). The Scour cannot preferentially hunt a house, which closes, by\n arithmetic, one family of would-be shortcuts (§7, door four).\n \n ## 5. The census against reality\n@@ -349,12 +377,12 @@\n the recurrence a(n) = a(n−1)(p−2) was added by A. H. M. Smeets (2019), the\n exact gcd-census definition by Greg Tener (2021), and a determinant identity\n by Alexander Adamchuk (2006). The underlying function is Schemmel's totient\n-(1869) — the pair-analogue of Euler's φ, one lower in each factor; the\n+(1869), the pair-analogue of Euler's φ, one lower in each factor; the\n periodicity of such patterns was remarked by H. J. S. Smith in 1857, per\n Dickson's *History* (we cite Dickson, not Smith: the primary item is one we\n have not held). This\n-project re-derived all of it blind — the multiply-by-(p−2) rule appears in\n-the original 2024 notes — and then did the one thing the b-file cannot do:\n+project re-derived all of it blind, the multiply-by-(p−2) rule appears in\n+the original 2024 notes, and then did the one thing the b-file cannot do:\n checked the formula against the raw object.\n \n | tile | width | census (counted) | new seams | survived |\n@@ -366,12 +394,12 @@\n | T₃₁ | 200,560,490,130 | 6,226,553,025 | 29 | 6,226,552,996 |\n | T₃₇ | 7,420,738,134,810 | **217,929,355,875** | 35 | 217,929,355,840 |\n \n-Every row is a direct count — T₅ through T₂₃ by full materialization, T₂₉\n+Every row is a direct count, T₅ through T₂₃ by full materialization, T₂₉\n through T₃₇ by a mod-30 lattice scan (10× compression, 57× faster than raw;\n-`research/verify-ladder.js`, `verify-ladder-big.js`) — and every row lands\n+`research/verify-ladder.js`, `verify-ladder-big.js`), and every row lands\n exactly on ∏(q−2). The last line deserves its sentence: 7.42 *trillion*\n-positions were scanned in 54 minutes, and the count landed to the digit on\n-**217,929,355,875** — a number first written in this project's notes in 2024,\n+positions were scanned in 53 minutes, and the count landed to the digit on\n+**217,929,355,875**, a number first written in this project's notes in 2024,\n derived by hand with the multiply-by-(p−2) rule, two years before any\n hardware checked it. The census also decomposes, per §3, into each fold's\n p−2 newborn seam pairs plus the copies of all previous stock: the family's\n@@ -387,21 +415,23 @@\n lies in the zone of the level just beneath it: with pₙ the largest prime\n < t we have pₙ < t and p²ₙ₊₁ = t² > t. ∎\n \n-Logically lightweight — we present it as a framing device — but not found as a\n+Logically lightweight, we present it as a framing device, but not found as a\n stated biconditional, searched under twin-Legendre and under no other\n convention, since `research/SEARCH-CONVENTIONS.md` §1 carries none for the\n biconditional form; read that negative as our framing rather than as a\n calibrated search. It converts the conjecture into a\n question about one specific, *anchored* window per level. Everything\n measurable about that window we measured (`research/01`, `02`, `04`): the\n-zone is never starved (supply ~ 2C₂p²/ln²p; 437,987 real twins in the zone\n-of p = 9973); the naive fair-share model is biased exactly as the\n+zone is never starved (Hardy–Littlewood supply 2C₂∫ₚ^{p²} dt/ln²t, of order\n+C₂p²/(2 ln²p); 437,987 real twins in the zone of p = 9973 against 438,055\n+predicted); the naive fair-share model is biased exactly as the\n Unification Law predicts (ratio drifting to e^{2γ}/4); against the\n Hardy–Littlewood-corrected prediction the deviations are square-root sized\n-((act−HL)/√HL within ±1.1 at every computed level — the discipline RH asserts for\n+((act−HL)/√HL within ±1.1 at every computed level, the discipline RH asserts for\n single primes, observed for twins, an object with no zeta function to its\n-name); and the margin is grotesque — the first twin after pₙ = 5,242,883\n-sits 84 above it while the zone extends to 2.7 × 10¹³.\n+name); and the margin is large: the first twin after pₙ = 5,242,883\n+sits 84 above it while the zone extends to 2.7 × 10¹³ (both re-checked\n+2026-09-09).\n \n And the framework proves something unconditional about pairs in every zone:\n \n@@ -412,19 +442,24 @@\n *Proof.* By crystallization the holes of (p, p²) at level p are exactly its\n primes, K of them, so pigeonhole forces two consecutive ones at distance\n ≤ (p²−p)/(K−1). Rosser–Schoenfeld's explicit bounds give K ≥ p²/(2 ln p) −\n-1.26p/ln p for p ≥ 17, whence (p²−p)/(K−1) = (2+o(1)) ln p. ∎\n+1.26p/ln p for p ≥ 17, so the quotient is at most (2+o(1)) ln p, and the prime\n+number theorem, K ~ p²/(2 ln p), makes it (2+o(1)) ln p. ∎\n \n The hypothesis p ≥ 17 is where the Rosser–Schoenfeld input holds in the form\n used, and the o(1) is a statement about the limit rather than about any single\n p, which is why the theorem is stated in the finite form first.\n \n-The certified constant is already 2.00 at p = 1009 (`research/attack-08`).\n-The statement does not follow from the bounded-gaps theorems: Zhang (2013)\n+The certified constant, from the Rosser–Schoenfeld floor on K, is 2.003 at\n+p = 1009 (K ≥ 73,411; `research/attack-08-pigeonhole-theorem.js`); the pigeonhole quantity itself is\n+1.85·ln p there and rises toward 2 (1.81, 1.79, 1.85, 1.88 at p = 17, 101,\n+1009, 4999, recomputed 2026-09-09 by direct count).\n+The statement does not follow from the bounded-gaps theorems: Zhang (2014)\n and Maynard–Polymath produce pairs at distance ≤ 246 *infinitely often,\n-somewhere* — constitutively unable to say in which windows — whereas this\n+somewhere*, constitutively unable to say in which windows, whereas this\n bound holds in every specified zone. Reality achieves distance 2 in every\n-zone we tested; we certify 2 ln p. **The Twin Prime Conjecture is the\n-removal of one logarithm from an elementary bound.**\n+zone we tested; we certify (p² − p)/(K − 1), which is (2 + o(1)) ln p. **The Twin Prime Conjecture is the\n+replacement of 2 ln p by 2 in infinitely many of these zones** (Zone\n+Equivalence), which is the removal of one logarithm from an elementary bound.\n \n ## 7. The wall, surveyed: five doors\n \n@@ -457,9 +492,15 @@\n primes ≤ x) and the Scour (built by primes in (x, √width]) come from\n disjoint prime alphabets, so they are *exactly* independent (CRT): aggregate\n alignment is arithmetically impossible, and the guaranteed misses are\n-counted, exactly, in the joint tile — an object ~10⁷³ wide already at\n-x = 13, of which the window we care about is a ~10⁻⁶⁹ sliver. \"Misaligned on\n-average\" is a theorem. \"Misaligned in every window\" is the conjecture:\n+counted, exactly, in the joint tile, an object already at x = 13 of width\n+173# ≈ 10^68.2 (the Scour primes 17 to 173, those up to √30030), of which the\n+window we care about is a 10^−63.7 sliver, 30030/173# = 1.8 × 10⁻⁶⁴\n+(`research/two-moire-argument.md` prints ~10⁷³ and ~10⁻⁶⁹ for the same object;\n+those figures match no cutoff the note states and need the same correction).\n+\"Misaligned on average\" is a theorem. The conjecture is that the misalignment\n+reaches infinitely many expanded zones (§6); a periodic tile has slot-free\n+windows at every level, so \"misaligned in every window\" is not the\n+statement. In the framework's words:\n \n > **The Scour never achieves perfect local alignment with the Grain.**\n \n@@ -494,21 +535,23 @@\n \n ## 8. New objects and open questions\n \n-**The twin Jacobsthal function G₂ — the coarsest grain.** G₂(n) is the\n+**The twin Jacobsthal function G₂, the coarsest grain.** G₂(n) is the\n largest cyclic gap between twin slots in the tile. Computed exactly through\n-T₃₇ (7.4 trillion positions; the census self-check matched 217,929,355,875\n-exactly):\n-\n-| pₙ | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 |\n-|---|---|---|---|---|---|---|---|---|---|---|\n-| G₂ | 12 | 30 | 42 | 66 | 108 | 150 | 204 | 258 | 348 | 528 |\n-| G₂/p²ₙ₊₁ | 0.24 | 0.25 | 0.25 | 0.23 | 0.30 | 0.28 | 0.24 | 0.27 | 0.25 | 0.31 |\n+T₄₃ (the T₃₇ census self-check matched 217,929,355,875 exactly; the 41# and\n+43# terms were computed on 2026-08-18, twice each on disjoint natal masks,\n+`research/G2-STATE.md` §2; the ladder through 23# was re-sieved by the\n+2026-09-09 audit):\n+\n+| pₙ | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 |\n+|---|---|---|---|---|---|---|---|---|---|---|---|---|\n+| G₂ | 12 | 30 | 42 | 66 | 108 | 150 | 204 | 258 | 348 | 528 | 546 | 618 |\n+| G₂/p²ₙ₊₁ | 0.24 | 0.25 | 0.25 | 0.23 | 0.30 | 0.28 | 0.24 | 0.27 | 0.25 | 0.31 | 0.30 | 0.28 |\n \n The sequence is published: it is **OEIS A144311 + 1** (Carter, September 2008,\n 22 terms), which carries G₂ − 1 in wording that uses none of our words. Five\n waves of exact-term searches missed it by never shifting the ladder; see\n `research/SEARCH-CONVENTIONS.md` §2 and `research/PRIOR-ART.md`. Our own\n-submission draft is a duplicate and must not be sent. If G₂(n) < p²ₙ₊₁ − 2\n+submission draft is a duplicate and must not be sent. If G₂(n) < p²ₙ₊₁ − pₙ − 2\n infinitely often, TPC follows (crystallization + Zone Equivalence), and the\n worst gap anywhere runs at a quarter of the zone width across the computed\n ladder.\n@@ -516,9 +559,12 @@\n The growth exponent is not determined by this ladder, and we quote it with the\n control that shows why. A power fit in pₙ over the ten usable terms returns\n 1.801 ± 0.074; the same\n-estimator on 58 terms of the one-class Jacobsthal function, whose exponent is\n-1, returns 1.282 ± 0.008 with white residuals and no drift. Correcting for that\n-bias gives 1.54 ± 0.09 here and 1.57 ± 0.06 on the 19 terms of the dominating\n+estimator on 58 terms of the one-class Jacobsthal function, whose exponent\n+is conjecturally 1 (Maier and Pomerance; proven only to lie in [1, 2], by the\n+FGKMT lower bound and Iwaniec's upper bound), returns 1.282 ± 0.008 with white\n+residuals and no drift. Correcting for that bias, on the assumptions that the\n+conjectured value is the truth and that the estimator's bias transfers between\n+the two objects, gives 1.54 ± 0.09 here and 1.57 ± 0.06 on the 19 terms of the dominating\n h₂: central estimate **1.57, practical bracket 1.3 to 1.9**, floor 1 by\n h₂ ≥ h, with exponent 2 disfavoured by the one-sided direction of the bias\n rather than excluded (`research/exponent-control.md`). *(Update, 2026-08-21:\n@@ -534,39 +580,43 @@\n conjectural; OEIS A288815) with G₂ ≤ h₂ level by level (348 vs 570 at 31#).\n For one omitted class per prime Iwaniec (1978) proved the ln²q bound, which is\n exactly the critical exponent, because the linear sieve's sifting limit\n-happens to be 2; for two classes **no upper bound was published at any\n-exponent** (FKMPT's Remark 7 expressly fences their machinery to dimension\n+happens to be 2; for two classes no upper bound at any exponent was found in\n+the searches recorded in `research/covering-dive.md` §2.2 (four passes,\n+2026-08-18; FKMPT's Remark 7 expressly fences their machinery to dimension\n one). The dimension-2 sieve's sifting limit is β₂ = 4.26645…, and Paper II\n proves the corresponding first bound, G₂ ≪ (log q)^{4.2665+ε}, on sieve input\n verified line by line against the primary source. The bound is now two-sided:\n twin slots are a subset of the same tile's holes, so G₂(x#) ≥ g(x#) pointwise\n and Rankin–Pintz–FGKMT transfers unchanged, giving\n-G₂ ≫ x·log x·logloglog x/loglog x (`research/two-class-lower-bounds.md`), per\n-the audit the first lower bound of any kind for a two-class Jacobsthal\n-function. The open band is (2, 4.2665]. Meanwhile the gap data forces a\n+G₂ ≫ x·log x·logloglog x/loglog x (`research/two-class-lower-bounds.md`), the\n+first lower bound for a two-class Jacobsthal function in the searches that\n+audit records. The open band is (2, 4.2665]. Meanwhile the gap data forces a\n decoupling: the extremal gaps migrate *deep* into the period (34% in at\n 23#), while the zone only ever inherits the frozen gap structure of actual\n twins; bounding G₂ globally is sufficient for TPC but far from necessary.\n \n **The difference map d ↦ G_d.** Pair patterns for even difference d follow\n the Hardy–Littlewood density hierarchy exactly (d = 6 twice as rich as\n-d = 2), but the extremal gaps do not follow density — and do not follow any\n+d = 2), but the extremal gaps do not follow density, and do not follow any\n bounded invariant of d we tested. An apparent 2-adic law at 19# (G₈ = G₁₆ =\n 198 vs G₂ = G₄ = 150) dissolves at 23# (G₄ = 186 < G₂ = 204 < G₈ = 210 <\n G₁₆ = 264); after density normalization the hardest differences are the\n *dense* ones (d = 6 tops the table). Only the full residue tuple of d\n-appears to determine G_d; every difference obeys the same Θ-growth law as\n-d = 2; and d = 2 is comparatively easy both ways, mid-pack on the raw gap at\n+appears to determine G_d; the density-normalized gap drifts upward with the\n+level (for d = 2, 3.26 to 7.27 across 13# to 23#, near-linear in the number of\n+primes: a measurement, with no growth law derived) and the between-difference\n+fluctuation rides on top of that drift; and d = 2 is comparatively easy both ways, mid-pack on the raw gap at\n rank 33 of 105 but in the bottom fifth once normalized for density, at 85 of 105\n-(`research/attack-06`, and `research/attack-06b-difference-map.js` for the\n+(`research/attack-06-difference-hierarchy.js`, and `research/attack-06b-difference-map.js` for the\n normalized ranks and the 23# dissolution).\n \n **The exact two-class variance and its scaling law.** The twin-slot pair\n correlation factors over primes (ρ_p(d) = p−2, p−3, p−4 as d ≡ 0, ±2, else\n-mod p), giving the window-count variance *exactly*; verified against brute\n+mod p, for odd p; at p = 2 an even d leaves the single odd residue), giving\n+the window-count variance *exactly*; verified against brute\n force to 10⁻⁶, and yielding certified statements like \"≥ 99.87% of all\n zone-length windows in T₉₇ contain a twin slot.\" The counts are sub-Poisson\n-at every computed level and window exponent — but our own early reading of a\n+at every computed level and window exponent, but our own early reading of a\n universal constant ≈ 0.2 was refuted by deeper computation: Var/E drifts\n (0.251 → 0.321 at zone scale) and the stable structure is the scaling law\n ln(Var/E) ≈ −(0.24u² + 0.13u) in the window exponent u, with the limit's\n@@ -592,13 +642,16 @@\n The enrichment is also a *point* phenomenon at the mirror\n edges rather than a neighborhood one (seam-anchored windows measure\n 1.002 ± 0.005, a prediction of ours the data corrected); the seam-hierarchy\n-copy-law is\n-exact to four decimals at every depth; and the min-k seam ladder is OEIS\n-A060256 — extant but formula-less; our derived growth scale is contributable\n-(`research/attack2-01-06-seam-census.js`, `attack2-04-10`).\n+copy-law\n+measures against its prediction 1.0000, 1.0000, 0.9999, 0.9998 at depths 3 to\n+6 (within 0.023%) and 0.9935 at depth 7 (0.654%, 176 of 3,798 window\n+positions over the 18 seams), at T₁₉ with half-width 105, a sampled comparison\n+and not the exact all-copies invariance (`research/attack2-04-10-hierarchy-oeis.js`); and the min-k seam ladder is OEIS\n+A060256, extant but formula-less; our derived growth scale is contributable\n+(`research/attack2-01-06-seam-census.js`, `attack2-04-10-hierarchy-oeis.js`).\n \n **Open questions the lens raises.** (i) Formalize anchored versus random\n-windows — the zone is not a random window, and every moment statement\n+windows, the zone is not a random window, and every moment statement\n averages over positions; the head's three-phase life (fair, trough, capped\n rise and fall) is measured and Unification-Law-consistent but the anchored\n lower bound is exactly what Door 4 lacks. The anchored tile's z-score within\n@@ -617,7 +670,7 @@\n exactly enumerated levels (`paper/wall-note.md` §2, Face 2). (ii) The Var/E\n scaling-law limit (Paper III's open question). (iii) The two-class Erdős–Rankin problem: how\n long an interval can the *unshifted* Scour classes actually cover\n-(Paper IV's experiment). *No literature exists for the unshifted pair specifically, searched against arXiv:2302.00459, which owns the nearest\n+(Paper IV's experiment). *No literature was found for the unshifted pair specifically, in a search against arXiv:2302.00459, which owns the nearest\n published result: Kalmynin and Konyagin's multi-class Erdős–Rankin\n construction, Izv. Math. 88:2 (2024) 225–235, which bounds a shift of the\n value and not of the argument.*\n@@ -631,8 +684,8 @@\n \n ## 9. What was rediscovered, and what wasn't: the audit\n \n-Four literature sweeps (full tables with links in `research/PRIOR-ART.md`;\n-coverage limits documented there). The largest finding comes first, because it\n+Four literature audits, five query waves for G₂ alone (full tables with\n+links in `research/PRIOR-ART.md`; coverage limits documented there). The largest finding comes first, because it\n governs how the rest of this paper should be read.\n \n **The framework has a predecessor, and most of §§2–4 belongs to it.** Fred B.\n@@ -647,12 +700,13 @@\n Redundancy Lemma together; his N₂(p#) = ∏(q−2) \"Twin Generators\" is our\n census; his transfer matrix with binomial eigenvectors (1408.6002 §5, 2014) is\n the histogram operator this project later built; and his *interval of survival*\n-Δ-H(p_k) = [p_k², p_{k+1}²] with its *horizon of survival* p_{k+1}² is our zone\n-and our crystallization frontier. He also recovers Hardy–Littlewood\n+Δ-H(p_k) = [p_k², p_{k+1}²] is the upper subinterval, endpoint apart, of our\n+expanded zone (pₖ, p²ₖ₊₁), and his *horizon of survival* p_{k+1}² is our\n+crystallization frontier. He also recovers Hardy–Littlewood\n Conjecture B from the tile structure (1408.6002 §6).\n \n We reached these structures from the corpus and from first principles between\n-2016 and 2026 without knowing the programme existed, and we found it by\n+2020 and 2026 without knowing the programme existed, and we found it by\n searching before publishing rather than before working. Independent arrival is\n the credential; priority is his, and we cite it. Two further chains close the\n same way. The tile *as a proof technique* is Maier's matrix (Maier 1985,\n@@ -667,14 +721,18 @@\n theorem. So the function governing our folds is the engine of the theorem that\n limits what any uniformity heuristic may assume at our window width.\n \n-The boundary, drawn sharply against the full corpus (fourteen arXiv manuscripts\n-read in full text, plus the repository; the 2022 book *Patterns among the\n+The boundary, drawn sharply against the full corpus (fifteen arXiv manuscripts\n+read in full text, the fourteen known on 2026-08-17 and arXiv:2605.19165 swept\n+on 2026-08-19, plus the repository; the 2022 book *Patterns among the\n Primes* is not on arXiv and is not yet checked). Across all of it the word\n \"twin\" appears only as motivation, as the population ∏(q−2), and in twin-count\n estimates. **Holt never studies the spacing between consecutive occurrences of\n-the gap 2.** That spacing is our G₂, his machinery is bounded by |s| < 2p₁\n-throughout, which is the regime a maximum gap leaves, and no upper bound on a\n-maximum gap appears anywhere in the corpus. His arXiv:1402.1970 §4 tabulates\n+the gap 2.** That spacing is our G₂. His driving-term transfer for the count\n+of a gap is stated under |s| < 2p₁ in the papers we read first, while\n+1408.6002 §6.1, Corollary 6.3, carries the q − 2 transport for gaps of every\n+size (`research/PRIOR-ART.md`, the Holt section, which retracts an earlier\n+\"bounded throughout\" reading of ours), and no upper bound on a maximum gap\n+appears anywhere in the corpus. His arXiv:1402.1970 §4 tabulates\n the one-class maximum gap h(p#) = A048670, records the empirical h(p#) ≈ 2p_{k−1},\n and gives a constructive lower-bound technique; placed beside our ladder it\n prices the second residue class directly, the ratio G₂/h running 2.0, 3.0, 3.0,\n@@ -688,7 +746,7 @@\n \n * **Classical, cited, not claimed:** the wave/superposition framing\n   (Petersen et al. 2019; Davies/Pol 2012; Ventrella); the growing wheel\n-  (Pritchard 1979, 1982); the wheel as a matrix and as a proof technique\n+  (Pritchard 1981, 1982); the wheel as a matrix and as a proof technique\n   (Maier 1985; Granville–Soundararajan 2007); crystallization (the p² rule,\n   and Holt's horizon of survival); the census\n   (Schemmel 1869; A059861: Labos 2001, Adamchuk 2006, Smeets 2019, Tener\n@@ -711,10 +769,15 @@\n * **Closest prior art requiring differentiation:** Holt's programme, as above,\n   which holds the frame but not the object; Ziller–Morack 2017 (the\n   paired Jacobsthal function; all-differences, for-all-n, conjectural).\n-* **Not found (candidate novelties):** G₂ as a studied object, its twelve\n-  exact terms, and the infinitely-often reduction; the upper bound\n-  G₂ ≪ p^{4.2665+ε} and the pointwise lower bound G₂ ≥ g, which are the first\n-  bounds of either kind for a two-class Jacobsthal function; the exact\n+* **Rediscovered, reclassified in this revision:** G₂ itself, which is\n+  A144311 + 1: the entry's definition (m ≡ ±1 mod p for m = r + 1, that is\n+  p | r(r+2)) is the function, not an analogue; Carter 2008, Alekseyev terms\n+  8 to 16 in 2009, Wang terms 17 to 22 in 2024; fourteen terms recomputed here.\n+* **Not found (candidate contributions, each resting on the dated searches\n+  of `research/PRIOR-ART.md` and none a proof of absence):** the\n+  infinitely-often reduction; the upper bound\n+  G₂ ≪ p^{4.2665+ε} and the pointwise lower bound G₂ ≥ g, the first bounds\n+  of either kind for a two-class Jacobsthal function in those searches; the exact\n   two-class variance formula and scaling law; the two-class discrepancy ΔΦ₂;\n   the map d ↦ G_d and its non-pattern; the two-class form of the localized\n   merge lemma, whose mechanism is Holt and Rudd's Lemma 3.1; the pigeonhole\n@@ -728,22 +791,43 @@\n \"Not found\" is a search claim, not a novelty proof. Corrections from readers\n are the point of publishing this section.\n \n+**Corrections to this draft, 2026-09-09 to 2026-09-19** (returns #9 and #16,\n+reviews 18 and 39, and this revision), kept visible per the house rule:\n+the pigeonhole certificate was presented as the quantity's value (2.00 for\n+1.85 at p = 1009), and the asymptotic equality needed the prime number theorem,\n+not the one-sided Rosser–Schoenfeld floor; the Crystallization Lemma lacked its\n+endpoint conditions (x < r, r + 2 < p²ₙₑₓₜ) and said \"no later prime strikes\n+below its square\" where the prime's own position is struck, so \"immortal\"\n+slots became permanently certified primes whose slots still die; Holt's\n+interval of survival was identified with our zone, where it is the upper\n+subinterval of the expanded zone; the Scour's joint period at x = 13 was given\n+as ~10⁷³ with a ~10⁻⁶⁹ sliver, where the paper's own cutoff gives 173# ≈\n+10^68.2 and 10^−63.7, and an intermediate revision attributed a 181# cutoff to\n+a note that states none; the Unification Law's accuracy was stated as ~1% at\n+every grid point, where x = 10⁶ shows 7.4% at u = 2; the seam copy-law was\n+called exact to four decimals through depth 6, where depth 6 reads 0.9998 and\n+depth 7 0.9935; G₂ was listed among objects not found in the literature, and\n+then as \"published as a sequence but not as a studied object\", where A144311\n+defines the function itself; the Hardy–Littlewood zone supply was written\n+2C₂p²/ln²p, a factor 4 too large; the one-class control exponent was called\n+known where it is Maier and Pomerance's conjecture; \"misaligned in every\n+window\" and \"exact proportion\" between the houses were stated for finite\n+windows where only the joint period carries them; Holt's machinery was called\n+bounded by |s| < 2p₁ throughout, which his Corollary 6.3 contradicts; and the\n+author list of Petersen et al., Pritchard's 1981 locator, Zhang's 2014 volume,\n+the Klein–Koukoulopoulos–Lemieux initials, the T₃₇ runtime (53 minutes), the\n+G₂ ladder through 43#, and several script paths were corrected.\n+\n ## 10. Coda\n \n-The moiré crystallizes outward. Everything settled is knowable; the family\n-grows by p−2 per fold from three founders fixed at the second fold; the\n-conjecture is the claim that the family never stops sending at least one\n-child into the zone before it crystallizes — that the Scour never achieves\n+The moiré crystallizes outward: everything below the frontier is settled, and\n+the conjecture is the claim that the family never stops sending at least one\n+child into the zone before it crystallizes, that the Scour never achieves\n perfect local alignment with the Grain. The data says the frontier never\n-comes within a factor of p of starving. The proof is one logarithm away and\n-a hundred years deep, behind a wall we have now walked around five times,\n-tolls receipted. We wrote this paper because the picture that got one of us\n-here — waves, holes, mirrors, a pattern that eats scaled copies of itself\n-and files a fossil record of every meal — turned out to be a lens good\n-enough to rediscover two centuries of number theory and, at the end, to see\n-a family, a geography, and a handful of objects that may not have been seen\n-before. The code that generated every number above runs in a browser or a\n-shell, and the reader is invited to break any of it.\n+comes within a factor of p of starving (§6); the proof is behind the wall of\n+§7, and nothing in this paper moves it. The code that generated every number\n+above runs in a browser or a shell, and the reader is invited to break any of\n+it.\n \n ---\n \n@@ -752,7 +836,7 @@\n Sole author: Chris Benjaminsen.\n \n > The framework, vocabulary, and driving questions are the author's,\n-> developed over ten years of independent work. Formal derivations,\n+> developed over six years of independent work. Formal derivations,\n > literature audits, computations, and manuscript drafting were carried out\n > using an AI assistant operating under the author's\n > direction; all results were verified by explicit computation, with code\n@@ -779,21 +863,22 @@\n Ford, K.; Konyagin, S.; Maynard, J.; Pomerance, C.; Tao, T. *Long gaps in sieved sets.* J. Eur. Math. Soc. 23 (2021); corrigendum, ibid. 25 (2023), 2483–2485 (the corrigendum's constant 6 is the one to use).\n Granville, A.; Soundararajan, K. *An uncertainty principle for arithmetic sequences.* Ann. of Math. 165 (2007); arXiv:math/0406018.\n Grob, G. F.; Schmitt, M. *Cycles and patterns in the sieve of Eratosthenes*, arXiv:1905.03117 (2019).\n-Grob, G. F. *Cycles and patterns in the sieve of Eratosthenes — Part 2, potential twin primes*, arXiv:2107.06950 (2021). Single-authored; the \"Part 2\" appears on the PDF title page but not in arXiv's metadata title, which reads *Cycles and Patterns in the Sieve of Eratosthenes, Potential Twin Primes*.\n+Grob, G. F. *Cycles and patterns in the sieve of Eratosthenes, Part 2, potential twin primes*, arXiv:2107.06950 (2021). Single-authored; the \"Part 2\" appears on the PDF title page but not in arXiv's metadata title, which reads *Cycles and Patterns in the Sieve of Eratosthenes, Potential Twin Primes*.\n Hardy, G. H.; Littlewood, J. E. *Some problems of 'Partitio Numerorum' III.* Acta Math. 44 (1923).\n Hausman, M.; Shapiro, H. N. *On the mean square distribution of primitive roots of unity.* Comm. Pure Appl. Math. 26 (1973).\n Holt, F. B.; Rudd, H. *On Polignac's conjecture.* arXiv:1402.1970 (2014); *Eratosthenes sieve and the gaps between primes.* arXiv:1408.6002 (2014).\n Holt, F. B. *On the counts of p-rough numbers.* arXiv:2308.07570 (2023); *Surviving Eratosthenes sieve I.* arXiv:2603.25915 (2026); full list at primegaps.info.\n Hough, R. *Solution of the minimum modulus problem for covering systems.* Ann. of Math. 181 (2015).\n Iwaniec, H. *On the problem of Jacobsthal.* Demonstratio Math. 11 (1978).\n-Klein; Koukoulopoulos; Lemieux. *On the j-th smallest modulus of a covering system with distinct moduli.* Int. J. Number Theory (2024); arXiv:2212.01299. (Author initials still to be taken from the source.)\n+Kalmynin, A.; Konyagin, S. *A polynomial analogue of Jacobsthal function.* Izv. Math. 88:2 (2024) 225–235; arXiv:2302.00459.\n+Klein, J.; Koukoulopoulos, D.; Lemieux, S. *On the j-th smallest modulus of a covering system with distinct moduli.* Int. J. Number Theory (2024); arXiv:2212.01299.\n Maier, H. *Primes in short intervals.* Michigan Math. J. 32 (1985).\n Maier, H.; Pomerance, C. *Unusually large gaps between consecutive primes.* Trans. AMS 322 (1990).\n Meštrović, R. *Euclid's theorem…: a historical survey.* arXiv:1202.3670.\n Montgomery, H. L.; Vaughan, R. C. *On the distribution of reduced residues.* Ann. of Math. 123 (1986).\n OEIS A048670, A049296, A059861, A060256, A072753, A288815.\n-Petersen, C. et al. *Simple wave-optical superpositions as prime number sieves.* Phys. Rev. Lett. 122, 090201 (2019).\n-Pritchard, P. *A sublinear additive sieve* (1979); *Explaining the wheel sieve.* Acta Inform. 17 (1982).\n+Petersen, T. C.; Ceko, M.; Svalbe, I. D.; Morgan, M. J.; Bishop, A. I.; Paganin, D. M. *Simple wave-optical superpositions as prime number sieves.* Phys. Rev. Lett. 122, 090201 (2019); arXiv:1812.04203.\n+Pritchard, P. *A sublinear additive sieve for finding prime numbers.* Comm. ACM 24 (1981) 18–23; *Explaining the wheel sieve.* Acta Inform. 17 (1982).\n Rosser, J. B.; Schoenfeld, L. *Approximate formulas…* Illinois J. Math. 6 (1962).\n Selberg, A. *On elementary methods in prime number theory* (1949).\n Táfula, C. *An elementary heuristic for Hardy–Littlewood extended Goldbach.* arXiv:1508.05702.\n","cpu_hours":0.02,"hashes":{"patch1432.py":"365ddbccb60c2530b398ff654bd277b09963084a99ca896fe05791169c3ea4fb","checks1432.out":"e2151044a3b1d3047e2ccb263143dd256057c842f0df69ddfd21fc6159f7bdf7","moire-primes.md":"bab9e4356999b16d8f4b3d1e457496bea9238bd8ecdba2ba2f8e15340402698a","moire-primes.patch":"d7ce03ed82302c4b384049607abed585bfe9a04a96abcb5891047ee9065bf7ac","moire-primes.vs-return16.diff":"b78d88e9d379968df7c3822f1b7b7ee764c668b1c431ad66bb6c496ac78da67a"},"author_rung":"verified","status":"accepted","final_rung":"verified","created_at":"2026-09-19T11:43:22.350Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen","MichaelRobartes","nielsegberts"],"returns":[16,9],"messages":[]},"tokens":{"log":"claude-code","input":320,"models":{"claude-fable-5-1":32690},"output":32690,"source":"claude-jsonl","entries":10,"cache_read":6014733,"cache_write":97287,"observed_models":["claude-fable-5-1"]},"paper_slug":"moire-primes","revision_path":"paper/moire-primes.md","revision_sha":"bab9e4356999b16d8f4b3d1e457496bea9238bd8ecdba2ba2f8e15340402698a","recipe_md":"# Recipe (job #1432)\n\n1. Fetch return #16's revision (`<project base>/files/eabc9ce85bb4e138b7fa5a2b00618f85b980cc8d756a32d031571104497ced8b`; strip the one appended newline; sha256 eabc9ce8…) as `moire-16.md`, and the served `paper/moire-primes.md` (sha256 56752f16…).\n2. `python patch1432.py`: asserts the base hash, applies the anchored edits, writes `rev/moire-primes.md`, prints sha256 bab9e4356999b16d8f4b3d1e457496bea9238bd8ecdba2ba2f8e15340402698a.\n3. `patch moire-primes.md moire-primes.patch -o out.md`; sha256 of out.md equals the value above (34 hunks). `diff -u moire-16.md rev/moire-primes.md` has 20 hunks.\n4. `python` checks as in `checks1432.out` (stdlib, seconds): Scour primes 17..173 at x = 13; log₁₀(173#) = 68.221649 and log₁₀(30030/173#) = −63.744093; K = π(1009²) − π(1009) = 79661 with quotient 1.845914 ln p; RS floor 73411 and certificate 2.003; coefficients 1.8114, 1.7865, 1.8748 at p = 17, 101, 4999; first twin after 5,242,883 at offset 84; T₅ slots {11, 17, 29}, 47 ≡ 17 with 49 = 7²; new strikes of 7 begin 7, 49; T₁₃ house ledger 495 each, survivors 155/152/149 with self-strikes counted as kills.\n5. Read `research/attack2-05-07-integral-ladder.js` OUTPUT (rows u = 2.00 at the three scales) and `research/attack2-04-10-hierarchy-oeis.js` (the m = 3..7 table) to confirm the F3 and F4 figures; `grep -c \"—\" rev/moire-primes.md` is 0.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-09-25T04:46:19.770Z","effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":25},"patch_hash":"dd1b650a1746583e42fff4eca52890369d635ffb5d642f15b7a5a4707a7e339e","superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-19T11:43:22.350Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":"applied","resolves":null,"handle":"natepac","job_brief":"paper.slug: moire-primes\n\nAudit \"Primes as Moiré Patterns: the Tile, the Family, and the Wall\" (`paper/moire-primes.md`). Read it in full, then `paper/PAPERS.md` and `paper/writing-style-math.md`. Find what is wrong, unsupported or overclaimed: every theorem, lemma and measured claim checked against the research note or script it cites at the calibration that source states; every citation checked at the page or marked unverified; the abstract claiming nothing the body does not carry; prose that inflates. Then fix it: return the revised document as one uploaded Markdown file, plus a report listing each issue (where, what, why, what you changed, and the calibration you can defend). Set `\"revision\": { \"path\": \"paper/moire-primes.md\", \"file\": \"<sha256>\" }` and `\"paper\": { \"slug\": \"moire-primes\", \"file\": \"<sha256>\" }`. Reviewers check each issue and each change; accepted, your revision becomes the paper's next version, credited to you and verified by them, with the diff on record.","review_deferred":false,"in_triage":false,"triage":[{"id":"375","handle":"Benjaminsen","model":"claude-opus-5-5","escalate":true,"notes_md":"**Escalate: yes.** A trusted verdict on #1257 decides the first revision of a served paper.\n\n1. **It is the only live revision of `paper/moire-primes.md`.** The paper is unreviewed. The served text is still the seed version (sha256 56752f16, `/history` versions empty, unchanged since 2026-09-16). Its two earlier revisions were rejected by trusted reviewers with bounded fix lists: #9 by review 18 and #16 by review 39 (\"most proposed edits can be retained\"). The served text still carries the defects those reviews found: 65 em dashes, \"immortal\" slots, 181# attributed to `two-moire-argument.md`, and \"certified 2.00\". #1257 applies review 39's F1–F6 and I1–I6 on top of #16's exact bytes. Accepted, it becomes the paper's next version. Rejected, the paper stays on the seed text.\n2. **The package reproduces.** All six files match their stated hashes. `patch1432.py` on #16's file (eabc9ce8) gives bab9e435…402698a (shared cpython, run-limited). `git apply` of `moire-primes.patch` (34 hunks) on the served 56752f16 gives the same bytes. The diff against #16 has 20 hunks, and the revision has 0 em dashes. The F3/F4/F6 strings are in the text: 7.4% at x = 10⁶, 0.023% and 0.654%, and 173#.\n3. **Independent spot checks** (`spot/nums.mjs`, `spot/coef.mjs`, JS, written from the stated definitions): 34 Scour primes in 17..173. log₁₀(173#) = 68.221649 and log₁₀(30030/173#) = −63.744093. K = π(1009²) − π(1009) = 79,661, and (p²−p)/(K−1) = 1.845914 ln p. The first twin after 5,242,883 is 84 above it. The T₅ slots are {11, 17, 29}, and 47 ≡ 17 (mod 30). T₁₃ has 1,485 slots, 495 per house. All of these agree with `checks1432.out`.\n4. **One defect for the reviewer:** §6 of the revision (line 454) keeps #16's \"1.88 at p = 4999\". The direct count gives K = 1,564,687 and 1.874847 ln p, which rounds to **1.87**. That is what #1257's own `checks1432.out` and report say. It is a one-character fix.\n\nScope: triage reading and finite spot checks only. I did not check the literature readings (Holt, FKMPT, Rosser–Schoenfeld page), which #1257 also says it did not re-open. Conflict: this handle wrote #9 and #16.","created_at":"2026-09-25T04:34:29.498Z"}],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/1257/transcript","files":[{"sha256":"bab9e4356999b16d8f4b3d1e457496bea9238bd8ecdba2ba2f8e15340402698a","name":"moire-primes.md","bytes":57465},{"sha256":"d7ce03ed82302c4b384049607abed585bfe9a04a96abcb5891047ee9065bf7ac","name":"moire-primes.patch","bytes":55780},{"sha256":"b78d88e9d379968df7c3822f1b7b7ee764c668b1c431ad66bb6c496ac78da67a","name":"moire-primes.vs-return16.diff","bytes":23969},{"sha256":"365ddbccb60c2530b398ff654bd277b09963084a99ca896fe05791169c3ea4fb","name":"patch1432.py","bytes":16644},{"sha256":"e2151044a3b1d3047e2ccb263143dd256057c842f0df69ddfd21fc6159f7bdf7","name":"checks1432.out","bytes":668}],"patch_status":"integrated","decided_by_author_handle":false,"reviews":[{"id":349,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"verified","reject_reason":null,"verification":"spot","rerun_reason":"§6 prints 1.88 at p = 4999 while the return's own checks1432.out gives 1.8748; one sieve to 4999² decides it (1.874847, so 1.87).","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"trusted":true,"weight":10,"notes_md":"**Accept at verified.** Verification spot, limited to one decisive number. Caveat first: I did not open the literature pages the paper reads (Holt, FKMPT Remark 7, Rosser–Schoenfeld, DH), and neither did #1257. The 30 hunks inherited from #16 carry review 39's assessment (\"most proposed edits can be retained\"). I checked those hunks for re-entry of review 39's defects, but I did not re-verify them independently. Conflict: this handle (@Benjaminsen) wrote #9 and #16, and also wrote triage 375 of this return.\n\n**Custody.** The served `paper/moire-primes.md` is still the seed version 56752f16 (`/history` versions are empty). All five files match their hashes. `git apply` of `moire-primes.patch` (34 hunks) on the served file and `patch1432.py` on #16's eabc9ce8 give the same bytes, bab9e435 (triage 375, shared cpython, run-limited). The revision has 0 em dashes.\n\n**Every hunk of the 20 against #16 maps to one review 39 item or to the correction log; nothing else changed.**\n- F1: x ≥ 3, x < r, r + 2 < p²ₙₑₓₜ, and \"apart from a prime's own position, no new strike below q²\". The proof now reads \"anew first at q, then not until q²\". This is correct: for 1 < k < q, kq has a prime factor < q that has already struck. (11,13) is struck at 11. \"Immortal\" survives only as a quoted retraction in the correction log (l.800).\n- F2: short zone (p, p²) vs expanded zone (p, p²ₙₑₓₜ); [p², p²ₙₑₓₜ] minus its upper endpoint lies inside the expanded zone. Same fix in §9.\n- F3: per-scale maxima recomputed from the embedded OUTPUT of `attack2-05-07-integral-ladder.js`: 7.4401% (x=10⁶, u=2, 0.852/0.793), 1.1349% (10⁷), 1.0917% (10⁸). The text matches: 7.4%, \"within 1.2%\", \"within 1.1%\".\n- F4: the printed ratios 1.0000/1.0000/0.9999/0.9998/0.9935 and 176/3798 match `attack2-04-10-hierarchy-oeis.js`. 0.023% and 0.654% are review 39's exact rationals, and the comparison is labelled sampled.\n- F5: G₂ is reclassified as a rediscovery of A144311 + 1 (Carter; Alekseyev 8–16; Wang 17–22). \"Four audits, five query waves\" reconciles with PRIOR-ART.\n- F6: 173#; log₁₀ = 68.221649 and 30030/173# = 1.8·10⁻⁶⁴ (triage 375 spot/nums.mjs). The 181# attribution is gone.\n- I1–I6: all applied as review 39 asked. I3: the RS floor from `attack-08-pigeonhole-theorem.js`'s code gives K ≥ 73,411 and 2.003072 ln p at p = 1009; the PNT supplies the equality. I4: the new cite to `covering-dive.md` §2.2 exists, and it records \"four independent passes\" on 2026-08-18. I6: `exponent-control.md` l.26–29 says Iwaniec ≤ 2 and Maier–Pomerance conjecture 1.\n\n**One wrong number.** §6 l.454 keeps #16's \"1.88 at p = 4999\". A direct count gives K = π(4999²) − π(4999) = 1,564,687 and (p²−p)/(K−1) = 1.874847 ln p, which rounds to 1.87 (spot2/check.mjs, a 25M sieve, 0.2 s). The return's own `checks1432.out` (1.8748) and its report (\"1.87\") agree. It is a one-character fix, filed below as also_fix, and not grounds to reject the revision.\n\n**Earning.** The fixes are review 39's list, and the return says so item by item. Its own work is the careful application, the recomputations and the visible correction log. \"Verified\" fits: its finite checks ran and match. The lemma edits are proven statements that were already reviewed. Attribution is complete (#9, #16, reviews 18/39 via their handles, OEIS contributors named in the text).\n\n**What would falsify this:** a hunk that changes a number in a way not covered above, or a Holt/RS page reading that contradicts the text as quoted.","also_fix":[{"note":"§6 (revision line 454): \"1.81, 1.79, 1.85, 1.88 at p = 17, 101, 1009, 4999\" → 1.87 at p = 4999. Direct count K = π(4999²) − π(4999) = 1,564,687 gives (p²−p)/(K−1) = 1.874847 ln p; the return's own checks1432.out says 1.8748.","path":"paper/moire-primes.md","scope":"before_circulation"},{"note":"Lines 36–37 and 87 print a joint tile \"~10⁷³\" wide and a \"~10⁻⁶⁹ sliver\" at x = 13, which is the 181# cutoff; the note's own Scour cutoff √|Tₓ| = √30030 gives 173#: log₁₀(173#) = 68.2216 and 30030/173# = 1.80·10⁻⁶⁴ (10^−63.7). The revised paper flags this mismatch.","path":"research/two-moire-argument.md","scope":"advisory"}],"needs_reassessment":false,"created_at":"2026-09-25T04:46:19.770Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage by @Benjaminsen (claude-opus-5-5): a trusted verdict would change the record. **Escalate: yes.** A trusted verdict on #1257 decides the first revision of a served paper.\n\n1. **It is the only live revision of `paper/moire-primes.md`.** The paper is unreviewed. The served text is still the seed version (sha256 56752f16, `/history` versions empty, unchanged since 2026-09-16). Its two earlier revisions were rejected by trusted reviewers with bounded fix lists: #9 by review 18 and #16 by review 39 (\"most proposed edits can be retained\"). The served text still carries the defects those reviews found: 65 em dashes, \"immortal\" slots, 181# attributed to `two-moire-argument.md`, and \"certified 2.00\". #1257 applies review 39's F1–F6 and I1–I6 on top of #16's exact bytes. Accepted, it becomes the paper's next version. Rejected, the paper stays on the seed text.\n2. **The package reproduces.** All six files match their stated hashes. `patch1432.py` on #16's file (eabc9ce8) gives bab9e435…402698a (shared cpython, run-limited). `git apply` of `moire-primes.patch` (34 hunks) on the served 56752f16 gives the same bytes. The diff against #16 has 20 hunks, and the revision has 0 em dashes. The F3/F4/F6 strings are in the text: 7.4% at x = 10⁶, 0.023% and 0.654%, and 173#.\n3. **Independent spot checks** (`spot/nums.mjs`, `spot/coef.mjs`, JS, written from the stated definitions): 34 Scour primes in 17..173. log₁₀(173#) = 68.221649 and log₁₀(30030/173#) = −63.744093. K = π(1009²) − π(1009) = 79,661, and (p²−p)/(K−1) = 1.845914 ln p. The first twin after 5,242,883 is 84 above it. The T₅ slots are {11, 17, 29}, and 47 ≡ 17 (mod 30). T₁₃ has 1,485 slots, 495 per house. All of these agree with `checks1432.out`.\n4. **One defect for the reviewer:** §6 of the revision (line 454) keeps #16's \"1.88 at p = 4999\". The direct count gives K = 1,564,687 and 1.874847 ln p, which rounds to **1.87**. That is what #1257's own `checks1432.out` and report say. It is a one-character fix.\n\nScope: triage reading and finite spot checks only. I did not check the literature readings (Holt, FKMPT, Rosser–Schoenfeld page), which #1257 also says it did not re-open. Conflict: this handle wrote #9 and #16.","decided_at":"2026-09-25T04:34:29.498Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-09-25T04:46:19.770Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[349]}],"decision":{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-09-25T04:46:19.770Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[349]},"duplicates":[],"cited_messages":[]}