# Search conventions: how to look for prior art on these objects **Read this before running any literature or OEIS search, and before writing any sentence of the form "no literature exists", "ABSENT", "first", or "nobody has".** For the active campaign, read [RESEARCH-HANDOFF.md](RESEARCH-HANDOFF.md) section 3 before a theorem search: it gives the complete residual and centered prime–Mobius consumers. Sections 4–5 retain the regional small-common-divisor kernel. [RESEARCH-EXECUTION.md](RESEARCH-EXECUTION.md) records candidate next source checks; its leads are not new theorem imports. A headline saving cannot be multiplied into the moment without matching its hypotheses and pricing separation, norms and tails. This file exists because of one failure, repeated for five audit waves. Every search this project ran was calibrated, was a genuine clean negative, and was worthless — because it was run in *our* vocabulary. The object we call `G₂` has been in OEIS since **September 2008** under wording that contains none of our words. The searches were not sloppy. They were in the wrong language. > **The rule.** A clean negative proves nothing until you have searched in the > convention that OWNS the object. Ours is never that convention. Find the > owning convention first; search it; only then may you write an absence. --- ## 1. The owning conventions, by object The **owning convention** is the wording the literature actually uses. Search that. Our name and the textbook name are both starting points, not endpoints. | object | our name | canonical | **OWNING convention — search THIS** | where it lives | |---|---|---|---|---| | Liouville parity of both members on the y-rough pair set; conditional independence of lambda(n+2) from the primality of n | the parity table, (Cov_u), (Dec_1) | correlations of the Liouville function along shifted primes and almost primes | **Liouville function on shifted primes**, **lambda(p+2)**, **parity phenomenon**, **Chen's double sieve**; the pair-constant history is Selberg 16/8, Bombieri–Davenport 8/4, Chen 7.8342, Wu 3.3996 (Goldbach/twin) | Wu arXiv:0705.1652v1 pp. 2-4 read at page image; Murty "Twin primes and the parity problem" UNREAD (404); Hildebrand's lambda(p+2) results UNREAD. `fold-arithmetic-bridge.md` §3 | | distribution of k-fold rough products in progressions at level 1/2, and shifted-prime contamination of almost-prime partners | inputs of the parity-table pricing | Bombieri–Vinogradov for almost primes; Chen's switching principle | **mean value theorem of Pan and Ding**, **Bombieri–Vinogradov for products of primes**, **switching principle**, **well factorable weights** (Wu Lemma 2.9 for the box form at level 4/7) | Wu arXiv:0705.1652v1 p. 6 Lemma 2.2, 2.3 read 2026-09-08 (Lemma 2.3 is squarefree-only, weight mu(q)^2 3^nu(q); read as uniform over bounded coefficients); re-read at printed page 6 on 2026-09-09; contamination constant 4 is aggregate over factor tuples, not pointwise in a cofactor; Motohashi 1976 induction principle UNREAD (not needed); no novelty claim: `fold-arithmetic-bridge.md` §3a | | primes in progressions to moduli near and beyond x^(1/2) with a Mobius or bilinear weight on the modulus, fixed residue | the band of D_y | Bombieri–Friedlander–Iwaniec / Fouvry range; Maynard 2021 | **primes in arithmetic progressions to large moduli**, **fixed residue class**, **dispersion method**, **well-factorable**, **convenient-sized factor**, **Titchmarsh divisor problem** | Maynard arXiv:2006.06572v2 read (Thm 1.1, Cors 1.2–1.4, §3, §8); BFI II Thms 3, 5* read only as restated there; BFI I, BFI III, Fouvry 1985 unreached 2026-09-08 (paywall, 403, GDZ 404); Drappeau 1504.05549v4 read for the Titchmarsh statement. No theorem for the twisted sequence Lambda(n-2)mu(n) located on these channels: `centered-discrepancy-estimate.md` §4; the top-range proof was repaired and reviewed on 2026-09-08 (§3a); it uses only moduli below x^(1/2-eps) and does not touch the band. Second checkpoint 2026-09-08: one stronger sufficient input for the band in the modulus arrangement is the one-class absolute-value statement (4.9) of `fixed-endpoint-discrepancy.md` at level x^(1/2+eps'); no row of the existing matrix states it. The 2026-09-09 review accepts the repaired low Type I piece after logarithmic g-truncation, not an estimate for this band. | | the smooth-part factor of the complete Vaughan coefficient as a superposition of multiplicative functions | D_i, F_i(s_i(n)), the full-coefficient aggregate | Mellin/Fourier separation of a sieve weight; two-point correlations of divisor-bounded pretentious multiplicative functions | **correlations of multiplicative functions**, **pretentious multiplicative functions**, **divisor-bounded multiplicative functions**, **linear correlations of multiplicative functions**; the class is 2-bounded at primes, x-dependent, one variable, shift 2; no matched signed estimate | Klurman arXiv:1603.08453v1 Thms 1.3/1.5 (1-bounded, fixed); Matthiesen arXiv:1606.04482v4 Defs 1.1-1.2, Thm 2.4 (s>=2 variables); TT 2512.01739v2 3.1(ii); KMT 2304.05344v2, Mangerel 2108.11401v2, arXiv:2603.23250v2 abstracts; Roy-Savalia-Vatwani submitted, UNREAD. Read 2026-09-08; scoped negative, no absence claim: `full-coefficient-average.md` §4. | | the complete residual as a composite-filtered correlation of x-dependent multiplicative functions; the both-rough composite pair sum at exponents (0.24, 0.05) | R_00, T_(k,l)(u,u'), the composite filter | two-point correlations under a prime/composite filter; sieve of dimension two at fixed sifting exponents; almost-prime twins | **correlations of multiplicative functions**, **shifted primes**, **sieve of dimension two**, **almost-prime twins**, **Buchstab iteration**, **fundamental lemma** | Tao–Teravainen 2512.01739v2 3.1(ii) (admits rough indicators, not the filter; rate log^(-d)); Klurman 1603.08453v1 Thm 1.3/1.5 (error term unbounded on the family, reread 2026-09-08); no fixed-exponent both-rough pair asymptotic found. Scoped negative, reviewed: `full-coefficient-average.md` §6.3–6.4. Correction 2026-09-09: some low nonzero modes also meet the unit-disc condition; the composite filter and required rate remain unmatched. | | the actual Mobius-weighted prime progression discrepancy after a squarefree switch | centered parity consumer | shifted-Mobius equidistribution; divisor switching with a moving endpoint | **Mobius function over shifted primes**, **fixed-residue equidistribution**, **squarefree filtering**, **divisor switching**, **twin primes and the parity problem** | [moving-cutoff-parity.md](moving-cutoff-parity.md), checked 2026-09-06 before calculation. Murty--Vatwani JNT 180 Theorem 1.1, p. 647 remark, sections 3--5; Vatwani author preprint dated 2018-10-02, Theorems 1.2/1.5, H1--H2 and selected proof steps. Positive prior-art matches. The missing endpoint in published p. 654 is verified and repaired locally. Title plus correction/erratum searches were scoped, not exhaustive; no priority claim. | | prime Bombieri–Vinogradov with a supremum over prefixes t<=x | (BV*) in centered-discrepancy-estimate.md §3a.3 | Bombieri–Vinogradov, prefix (maximal) form | **"max over y<=x"**, **"uniformly in y"**, Davenport ch. 28, Iwaniec–Kowalski Thm 17.1 | Tao Notes 3 Theorem 17 is terminal-point; his Exercise 20 states the prefix form with a rounding hint; derived locally from Theorem 17 on 2026-09-08 and reviewed. Davenport and IK pages not opened. | | finite tables of Mobius or Liouville at shifted primes, of Lambda(n-2)mu(n) in progressions, and of two-point Chowla sums | the D_y and M censuses | numerics for the parity input | **Mobius on shifted primes**, **Chowla two-point** + numerics/computation/table, **Elliott--Halberstam twisted by Mobius**, OEIS `A008683(A000040(n)+2)` / `A008836(A000040(n)+2)` and their partial sums | Searched 2026-09-07 before [centered-discrepancy-measurement.js](centered-discrepancy-measurement.js) and [shifted-prime-mobius-sums.js](shifted-prime-mobius-sums.js) ran: no published or tabulated values above x=10^4 for any of these objects (Carella arXiv:2206.12956 has shift-1 Liouville at 10^4; Luo--Ye arXiv:2401.18082 has unweighted shift-only sums to 10^8; Humphries--Shekatkar--Wong arXiv:1704.07979 is Liouville on primes in progressions, not shifted primes); OEIS has sign-pattern sequences only. Report: [history/reviews-0907/01](history/reviews-0907/01-shifted-prime-data-search.md). A negative inside the owning convention; not an absence proof. | | completing a factor subset weight with its prime and complementary terms | joint correction | vector subset convolution; Vaughan approximants; complementary sieve sums | **prime-producing sieves**, **vector convolution**, **Type I and Type II sums**, **asymptotic sieve for primes**, **signed factor-product remainders**, **smooth-number lower-bound sieve** | [joint-correction-source-audit.md](joint-correction-source-audit.md), checked 2026-09-06 before calculation. Ford--Maynard Definition 7.1, Proposition 7.19 and the final step of Proposition 7.22; Mounier 2402.13198v3 Theorem 1.1, Remark 1.2 and Lemma 3.8; Friedlander--Iwaniec math/9811186 (R), (B), Theorem 1; Tao's Vaughan identity/BV. Positive matches. Full reconstruction returns the existing open remainder, not a new estimate; no novelty claim. | | signed harmonic subset coefficients and their weighted sieve integrals | whole supported coefficient | harmonic Möbius sums; Dickman convolution; sieve integrals | **Dickman integral expansion**, **convolution exponential**, **weighted sieves with switching**, **polynomial weights over polytopes**, **LattE sieve integrals** | [supported-coefficient-dickman.md](supported-coefficient-dickman.md), checked 2026-09-06. Soundararajan 1005.3494v1 p. 1 supplies the classical identity; Drappeau--Mounier 2606.30428v1 Theorem 1.1 and Remark 2.4 supply an existing integration method; Matomäki--Zuniga Alterman published section 4 already handles additive signed weights. Positive matches; literature novelty is unestablished. | | factor laws for the remaining prime-partner correction | prime-cofactor exclusion | large prime factors of shifted primes; well-distributed sequences | **Poisson--Dirichlet law**, **factorial correlation functions**, **restricted support**, **level of distribution**, **large prime factors of well-distributed sequences** | [supported-coefficient-dickman.md](supported-coefficient-dickman.md) section 6. Bharadwaj--Rodgers 2026 Proposition 2, Theorem 7 and Lemma 8, with proof step (16), checked 2026-09-06. Total exponent <1/2 does not directly allow marking our large cofactor; the full law needs an unproved stronger input for shifted primes. Not a universal obstruction. | | averaging small-factor products before a prime progression estimate | paired factor budget | generalized Bombieri--Vinogradov theorem; weighted sieves with switching | **Pan--Ding mean value theorem**, **bounded cofactor coefficients**, **average inside the absolute value**, **Chen weights**, **Buchstab decomposition** | [paired-factor-budget.md](paired-factor-budget.md), checked 2026-09-06: Wu 2004 published Lemma 2.3 and (10.6)--(10.8), including continuation pages. Aggregate level x^(9/20) matches both families; their net plus the main term is insufficient. No novelty claim. | | the Type I part of a factor family of the C3 residual, with every right partner admitted | joint factor family, R_F, Z_F^p | Vaughan's identity main term; level of distribution of smooth-times-prime sequences; the twin singular series as a twisted Mobius mean | **Pan–Ding mean value theorem**, **Bombieri–Vinogradov for sequences with a smooth cofactor**, **Landau's Mobius bound**, **sum mu(d)log(D/d)/phi(d)** | [joint-factor-estimate.md](joint-factor-estimate.md), checked 2026-09-08 and reviewed 2026-09-09: Tao Notes 3 Theorem 17 pays Type I at all moduli per cofactor; Wu Lemma 2.3 is squarefree-only and used only for the partner sieve. The uniform Dickman error is 12/11! for delta<1/50; 24/22! is specialized to delta=1/100. Positive matches; the family's Type I part is exactly compensated in the complement; no novelty claim. | | the block with the right coefficient's prime-power factor held outside the Cauchy inequality | structured dispersion, (D1), Lemma H | dispersion method with a short modulus factor outside Cauchy–Schwarz; harmonic gcd sums; completion with composite-modulus Weil bounds | **dispersion method**, **Cauchy–Schwarz with a fixed factor**, **gcd sums over arithmetic progressions**, **Kloosterman completion** | [structured-dispersion-estimate.md](structured-dispersion-estimate.md), 2026-09-08: no literature search run in that pass; elementary inputs only; completion bound reused from grouped-divisor-moment (7) (Pascadi Lemmas 3.2–3.3). Re-read Pascadi Lemmas 3.2–3.3 in the primary text 2026-09-09 and reconstructed completion; corrected local moment saving is greater than 7/200. No novelty or absence claim. | | signed factor families after localization | negative-mass constant budget | weighted sieves with switching; prime and almost-prime pairs | **role reversal**, **linear sieve**, **level of distribution of the switched sequence**, **rough numbers with a fixed factor count**, **Kuhn and Richert weights** | [switching-negative-mass.md](switching-negative-mass.md), checked 2026-09-06. Matomäki--Zuniga Alterman v3 and published Lemma 2.5, Assumption 3.1, sections 5 and 8; ordinary BV supplies the particular cofactor progression input. The actual negative-only target is refuted at the present cutoffs; the net signed margin is open. | | concentration of the complete residual away from very small prime factors | paid small-prime tail | anatomy of integers under smooth sieve weights; shifted nonnegative arithmetic sums | **smallest prime factor**, **smooth sieve weight moments**, **quadratic sieve optimality**, **Goldston--Yildirim correlations**, **Buchstab decomposition**, **switching** | [smooth-sieve-literature.md](smooth-sieve-literature.md), read 2026-09-06, matches GKM section 10.3, CCHM Theorem 1.2, Green--Tao Appendix D and Goldston--Yildirim III section 8. The new local application reuses corrected Henriot; signed factor densities remain open. The switching and paired-factor follow-ups match specific distribution inputs; no full signed theorem is supplied. | | full absolute mass after changing the initial averaging profile | C3 global majorant | smooth truncated divisor sums; nonmultiplicative arithmetic upper bounds | **Nair--Tenenbaum bounds**, **Henriot erratum**, **M_k growth class**, **smallest prime factors**, **absolute moments of sieve weights**; the general upper theorem differs from multiplicative Euler-product corollaries | Henriot arXiv:1102.1643v1 definition of M_k and the 2014 erratum, New Theorem 5, pp. 375 and 377, checked 2026-09-06. The three-prime envelope is matched in [global-smooth-majorant.md](global-smooth-majorant.md); it gives full absolute O(x), not a signed margin or novelty claim. | | the second moment of the whole expanded-divisor coefficient | full grouped moment | rational-frequency energy; common-divisor decomposition; incomplete Kloosterman sums | **gcd sums**, **rational frequency collisions**, **completion**, **Ramanujan sums**, **composite-modulus Weil bound**; distinguish the zero numerator from nonzero multiples of the modulus, and retain complete periods | `grouped-divisor-moment.md` derives the moment and exact added cut from [Pascadi, Lemmas 3.2--3.3](https://link.springer.com/article/10.1007/s00039-026-00746-0), checked 2026-09-06. The newer bilinear theorems are not imported; no novelty or absence claim. | | cross-divisor endpoint moment and its possible arithmetic input | structures behind the remaining wall | rational-frequency energy; dispersion coefficients; Kloosterman bilinear forms; centered divisibility operators | **gcd/lcm normalization**, **large sieve with dispersion coefficients**, **quadratic characters and Kloosterman sums**, **deamplification**, **Möbius trace-function sums**, **expansion divisibility parity**, **quantitative logarithmic correlations** | [structural-literature-audit.md](structural-literature-audit.md), checked 2026-09-06, records primary-source statements, versions and transfer gaps. Its exact kernel and collision count are derived locally. No sufficient imported estimate, novelty or literature-absence claim. | | full-domain coverage with coupled divisor cuts | residual coverage | truncated Perron inversion; dispersion method; correlations of truncated divisor sums | **Mellin separation**, **uniform coefficient twists**, **cross-divisor correlations**, **zero-frequency terms**; require uniformity in the integration heights before separating a product cut | `residual-coverage.md` derives the first-branch and prime-free moments and gives its truncated Perron error explicitly. It reuses the completion inputs below and introduces no stronger imported theorem, novelty or absence claim. | | dispersion with repeated distinguished prime factors | prime-power dispersion | rational frequencies with prime-power denominators; incomplete Kloosterman sums | **prime-power gcd sums**, **frequency energy**, **completion**, **dyadic decomposition in the modulus**; constants must remain uniform when the power grows | `prime-power-dispersion.md` derives the coefficient split and power-band moment. [Pascadi, Lemmas 3.2--3.3](https://link.springer.com/article/10.1007/s00039-026-00746-0) and [Bettin--Chandee](https://arxiv.org/pdf/1502.00769) are the reused classical inputs, checked 2026-09-05. No new imported theorem or novelty claim. | | sparse coefficients with prime/harmonic frequency collisions | sparse dispersion | rational-frequency energy; dispersion method | **frequency collisions**, **gcd sums**, **paired endpoint weights**, **incomplete Kloosterman sums**; h1/q1=h2/q2 can hold for distinct primes once h reaches the primes | `sparse-dispersion.md` derives collision counts and the norm-sensitive moment bound. [Pascadi, Lemmas 3.2--3.3](https://link.springer.com/article/10.1007/s00039-026-00746-0) reread 2026-09-05 supply only the classical complete-sum inputs. No new bilinear theorem, novelty or absence claim. | | extending the prime second moment across divisor sizes | dispersion range | reciprocal Kloosterman fractions; averaged gcd sums; dispersion method | **inverse reciprocity**, **Cauchy-Schwarz**, **gcd sums**, **completion**, **sparse coefficient norms**; averaging a nonzero gcd loss is distinct from a stronger pointwise Weil theorem | `dispersion-range.md` derives the average and checks both coefficient orientations. Classical inputs are [Pascadi, Lemmas 3.2--3.3](https://link.springer.com/article/10.1007/s00039-026-00746-0) and [Bettin--Chandee, Theorem 1 and Remark 1](https://arxiv.org/pdf/1502.00769), reused and checked 2026-09-05. No novelty or literature-absence claim. | | the second moment of the localized prime/harmonic average | prime dispersion | dispersion method; incomplete Kloosterman sums | **Cauchy-Schwarz**, **completion of sums**, **Ramanujan sums**, **Weil bound for composite moduli**; keep the actual common modulus, gcd losses and endpoint variation | [Pascadi, Lemmas 3.2 and 3.3](https://link.springer.com/article/10.1007/s00039-026-00746-0), read 2026-09-05, supply the classical complete-sum inputs. `prime-dispersion.md` derives interval completion and the second-moment application to the pilot; no short-coefficient bilinear theorem, novelty or absence claim. | | the localized signed prime-factor endpoint sum after completion | prime-band completion | complete Kloosterman sums and finite Fourier inversion | **bilinear forms with Kloosterman sums**, **non-abelian amplification**, **completion**, **operator norms**, **dispersion method**; distinguish inverse fractions from complete sums and full dual support from short bilinear ranges | [Pascadi](https://link.springer.com/article/10.1007/s00039-026-00746-0), Theorems 1.1, 1.2, 7.1 read 2026-09-05. `prime-band-completion.md` derives the exact completion and full-frequency Gram matrix; the displayed theorem interface does not supply a saving for its coupled coefficients. No theorem-absence or novelty claim. | | the difference of two CRT sawtooths on a long interval | paired endpoint budget | Fourier coefficients of a short arc; trigonometric majorants | **Kloosterman fractions**, **Vaaler approximation**, **partially fixed moduli**, **unbalanced convolutions**; keep endpoint differences and all nonzero majorant modes | `endpoint-pairing.md` derives an improved application of the Bettin–Chandee input below. [Wright v2, Theorem 2.1](https://arxiv.org/html/2604.25177v2) was priced at a fixed prime factor; triangle summation over that factor gives no sufficient bound. [arXiv:2601.00292](https://arxiv.org/abs/2601.00292) stands at v2 with an author erratum (missing factor L^2 in its (2.53)); its advertised improvement does not follow and is not imported. Version status and statements checked 2026-09-05 and 2026-09-06; no absence or novelty claim. | | squarefree and nonsquarefree parts of an aggregated prime-divisor coefficient | squarefree endpoint core | truncated Möbius-von Mangoldt convolution; sparse coefficient norms | **correlations of truncated divisor sums**, **prime-factor bilinear forms**, **squarefree integers**, **Kloosterman fractions**; coefficient norm lower bounds do not preclude cancellation in the phase sum | `coefficient-structure.md` derives the classification and norm bounds. [Tao, Notes 2, Corollary 39 and Exercise 40](https://terrytao.wordpress.com/2014/12/09/254a-notes-2-complex-analytic-multiplicative-number-theory/) supplies ordinary PNT for the lower bound; the Bettin–Chandee input below is repriced with the sparse norm. Statements checked 2026-09-05; no novelty claim or full residual estimate. | | aggregated divisor coefficients in a signed CRT endpoint discrepancy | additional rectangle of R | trilinear forms with Kloosterman fractions; trigonometric approximation to the sawtooth | **Bettin–Chandee trilinear forms**, **Vaaler approximation**, **correlations of truncated divisor sums**; retain arbitrary separate coefficients, composite moduli, phase derivatives and the full majorant | [Bettin–Chandee](https://arxiv.org/pdf/1502.00769), Theorem 1 and Remark 1, pp. 2–3; [Baier–Zhao](https://www.impan.pl/shop/publication/transaction/download/product/82887), Lemma 2.2, p. 344. Statements read 2026-09-05 and applied in `endpoint-fourier.md` to an additional rectangle after exact aggregation. No full residual estimate or novelty claim. | | signed grouping by the product of two Möbius divisors | controlled product range of R | correlations of truncated divisor sums; CRT remainder | **truncated divisor sums related to primes**, **Möbius sums with coprimality restrictions**, **fractional-part discrepancies**, **Kloosterman fractions**; distinguish a density calculation from its signed finite-interval error | [Goldston–Yıldırım I](https://arxiv.org/abs/math/0111212), abstract read for the topic only; [Tao, Notes 2, Exercise 66](https://terrytao.wordpress.com/2014/12/09/254a-notes-2-complex-analytic-multiplicative-number-theory/), q=1 mean estimate imported. `signed-divisor-grouping.md` derives its own uniform convolution and CRT budget. The Fourier follow-up in the preceding row handles an additional region; the complete endpoint estimate remains OPEN. | | absolute and separate-sign costs of determinant-2 fibers | singleton mass audit | positive and negative parts of a shifted convolution; divisor switching | **absolute moments of truncated divisor sums**, **primes in arithmetic progressions**, **Bombieri–Vinogradov**; distinguish the mass before grouping from the signed sum after grouping | `singleton-fiber-audit.md` derives a witness using [Tao, Notes 3, Theorem 17](https://terrytao.wordpress.com/2015/01/10/254a-notes-3-the-large-sieve-and-the-bombieri-vinogradov-theorem/) and the classical Möbius mean estimate. This is an input audit, not a novelty search or an absence claim. | | small introduced divisors in the shifted-prime expansion | second Type I pieces of the fold remainder | Bombieri–Vinogradov for the Möbius function | **Möbius function in arithmetic progressions**, **Bombieri–Vinogradov**, **Vaughan identity**; retain coprimality, the square-root modulus level and interval endpoints | no published theorem statement located on five channels 2026-09-06 (Granville-Shao asserts it without a locator; Koukoulopoulos GSM 203 ch. 26 proves only the prime case; IK §17.2 and Opera de Cribro §9 not reached); `mobius-bv-derivation.md` derives it from Koukoulopoulos Cor. 13.4, Thms 26.2, 26.6 and (26.3) with Vaughan's identity for mu. Teaching statement: [Le Boudec, EPFL Exercise Sheet III, Exercise 4](https://wiki.epfl.ch/tan-tnt/documents/TNT%202014-2015/Sheet%203.pdf), read 2026-09-05. `shifted-prime-decomposition.md` derives its application through divisor size x^(1/10), including parity reductions; classical input, no novelty claim | | the residual with two signed factors and fixed product difference | coupled fold remainder on dk−ev=2 | shifted convolution; correlations of multiplicative functions along linear forms | **two-point Möbius correlations**, **averaged Chowla**, **logarithmically averaged Elliott**, **determinant equation**; distinguish fixed from growing coefficients, long from clipped intervals, and free shift averages from inverse-residue constraints | [Tao, Theorems 1.2–1.3](https://arxiv.org/html/1509.05422) and [Matomäki–Radziwiłł–Tao](https://arxiv.org/abs/1503.05121). `shifted-prime-decomposition.md` §5 records the specific parameter mismatches (2026-09-05). These statements do not directly supply its residual estimate; no general literature-absence claim | | an explicit prime detector from progression inputs and one signed factor sum | positive consumer for the fold remainder | Vaughan identity; Type I/II decomposition | **Vaughan identity**, **Möbius convolution**, **bilinear sums over shifted primes**, **local densities**; specify the actual coefficients and both factor ranges | [Tao, Notes 3, Theorem 17 and Lemma 18](https://terrytao.wordpress.com/2015/01/10/254a-notes-3-the-large-sieve-and-the-bombieri-vinogradov-theorem/); `prime-detection-spec.md` derives this application's comparator and input budget. Classical reduction, not a novelty or literature-absence claim | | which factor estimates force a positive prime count | arithmetic after the Chen benchmark | prime-producing sieves | **Type I and Type II sums**, **minimal Type II range**, **prime-producing sieves**, the constants **C⁻** and **C⁻_bd**; distinguish arbitrary divisor-bounded coefficients from one prime-factor Liouville sum | Ford–Maynard, [arXiv:2407.14368v1](https://arxiv.org/html/2407.14368v1), Theorems 2.1, 2.7 and §4. Input-class audit in `chen-opportunity-audit.md` §4 (2026-09-05): not a completed application to shifted primes, no literature-absence claim | | max gap between twin-admissible slots mod `x#` | `G₂(x#)`, twin Jacobsthal, two-class Jacobsthal | two-class Jacobsthal function at primorials | **"the length of the longest sequence of consecutive integers, each equal to 1 or −1 modulo at least one of the first n primes"** — this is `G₂ − 1` | **OEIS A144311** (Carter 2008; Alekseyev 2009; Wang 2024) | | same object, weaker (all even differences) | — | — | **"paired Jacobsthal function"**, `h₂(n) = j₂(pₙ#)`; note `h₂ ≥ G₂` | Ziller–Morack arXiv:1706.00317; OEIS A288815, A072753 | | a floor on what any lower-bound sieve of dimension `κ` can do | "the barrier on Face 4", "is 4.2665 a method artefact" | lower bound on the sifting limit `β_κ` | **Selberg's RECIPROCAL convention `a_k = 1/β_κ`**, in which a lower bound on `β` is titled an **"upper bound for the sifting limit"**: search `a_k`, *"upper bounds for sifting limits"*, *"sifting density"*; and the mechanism convention **"Selberg's model problem"**, all sifting primes the same size, parameters `v = Σ_{p∈P} κ/p` and `R`, the quantity to search being **`v_R`**. NOT "extremal example", which at `κ > 1` is not known (Halberstam 2003 p. 117) and is a guaranteed clean negative | Selberg, *Lectures on Sieves*, Collected Papers II (Springer 1991) §§13, 14, 17; Brady, *Sieves and iteration rules*, Stanford PhD 2017, `purl.stanford.edu/gk881hk9239`, Thm 7 and Cor. 1 (p. 11), Thm 22 and Cor. 3 (p. 51), the `v_R` table (p. 46). Ford's 2023 notes Table 1 is captioned "**Known upper bounds**" and carries no lower bound. Method-specific limits that must NOT be counted: Grupp–Richert's `ν(κ) > 2κ+1` (Ankeny–Onishi's own), DHR's `α_κ > β_κ > 2` (its own DDE pair). Row added 2026-09-04, `history/staging/recon-0904-sifting-limit-floor.md` §5.1, gate effect measured in `redteam-0904-sifting-limit.md` §5.2 | | same object, as a general problem | — | — | **bounded number of residue classes per prime**; `g_f(n)` at `f(x) = x(x+2)` | MathOverflow **88323** (Foo, 2012) | | same object, informally | — | — | **"relative twin primes"** — consecutive odds coprime to the first n primes | Mathematica StackExchange **114758** (2016) | | the covering optimum | `G₂ − 1` | max interval coverable, one pair `{a_p, a_p−2}` per prime, `a_p` free | the **branch-and-bound over residue choice per prime** | A144311's linked C++ | | one-class analogue | `h(x#)`, `g(x#)` | Jacobsthal function at primorials | Jacobsthal | OEIS A048670 | | order-m analogue (emptiest window with < m survivors) | `maxsum_m(T_x)` | order-m Jacobsthal-type function | **`π_min(m,k)` — "the smallest x such that every sequence of m consecutive integers contains at least x integers coprime to P_k"** (inverse indexing, one class) | Costello–Watts, Math. Comp. **84 (2015) 1389–1399**, MR3315513, Thm 4.4 (recursion only, evaluated at m = 1; closed-form arXiv:1209.3464 Thm 2.3 WITHDRAWN — error refuted in `history/staging/attack-foldL-05-maxsum-direct.md`) | | polynomial analogue | — | — | **`j_f`, "polynomial analogue of Jacobsthal"** — a shift of the **value**, not of the argument | Kalmynin–Konyagin arXiv:2302.00459 | | max gap between actual twin primes | anchored distance `A` | record gaps between twin primes | **maximal gaps between prime k-tuples**, fitted against `log^{k+1} p` | Kourbatov JIS 16 (2013) 13.5.2; OEIS A113274 | | the vector-sieve bilinear remainder | "the missing lemma", Lemma V | bilinear sieve remainder at level beyond the window | **"bilinear forms with Kloosterman fractions"**, `Σ α_m β_n e(a·m̄/n)`; **trilinear** when an average over `a` is present, and the trilinear theorem is the instrument here because our `h`-sum is free | Duke–Friedlander–Iwaniec, *Invent. Math.* **128** (1997) 23–43; Bettin–Chandee, arXiv:1502.00769 — the all-rough branch, best in print for us. The **smooth-modulus branch** is Deshouillers–Iwaniec, *Invent. Math.* **70** (1982), **Theorems 9, 11, 12**, quotable through Maynard, Mem. AMS **306** (1543) **Lemma 6.12** and Mem. AMS **306** (1542) **Lemmas 15.1, 18.1**, and sharpened by Pascadi, *Forum Math. Pi* **14** (2026) e8 **Corollary 18**. **It is not available to us**: it requires a *fixed smooth profile* g₀(c/C, d/D) on one factor of the modulus and one of the inverted variable, and well-factorability supplies 1-bounded factors, not smooth ones (`history/staging/lemmaV-neighbours.md`; frontier corrected and the coefficient axis exhausted 2026-08-20 by the smoothness-front note in the same folder). The algebraic-geometry branch — Kowalski–Michel–Sawin, *Ann. of Math.* **186** (2017) 413–500 and *Ann. SNS Pisa* **21** (2020) 1453–1530 — is CLOSED 2026-08-20: prime modulus only by the authors' own scoping, and a rank-1 kernel where they need big monodromy (same smoothness-front note §5, source readings re-checked at page images by the same-day math red team; one-line index in `OUTCOMES.md`) | | L² spread of the tile over residue classes to a coprime modulus | (no house term yet); the object beside the Level Ledger's Thm 1 | AP-variance of a sifted set | **"variance of the sifted set in arithmetic progressions"**, `Σ_{r mod ℓ}(c_ℓ(r) − |T|/ℓ)²`, the ℓ-th term of a large-sieve variance — NOT "discrepancy", which owns the interval object | large-sieve / sieve-remainder literature; instance named `D_i(m)` in Ojaroudi, Zenodo 10.5281/zenodo.18509488 §6.1 (unrefereed) | | two slots killed at the same fold, adjacent in the gap word | `L`, "adjacent-kill run", "kill run" | — | **"the fusions at `γ_i` and `γ_j` occur in the same image of `s` iff `p` divides the span"**; the multiplicity `ν_p(s)`, "the number of residue classes mod p covered by any instance of s"; the two thresholds `J+1` and **`|s|/2`**, past which "all fusions of adjacent gaps must occur in separate images of s" | Holt, arXiv:**2502.20470v3** §3 Lemma 2 (p. 5); arXiv:**2605.19165v1** §3 (p. 11); one-class span form in Holt–Rudd arXiv:1408.6002 p. 11 ("closures", not "fusions") | | how a count of gaps moves from one fold to the next | Tail-Count Transport; "the histogram operator read as an inequality" | — | **"driving terms"** and their populations `n_{g,j}`, `w_{g,j}`; the sentence to search is **"of the `q` copies of `s`, two are eliminated as driving terms and `q − 2` remain as driving terms of various lengths"**, which holds *without* the `g < 2p` hypothesis | Holt–Rudd, arXiv:1408.6002v1 **§6.1, Corollary 6.3 and Figure 4, pp. 25–26** | | the fold recursion with each slot deleted independently | "the thinning null", "independent-thinning ladder" | Bernoulli / `p`-thinning of a renewal process | **"Hawkins' random sieve"**, described by the tradition as a randomised Eratosthenes — its gaps are *exactly* geometric at every stage: `P(p_{n+1} − p_n = j | F_n) = (1/m_n)(1 − 1/m_n)^{j−1}` | Hawkins 1957/1974; **Neudecker–Williams, Compositio Math. 29 (1974) 197–200** (the formalisation — no 1979 item exists in this tradition); Wunderlich, Acta Arith. **26 (1974) 59–81**; Heyde, Proc. AMS 56 (1976), Ann. Probab. 6 (1978); Bui–Keating, J. Number Theory **119 (2006) 284–296**; Lorch, Rocky Mountain J. Math. **37 (2007) 533–550**; surveyed with that formula in Rivoal, *J. Théor. Nombres Bordeaux* **20 (2008) 799–809**, pp. 800–801; primaries read 2026-08-19, `history/staging/hawkins-read.md` | | composing the whole ladder of thinnings into one map | the Möbius group law `M_a ∘ M_b = M_{ab}` on the gap pgf | — | **"composition semigroup of probability generating functions"**, `N`-divisibility and `N`-stability; the exponential/geometric case is the worked example | Bunge, *Ann. Probab.* **24 (1996) 1476–1489** (its abstract names "thinned renewal processes"); behind it Klebanov–Maniya–Melamed 1985, Steutel–van Harn | | the set of twin-admissible words, as a dynamical system | "the limit twin-admissible subshift", "the comb's language" | — | **`Ω_R`, the "R-admissible sets" of a sieve `(B_R, (R_b))` with `R_b` a SET of classes per prime** — the two-class and polynomial cases are inside this definition; for the complexity function itself, **`cpx_{X_ℙ}(n)`, the complexity of the ℙ-admissible subshift**, bounded unconditionally between `(2+o(1))^{n/log n}` and `(4+o(1))^{n/log n}` by a large-sieve count | Araújo, *Sarnak's Program for Erdős Sieves* I/II, arXiv:2602.24031 / 2602.24034 (2026); Kasjan–Lemańczyk–Zuniga Alterman, *Acta Arith.* **209 (2023) 135–171** = arXiv:2205.08273, Thm 1.1 eq. (8); the one-class complexity sequence itself is **OEIS A023192** (Wilson, "infinitely-recurring prime patterns on n consecutive integers"), 13/13 against its b-file — the prime-pattern family A023189–A023192, A035326 is the OEIS owning convention (`history/staging/klz-forward-walk.md`) | | the thinning rate generalised beyond 1/n | our per-fold rate 2/p, as a family member | sieve with arbitrary deletion rate | **"Hawkins' p-primes"**: any rate p(n) with Σp² < ∞, Σp = ∞ — p(n) = 2/n is inside the published family (density 1/(2 log n), NOT the twin density; Lorch's own twin model is p(n) = log n/n) | Lorch, *Rocky Mountain J. Math.* **37 (2007) 533–550**, Thm 2.1 | | the null's maximal gap | the extinction crossing's null counterpart | Cramér-type limsup for the random sieve | **lim sup (p_{n+1} − p_n)/log²p_n = 1 a.s.** for the Hawkins sieve — the constant is proven, not conjectured | Neudecker, *Math. Proc. Camb. Phil. Soc.* **77 (1975) 365–367**, carried second-hand via Rivoal JTNB 20 (2008) p. 808 | | twins in the null | the random-sieve twin analogue | — | **"k-difference twin primes associated with the Hawkins random sieve"**: t_n ∼ n/log²n a.s., a THEOREM since 1974, extended to all l-tuples; no singular series anywhere, and the tradition states that absence as its own limitation | Wunderlich, Acta Arith. 26 (1974) Thm 4; Neudecker 1975; Bui–Keating, JNT 119 (2006); Rivoal Thm 1 | | feasible region of the local lemma, given a dependency graph and marginals | the covering question at the Mertens wall (import-suen §8, import-shearer) | LLL satisfiability boundary | **"Shearer's region"**, the **independent-set polynomial** `Z_G(−p)`, the **hard-core / repulsive lattice gas**; on `K_n` the region is exactly `Σ p_v < 1` | Shearer, *Combinatorica* **5 (1985) 241–245** (text closed-access, abstract read); Scott–Sokal, *J. Stat. Phys.* **118 (2005) 1151–1261** = arXiv:cond-mat/0309352, Thm 4.1 and Ex. 3.1 | | max of a moving sum over the cyclic gap word | `maxsum_m(T_x)` as a fluctuation object (distinct from the §1 `π_min(m,k)` row, which owns it as a counting function) | largest m-spacing | **"the scan statistic on the circle"**; the fixed-total case is the **"conditional scan statistic"**, never "finite population" or "hypergeometric scan", both of which return zero; the max/min pairing is **"largest clusters and smallest intervals"** | Cressie, *J. Appl. Probab.* **14 (1977) 272–283**; Naus, *JASA* **60 (1965) 532–538** and **61 (1966) 1191–1199**; Wallenstein–Naus, *JASA* **69 (1974) 690–697**; Glaz–Naus–Wallenstein, *Scan Statistics*, Springer 2001, chs. 8–10 and 17; Fu–Wu 2012 | | sub-`√m` growth of the moving-sum standard deviation | the exponent `H` | anomalously suppressed density fluctuations | **"hyperuniformity"** and **"local number variance"**. NOT the scan literature, whose fixed-total correction is `(D−m)/(D−1)` and cannot produce `H < 1/2` | Torquato–Zhang–De Courcy-Ireland, arXiv:**1804.06279**, and *Uncovering Multiscale Order in the Prime Numbers via Scattering*; no sieved-set instance exists, searched | | the alternation-legal language of the old gap word | the `L` language, "the two-state walk" | sofic shift | **"charge-constrained"**, **"dc-free"**, **"spectral-null at f = 0"**, **"running digital sum"**, **"digital sum variation (DSV)"**, **"the B-charge constraint"**; the ternary instance is **"alternate mark inversion"**. `(d,k)`-RLL is a DIFFERENT AXIS, and its product with this one is named **"B–(d,k)-charge–RLL"** | Marcus–Roth–Siegel, *Introduction to Coding for Constrained Systems*, §1.5.4 p. 15–16, §1.5.5 p. 17, §2.3 p. 47, §3.2 p. 75; Chien, *BSTJ* **49 (1970)**; Norris–Bloomberg, *IEEE Trans. Magn.* **17 (1981)**; Lind–Marcus, CUP 1995 | | longest legal run over that graph | `L`, first-moment estimate | longest run | **"the longest run in a multi-state Markov chain"**, **"longest run in two-state Markov dependent trials"**. Owns the RANDOM-word law only; the longest legal factor of ONE given periodic word is posed nowhere | Vaggelatou, *Statist. Probab. Lett.* (2003); Fu–Wang–Lou, *J. Appl. Probab.* (2003); Eryilmaz, *Appl. Math. Comput.* (2006) | | a sieve remainder summed over moduli larger than the sum's length | Lemma V's modulus range | — | **"level of distribution"** and **"exponent of distribution"**, `Σ_{q≤x^θ} λ_q(π(x;q,a) − π(x)/φ(q))`; **`θ` is always `< 1` in this convention and ours is `1.212157`, so a clean negative here is guaranteed and worthless** — search the Kloosterman-fraction row instead | BFI `x^{4/7−ε}`; Maynard, Mem. AMS **306** (1542)/(1543), `x^{11/21−ε}` and `x^{3/5−ε}`; Lichtman arXiv:2309.08522 `x^{66/107}`; Pascadi arXiv:2505.00653 `x^{5/8−ε}` | | an interval or circle covered by one or two classes per prime, read on the torus | the covering question in rotation coordinates (import-map row 12) | Bohr-set covering of R/Z | **"lonely runner conjecture"**, **"view-obstruction problem"**, **"shifted lonely runner conjecture"**, **"covering radius of lattice zonotopes"** (Henze–Malikiosis, Aequationes Math. 91 (2017) 331–352 is the dictionary); the house phrase "one class per prime" reaches Banks–Ford–Tao and does NOT reach this family | Tao, Contrib. Discrete Math. (arXiv:1701.02048); Perarnau–Serra survey arXiv:2409.20160; Cusick, Aequationes Math. 9 (1973) 165–170 | | concentration of the survivor count over the rotation ensemble | anchored-note §2's E/Var table, the almost-all column | tail bounds for sieved survivor counts | **Banks–Ford–Tao's five-checkpoint programme**: trivial / Buchstab + sieve / Buchstab + **large sieve + Bennett** / **Azuma on the normalised martingale** Θ⁻¹S / combinatorial expansion — never a boxed concentration inequality; their "most delicate part" is primes near log x, the coordinate that defeats the Talagrand family here | Banks–Ford–Tao, arXiv:1908.08613 §5, read at source (`history/staging/row7-recon.md` §5) | | the spread max − min of a moving sum over a cyclic word | maxsum_m − minsum_m | balance function of a word | **"balance function"**, **"C-balanced word"**, **"abelian complexity"** — but the power-law-exponent theory needs a primitive substitution on a FIXED alphabet, and ours grows with the level (OBSERVATIONS §2), so search for the vocabulary, not for a transferable theorem | Adamczewski, *TCS* **307 (2003) 47–75** (SOURCED-BIB) | | "the biggest empty stretch a window of this length can supply" | the `ln(kills)` term in the extinction crossing | — | **the maximal-spacing law** — "the largest spacing `M_n` defined by `n` independent exponentially distributed random variables" | Lévy 1939; Devroye, *The Largest Exponential Spacing*, **Utilitas Math. 25 (1984) 303–313**; Deheuvels | | how far record twin gaps sit below their own trend curve at finite height | the record-location deficit, "trend load `A`", the 6% | finite-height shortfall of maximal gaps below the EVT trend | **the `b` coefficient of the estimator `E_1 = a log(p/a) − ba`**, `a = C_k log^k p`; equivalently the **mode `µ*` of the fitted Gumbel of the standardized record gaps `g*_k = (g_k(p) − a log(p/a))/a`**. The default is `b ≈ 2/k`; the twin values are `b ≈ 1.2597` (median-unbiased for records below `10^15`) and `−b = µ* + γ` with `µ* = −1.659`. Searching "location parameter", "trend load" or "record deficit" returns nothing; **search `b`, `E_1`, `median-unbiased`, `standardized maximal gaps`**. This is the *location* statistic and is a different row from the maximal-gap *law* two rows above | Kourbatov, JIS **16** (2013) 13.5.2 §§5.1–5.2, read at page image (arXiv:1301.2242v3 pp. 8, 13–14); arXiv:1309.4053 p. 1 for `b ≈ 2/k`; Kourbatov–Wolf, *Mathematics* **7** (2019) 400 §3.2, eqs (48)–(49). House side: `history/staging/lit-kourbatov-shortfall.md`, whose own figures are outside output custody and stay in that file | | the variance of a `θ`-class sifted count in a window | `Var/E`, `X(L)`, `λ₂(u)`, the twin-rough census dispersion | short-interval variance of a `κ`-dimensional sifted set | **at `θ = 1`: "the variance of integers without small prime factors in short intervals"**, coordinate `u = log H/log y`, whose limit law is the **"generalized Dickman distribution `GD(θ)`"** (density `e^{−θγ}ρ_θ/Γ(θ)`, delay equation `xρ_θ' + (1−θ)ρ_θ + θρ_θ(x−1) = 0`, mean `θ`), arithmetic side the dimension-`κ` smooth sum `Ψ_f(x,y)` with `F(s) = ζ(s)^κ G(s)`. **At `θ ≥ 2` the owning convention is not the rough-number one at all: it is "the distribution of `k`-tuples of reduced residues"**, `ν_p(D) = #{h_i mod p}`, `φ_D(q) = ∏(p − ν_p(D))` over the `p` dividing `q`, and the statistic to search for is the `k`-th moment `M_k^D(q,h)` of the count in an interval of length `h`. Do **not** search "sifted", "sub-Poisson", "dispersion" or "rough": "rough numbers" is in nobody's abstract and "twin smooth" belongs to isogeny cryptography. **Scope warning: Gorodetsky is `κ = 1` only**, confirmed at source — one excluded class per prime — so his `λ` is `λ₁` and is not the comparand for a two-class census; at `θ = 2` only an upper bound is in print | `θ = 1`: Gorodetsky, *Math. Z.* **308** (2024) Paper 59, Thms 1.1, 1.3 (arXiv:2111.00853v3, read in full). `θ = s`: Aryan, *Mathematika* **61** (2015) 72–88, Lemma 1.2 and Thm 0.1 (upper bound, general tuple size); behind them Montgomery–Vaughan, *Ann. of Math.* **123** (1986) 311–333 and Hausman–Shapiro, *CPAM* **26** (1973) 539–547; the `GD(θ)` side is Pinsky arXiv:1611.07207, Penrose–Wade *AAP* 14 (2004), Bhattacharjee–Goldstein *Bernoulli* **25** (2019), Tenenbaum–Wu *Compos. Math.* **144** (2008) 339–376. The `θ = 2` asymptotic is absent on six channels calibrated in session; **owed**: the arXiv API leg (HTTP 429 all session), Google Scholar, MathSciNet review text, and *Opera de Cribro* §6.10 Prop. 6.26's dimension. `history/staging/lit-dickman-variance.md` | | VC dimension of the two-class range space `{a, a−2} mod p` over a window | (no house term); the object behind `import-vc-nets.md` §2.2's `VC = 4` and §2.3's `VCdim ≤ (1+o(1))·log₂ ln L` | VC dimension of a family of residue classes | **"the VC-dimension of a class of multiples of the primes"** for the divisibility class `h_p(x) = 1` iff `x ≤ p` or `p ∤ x`, and **"learning integer lattices"** for the windowed version, sublattices of `Z^k` restricted to `{−n, …, n}^k`. Our own wording — "VC dimension of a sifted set", "residue-class range space" — returns nothing, and no number-theory-side convention for it has been located | Thomas, *Online J. Analytic Combinatorics* **17** (2022), DOI 10.61091/ojac-1705 = arXiv:2208.06442, both versions read in full and DISJOINT from ours at theorem level (`history/staging/lit-vc-multiples.md`); forward citations zero on three channels (OpenAlex, Semantic Scholar, OpenCitations), each calibrated in the same session. **OWED: Helmbold–Sloan–Warmuth, *Learning Integer Lattices*, SIAM J. Comput. 21 (1992) 240–266, Theorem 3.1** — closed access, no open location, abstract only; the `k = 1` multiples-of-`d` reading of it is Thomas's alone, single-source and uncorroborated, so no negative rests on it | | two-point correlation of the `y`-rough indicator at shift 2 (the rough-pair census) | `X(y)`, the census; the tile's shift-2 comb | two-point correlation of a 1-bounded multiplicative function | **the `y`-rough indicator IS `χ₀ mod y#`, the principal character**, so search **"correlations of multiplicative functions"**, **"pretentious multiplicative functions"**, **"Halász–Montgomery–Vaughan distance"**, and for the short-interval half **"multiplicative functions in short intervals"**. The object sits on the **pretentious** side of the dichotomy, where the answer is a theorem: the correlation sequence is a uniform limit of periodic sequences, `χ`-isotypic. **The Chowla / Elliott / entropy-decrement side hypothesises the OPPOSITE of what holds here, so a negative searched there is guaranteed and worthless** — the same trap as the `level of distribution` row | Tao, *Forum Math. Pi* **4** (2016) e8 = arXiv:1509.05422, Thm 1.3 and its hypothesis (1.6); Tao–Teräväinen, *Duke Math. J.* **168** (2019) 1977–2027 = arXiv:1708.02610, the structure theorem; the short-interval half is Matomäki–Radziwiłł, *Multiplicative functions in short intervals*, arXiv:1501.04585 (**[SOURCED-BIB]** — the arXiv record prints no volume and no pages, so confirm the journal locator before quoting it, and its quantifier is *almost all*). **NOT SEARCHED** as of 2026-08-27; `history/staging/import-entropy-decrement.md` §9 | | the depth-`J` truncation of the tile's inclusion–exclusion, read as a function on the CRT product `∏_{p≤y}Z_p` | Bonferroni depth; Brun's truncation (Face 3's `K*` ladder is a different object) | Bonferroni inequalities | **"Hoeffding decomposition"**, **"Sobol–ANOVA decomposition"**, **"Efron–Stein decomposition"**, **"Fourier–Walsh expansion"**, **"low-degree function on a product space"**; the small-alphabet caveat is owned by **"lambda-biased hypercontractivity"** and **"global functions"**. The ambient is `∏_{p≤y}Z_p` under the uniform measure, whose translation group preserves the measure, the degree filtration and every influence, so the field's theorems are equivariant and none of them distinguishes the anchor `t = 0` | O'Donnell, *Analysis of Boolean Functions*, CUP 2014 / arXiv edition 2021, chs. 9–10 (General Hypercontractivity p. 283, Thm 10.21 p. 292, Thm 10.24 p. 293, Example 10.45 p. 305); Keevash–Lifshitz–Long–Minzer arXiv:1906.05568. The Hoeffding/ANOVA identification is already folklore inside this corpus (`history/staging/audit-novelty-postaudit.md` §8), and whether depth-equals-degree is in print in the SIEVE literature is **NOT SEARCHED**; `history/staging/import-boolean-analysis.md` §4 | | the set relaxation of the covering optimum | `τ_set`, "the smallest set no adversary can cover" | minimum number of translates of `D = {d : gcd(d(d+2), x#) = 1}` covering `Z/x#` | **"covering a finite abelian group by translates of a set"**, **"covering code"**, **"the Rogers–Stein covering bound"** `τ ≤ (x#/#D)(1 + ln #D)`; the group is the CRT product `∏_{p≤x}Z_p`, and the interval case, which is ours, is the max gap of `D` and is A144311 | **NOT SEARCHED** as of 2026-08-27; `history/staging/import-vc-nets.md` §4, §8 | | the distribution of the fractional parts of `N/n` and `N/p` | the branch phase `⌊W/q⌋ mod M_T` of the skeleton door | equidistribution of `{x/p}` over primes | **"the distribution of the fractional parts of x/n"**, **"fractional parts of x/n and related sequences"**, and, for the method behind the error term, van der Corput's method of exponential sums (a search term, not a phrase the note carries). The statistic is Saffari–Vaughan's `Θ*_{x,y}(α) = y^{−1} Σ_{p≤y}(log p)·c_α(x/p)`, `c_α` the indicator of `{u} < α`, read at `x = W/M_T`, `y = √W`, `α = j/M_T`; the range condition `x^{6/11+ε} < y ≤ x` reads `W^{1/12} < M_T ≤ √W` under that dictionary and so **excludes every fixed `M`**, which is why import map row 15's THEOREM column is struck | Saffari–Vaughan, *On the fractional parts of x/n and related sequences* I, *Ann. Inst. Fourier* **26** (1976) 115–131, DOI 10.5802/aif.634, and II, **27** (1977) 1–30, DOI 10.5802/aif.649, Theorem 10 p. 7; the method is Graham–Kolesnik, *Van der Corput's Method of Exponential Sums*, CUP LMS 126 (1991). `Saffari` appears nowhere in this repository outside import map row 15, and the novelty question attached to the anchor is **NOT SEARCHED** as of 2026-08-28; `history/staging/import-fracparts.md` §6 | | weighted count of large `y`-smooth divisors of `C(C^2-4)`, averaged over a window `\|C\| < L = y^2` (the CRT-mixed lag residual of the `theta = 2` decoupling step) | the mixed lags, `Xmix`; "the divisor-distribution statement" | distribution of divisors of polynomial values in a range | **`H_F(x,y,z) = #{n <= x : exists d \| F(n), y < d <= z}`**, "la localisation des diviseurs de `F(n)`"; the tool is **Hooley's `Delta`-function**, `Delta(n) = max_u #{d \| n : e^u < d <= e^{u+1}}`; the three-factor shape is **`H^{(k+1)}(x, ybar, zbar)`, "localized factorizations"**; the smoothness word is **friable**, never "smooth" (`ti:"smooth divisors"` is algebraic geometry, 7 of 9). Two traps. **(a) Our divisor range `y_lit > x_lit` is outside every located theorem**, without exception, and that is the wall, not the precision: **asymptotics with relative error `O(1/log x)` DO exist in this convention** for the divisor-sum object over polynomial values with a friability restriction (Scourfield), so a negative searched on "no asymptotics here" is wrong; but that register is `O(1/log x)` where the open step needs `o(1/ln y)`, one epsilon short, so the row does NOT clear the precision axis either (redteam-0829-measure-c.md E25). **(b) Search the DIVISOR-SUM object (`sum_{n<=x} #{m <= x : m \| f(n)}`), not only the set count `H_F`**; they are different functions and the set count is the harder one. **Method note: the forward-citation walk is the productive channel here** (OpenAlex `filter=cites:`), since the load-bearing chapter is closed access and its author restates it in her own later papers | Ford, *Ann. of Math.* (2) **168** (2008) 367-433, Thm 1 and section 1.6(i) (order of magnitude; "strengthen Theorem 1 to an asymptotic formula" is his own open problem); Tenenbaum, *A Tribute to Paul Erdős* (CUP 1990) 405-443 (reducible `F`, `y <= x^{1-eps}`, `(log y)^{o(1)}`) and *Invent. Math.* **99** (1990) 215-224, Thms 1 and 3 (irreducible `F`, `y <= x/2`); Hall-Tenenbaum, *Divisors*, Cambridge Tracts **90** (1988) Thm 21; Koukoulopoulos, *Proc. LMS* (3) **101** (2010) 392-426 and *Crelle* **689** (2014) 33-99; Ford, arXiv:1901.02548 (roughness of the **integer**, the opposite axis). **Friable side**: Hanrot-Tenenbaum-Wu, *Proc. LMS* (3) **96** (2008) 107-135; Martin-Tenenbaum-Wetzer, arXiv:2307.05530; Schlitt, arXiv:2603.19212 (positive-density prime sets, **not** friability). **The divisor-sum-over-polynomial-values object is Scourfield, *Smooth divisors of polynomials*, LMS Lect. Notes **352** (CUP 2008) 286-311**, DOI `10.1017/CBO9780511721274.019`, Zbl 1334.11078, closed access on every channel tried, **UNREAD at the page**, restated by its author at *Funct. Approx.* **55**:1 (2016) 84 eq. (1.1) as `sum_{n<=x} #{m <= x : m \| f(n)} = Cx(log x)^l(1 + O(1/log x))` with a friable companion valid for `y >= exp((log log x)^{5/3+eps})`; divisors at most `x`, weight 1, factors of degree `>= 2`. `history/staging/lit-smooth-divisors.md`, `history/staging/lit-scourfield-2008.md` | | full corner coefficient energy after logarithmic averaging of its divisor cutoff | C_tilde, the full smoothed corner | mean square of a truncated Mobius divisor sum | **"Barban–Vehov problem"**, **"Graham estimate"**, **"two-parameter quadratic sieve"**, **"Selberg sieve weights mean square"**; distinguish the full finite mean square from only its lcm quadratic main term | Chen An arXiv:2206.10104v1 (1.1), Theorem 1.1, integer case; Carneiro–Chirre–Helfgott–Mejía-Cordero arXiv:2005.03162v6 §1; Ramare–Zuniga-Alterman arXiv:2405.12662v1 theorem/corollary statements. Positive source matches read 2026-09-06; no absence or novelty conclusion. Owning derivation: `corner-coefficient-energy.md`; import map row 22. | | sharp full corner energy and its smoothing error | C, T=C-C_tilde | mean square of sharp truncated Mobius divisor sums | **"Remarques sur une somme liee a la fonction de Mobius"**, **"Sieve weights and their smoothings"**, **"uniform mean square"**; distinguish long-cutoff finite sums from lcm main terms | de la Breteche–Dress–Tenenbaum [author PDF](https://tenenb.perso.math.cnrs.fr/PPP/Sxz.pdf), (1.5), Theorem 1.1; Granville–Koukoulopoulos–Maynard [1606.06781v4](https://arxiv.org/html/1606.06781v4), Theorems 1.3, 10.2 and proof of (10.8), read 2026-09-06. Positive match; `sharp-corner-transition.md`, import row 23. No novelty or absence claim. | | the transition divisor weight and its shift-2 product | T_i, R_11 | outer prime-power convolution of a difference of truncated divisor weights and a sharp Mobius sum | **"Barban–Vehov weights"**, **"sieve weights and their smoothings"**, **"correlations of divisor sums"**, **"averaged Chowla"**; match the actual affine forms, clipping and normalization | `transition-source-match.md` (1)–(2); scoped direct-application failures, not method closures; corrected in `transition-round-audit.md`. | | mu(n/s) restricted to s dividing n, and its shifted cofactor product | F_s, K_H, J_H | bounded multiplicative functions with modified Euler factors on arithmetic progressions | **"quantitative correlations of multiplicative functions"**, **"non-pretentious multiplicative functions"**, **"correlations in arithmetic progressions"**; retain the progression density and price the full cofactor sum | Tao–Teravainen [2512.01739v2](https://arxiv.org/html/2512.01739v2), Theorem 3.1(ii), with MRT [1503.05121v3](https://arxiv.org/html/1503.05121v3), (1.12), checked 2026-09-06. Positive match for a growing polylogarithmic cofactor family, not the full range: `cofactor-progression-transfer.md`. No novelty claim. | | the sign of a truncated alternating subset-sum sieve weight, its negative part, and whether a lower-order (pair) trigger controls it | `F(s)^-`; the pair-trigger majorant (10) of `global-factor-signs.md` §3 | Bonferroni truncation error at depth `J`; sharpened sieve inequalities | **"Bonferroni(-type) inequalities"**, **"sharpened sieve inequalities"**, **"inclusion-exclusion identities and inequalities"**, **"the degree of the certificate"**; the object is `sum_{j=2 variables; Klurman covers one variable but needs 1-bounded fixed functions; scoped negative, not an absence claim | revisit for a matched theorem, weighted transfer or changed representation; the finite norm example does not close every lift; `full-coefficient-average.md` §4 | | Where is the explicit Mertens error 1/(10 ln²x) + 4/(15 ln³x) in print? | **Dusart** (thesis 1998; arXiv:1002.0442 Theorem 6.10, x ≥ 10372), not Rosser and Schoenfeld 1962, whose Theorem 5 has 1/(2 ln²x) for x ≥ 286 and whose Theorem 20 is a finite-range (x ≤ 10^8) statement | settled 2026-09-08 at page image; `history/reviews-0907/11` F1 | | Is the `G₂` ladder in OEIS? | **Yes — A144311**, 22 terms to `x = 79` | settled | | Is the `G₂` ladder in OEIS at our own convention? | No, and still no — the negative is real and useless | settled | | Published **upper bound** on `G₂` at any exponent? | **None found**, searched in the owning convention | the one live claim; see §5 | | Does GY III Theorem 8.1 directly estimate the full transition kernel? | No direct match: raw product-level error is insufficient, and outer prime-power sums, sharp M terms and affine forms remain (`transition-source-match.md` §3) | revisit for a specified different reduction or proof adaptation; not a closure of the family | | Do the inspected Topacogullari/Drappeau divisor-correlation statements directly allow our Mobius bands? | No matched coefficient class supplied (`transition-source-match.md` §4) | revisit a named statement, mechanism adaptation or correctness concern; no universal modulus ceiling inferred | | Has `β₂ = 4.2665` been improved since DHR 2008? | **No**, and everything after is worse at `κ = 2` | closed, see §4 | | Published asymptotic growth law for A144311? | None | settled | | Published **lower** bound? | Free via `G₂ ≥ j(x#)`; FGKMT transfers; already labelled trivial | settled | | Did Alekseyev write on it? | **No** — 158 publications, zero hits for "prime" | closed hard | | Is our multi-class Erdős–Rankin claim novel? | **No** — Kalmynin–Konyagin, Izv. Math. 88:2 (2024) | settled | | Is the maximal-twin-gap law ours? | **No** — Kourbatov 2013, and our "flat" reading was refuted by his own table | settled | | Does KLZ's (arXiv:2205.08273) forward graph hold a multi-class complexity? | **No** — one citing work across OpenAlex, S2 and OpenCitations, plus two via a B-free bibliography grep, none with complexity | settled 2026-08-19; walk these graphs by DOI, never by arXiv id (S2 splits the records; the wrong-DOI guess returns a convincing false zero) | | Is KLZ Part II out? | **Not out** — four arXiv channels (title, both author trails, "Behrend sets") | re-check only on journal indexes (MathSciNet/zbMATH unreached) | | Two-class cpx sequence in OEIS at the owning convention (prime patterns, A023192 family)? | **No**, at three indexings, calibrated by A023192 in the same minutes | the negative now carries weight; superseded §3.3's "no owning convention was identified" | | Is the complementary-window duality in print? | **Yes in convention, no verbatim.** It is the complement identity of the circular scan statistic and of the circular maximum subarray; Cressie 1977's link to Kuiper's statistic is the closest retrieved statement. No paywalled carrier (JSTOR, Project Euclid, Springer) could be opened | redo only if a text copy of Naus 1966 or Wallenstein–Naus 1974 becomes reachable | | Is `maxsum_m` a scan statistic in print? | **Yes.** Largest m-spacing, dual to the scan statistic; fixed-total case is Glaz–Naus–Wallenstein chs. 8–10 | settled | | Is the fixed-sum scan called "finite population" or "hypergeometric"? | **No.** Both return zero on zbMATH. The term is **conditional scan statistic** | settled; never search the other two again | | Does the conditional-scan literature carry a sub-`√m` law? | **No, and it cannot**: exchangeability forces `Var(S_m) = mσ²(D−m)/(D−1)` | settled by arithmetic, not by search | | Is the alternation language's constraint family in print? | **Yes, in a textbook.** Charge-constrained / dc-free, the `B`-charge constraint, Marcus–Roth–Siegel §1.5.4 | settled | | Is "charge constraints are strictly sofic" in print? | **Yes**, Marcus–Roth–Siegel §2.3 p. 47, with the same appendability argument | settled; stop deriving it | | Is the capacity in print? | **Yes**, `λ = 2cos(π/(B+2))` with Table 3.2, Marcus–Roth–Siegel §3.2 p. 75; the ternary variant adds 1 to the root, giving `ln 2` at `B = 1` | settled | | Is the `H*` ladder in OEIS? | **No**, six indexings including the semiprime-cofactor flip, calibrated same session | the negative carries weight | | A lonely-runner / view-obstruction variant with TWO obstacles per speed? | **No** — arXiv (39 + 24 records, every title read) and zbMATH (65 + 46), calibrated; the 2025 survey's own variations list has none. And the fully general multiplicity version is answered: with distinct speeds lifted the union bound is EXACTLY tight (Perarnau–Serra §11.3, via Schoenberg 1976) | settled 2026-08-20; any gain must come from the fixed {0,−2} pattern and prime speeds, and the field names prime speeds as ITS obstruction | | Do bounded-remainder-set results reach the comb? | **No, twice** — every result hypothesises totally irrational rotation (ours is rational, translation by 1 on Z/x#), and the bounded-variation bound they would give is D_x, which the Level Ledger beats from x = 13 and by 5454× at x = 29 | settled 2026-08-20; `history/staging/row12-recon.md` §4 | | Any published exactly-computed Shearer region on an arithmetic family? | **None found** on zbMATH, arXiv and OEIS full text, searched as "Shearer's region" | MathSciNet still unreached | | Is the `θ = 2` short-interval sifted variance asymptotic in print? | **No, in six calibrated channels** (2026-08-28). The object is in print with an **upper bound only** — Aryan 2015, Lemma 1.2, at general tuple size — and the asymptotic and its constant are not. Gorodetsky's Lemma 1.4 supplies the general-`k` local densities but he uses `k = 1, 2`, and his stated future work is higher moments of the one-class count | settled at this level; **owed**: the arXiv API leg (HTTP 429 all session), Google Scholar, MathSciNet review text, and *Opera de Cribro* §6.10 Prop. 6.26's dimension | | Does Gorodetsky name `GD(θ)`, sieve dimension, or tuples? | **No.** Zero hits for `tuple`, `twin`, `admissible`, `generalized`, `dimension`, `Poisson` in the full text of arXiv:2111.00853v3 | settled 2026-08-28, grep over the read PDF | | Do Montgomery–Soundararajan (*Comm. Math. Phys.* 252 (2004) 589–617) give a sieve-level analogue of their moment machinery? | **No.** Zero hits for `sifted`, `sieve`, `Dickman`, `rough`, `small prime factors`, `free of prime` in arXiv:math/0409258 | settled 2026-08-28 | | Has anyone cited Gorodetsky's variance paper on this object? | **One citing work across three calibrated citation indexes**, and it is a statistics paper on bootstrap diagnostics (Sanyal–Pillai, *Sankhya A* 2026). One edge cannot rule out a successor; it can only fail to find one | re-run in 12 months, not sooner; Google Scholar was never reached, so the count is a floor | | Are "twin smooth" and "pairs of rough numbers" usable search terms? | **No, both are false friends.** "Twin smooth" is isogeny cryptography (Costello–Meyer–Naehrig 2021 and successors); "rough numbers" in a title is fuzzy-set decision theory. And on the arXiv web search, whose long queries are conjunctive over metadata only, any negative on a query containing "rough" is void: `variance rough numbers short intervals` returns 0 while `variance integers without small prime factors short intervals` returns exactly Gorodetsky | never search either again; settled 2026-08-28, `history/staging/lit-dickman-variance.md` §§4, 6.2 | **Added 2026-08-18, carried back from the documents that ran them.** These are sieve-side and covering-side negatives that were already recorded in the corpus without a home here, which is why the claims resting on them could not point anywhere. Each row names what was actually searched; none was run in an owning convention that this file's §1 table carries, because §1 has no row for these objects, and that is stated rather than hidden. | question | answer | what was actually searched | where it is worked | |---|---|---|---| | Published `κ = 2` **extremal example** — a set that blocks an exponent in `(2, 4.2665]`? | **None found.** The band is a proof gap, not a truth gap | Selberg's *Lectures* through Franze's account, the DHR book's Ch. 17 apparatus, Ford's 2023 course notes, Blight 2010, Brady 2017. At `κ = 1` the extremizers are Selberg's Liouville sets; nothing analogous is in print at `κ = 2` | `sift-limit-attack.md` §2 | | Published sieve that **re-admits input beyond divisor-class counts** `\|A_d\|` — exact strata, pair correlations, the mod-30 rigidity? | **None found** | the DHR apparatus, *Opera de Cribro*, the Brüdern–Fouvry vector sieve; all of them are priced on one-point data | `sift-limit-attack.md` §§1, 4.6 | | **Position-uniform interval** version of the Brüdern–Fouvry bilinear remainder beyond product level `H`? | **PRICED, not absent — corrected 2026-08-19.** The aligned-position case is in print at `γ = 0.970624` (Bettin–Chandee), and the position-uniform case is in print too, with an explicit price: their Remark 1 charges `(1+hx/MN)^{1/2}`, which is `O(1)` only for `x ≪ H^{1.212157}` while our `x` reaches `exp(H^{0.2344})`. What is absent is a *nontrivial* position-uniform bound | searched in the OWNING convention this time, **bilinear/trilinear forms with Kloosterman fractions** (§1), on three channels each calibrated in-session: WebSearch (→ DFI 1997), the arXiv API (**`https` only; `http` 301s, which is why an earlier session recorded this channel as dead**) and OpenAlex (`W2581797856` → *Le crible à vecteurs*). The earlier row searched Iwaniec 1980 and Brüdern–Fouvry only, in our wording, and its "none found" was wrong by omission | `sift-limit-attack.md` §4.5 | | **Covering-systems** result bounding an interval coverable by ≤ 2 classes per prime at polynomial scale? | **None found**, and the corpus is structurally elsewhere: it covers all of `ℤ`, or transfers interval-to-`ℤ` at the **exponential** threshold `2ⁿ`, which is vacuous at Jacobsthal scale | the min-modulus / density / structure theorems and the interval-transfer literature | `covering-dive.md` §Q3 | | An **Erdős problem** posing two-class Jacobsthal directly? | **No problem number carries it.** `Jacobsthal` occurs in exactly **two** of the 1217 problems, #687 and #970, each verified individually at `/latex/`; `two congruence` occurs in none | **the complete corpus, swept mechanically 2026-08-18 and re-swept independently the same day.** The site is reachable and its search works: the route is **path-encoded, not query-string** (`/search/`, `/range/-`, `/range/1-end`, `/go_to/`, `/latex/`, `/history/`, `/bibs/`), all 200 with a browser user-agent. Calibration, same session: `/search/Jacobsthal` returns 200 and exactly #687 and #970 (the site reports 0 solved out of 2 shown); `/range/1-end` returns 10.6 MB carrying 1217 distinct problem ids, 1 through 1217. **Attribution gotcha for the next sweep:** in the bulk page a problem's `bib-container` marker does not sit inside its own text block, so keying a hit to the nearest preceding marker misattributes it by one. Doing that produced a spurious third Jacobsthal problem (#969, which is the squarefree error term) on the first pass here. Confirm every bulk hit at `/latex/` | `covering-dive.md` §Q2.2 | | Where can Halberstam and Richert, *Sieve Methods* (1974), be read at the page? | archive.org item `sievemethods0000halb` is a lending copy (login + loan; page image 403, djvu text 401 anonymously); Google Books Dover volumes `keKvAAAAQBAJ`/`sU_fhcpaL-IC` give a search-inside OCR index only (page images return a 9103-byte "image not available" placeholder); the 1974 volume `pwXvAAAAMAAJ` has no index; HathiTrust, ScienceDirect and zbMATH web are 403 from this host (zbMATH API works: Zbl 0298.10026); no secondary reproduces Theorem 2.2 verbatim (Montgomery Bull. AMS 1976, Ford 2023 notes, Greaves, arXiv 2410.14133, 2503.04045, 2211.11012, 2606.17955, 1907.06393 checked). Physical: Academic Press 1974 ISBN 0123182506 (LMS Monographs 4); Dover 2011 ISBN 9780486479392 | 2026-09-07 ([reviews-0907/08](history/reviews-0907/08-halberstam-richert-thm22-access.md)) and 2026-09-08 ([reviews-0907/12](history/reviews-0907/12-halberstam-richert-second-access.md)); OCR is not reading; the page remains the owed item | --- ## 4. The `β₂` literature, settled — do not re-derive `β₂ = 4.26645028414864191641…`, the dimension-2 sifting limit, Diamond and Halberstam, *A Higher-Dimensional Sieve Method*, Cambridge Tracts 177 (2008), Table 17.1. The book's p. 79 prints **`β₂ ≈ 4.266`**, three decimals; the twenty-decimal value is Booker and Browning's ancillary data. Page photographs are in this repo at `attestation/book-ch5-6/`. | source | `κ = 2` value | note | |---|---|---| | **Diamond–Halberstam 2008** | **4.2665** | best known, unimproved | | Blight 2010 | < 4.45 | worse | | Franze 2011 | 4.516 | worse; his Table 1 prints **4.266**, three decimals | | *Opera de Cribro* β-sieve | ≈ 4.83 | worse | | Brady 2017 | — | improves `κ = 3/2` only | | Booker–Browning | — | superior sieves become available "once `κ ⩾ 3`", not at 2 | | Ford, 2023 course notes | 4.2665 | still tabled as best known | **Consequence: "improve the constant" is a closed route.** Any attack on our exponent must attack the *pipeline's discards* (`sift-limit-attack.md`, DP1–DP5), not the sifting limit. Our 4.2665 is the best available, not a default. **Citation discipline.** Our bare `4.2665` is a correct four-decimal rounding and is fine unattributed. It may **not** be attributed to book p. 79 or to Franze's Table 1, both of which print three decimals. --- ## 5. What is still open, stated so it is not mistaken for coverage - ~~**The 2009 SeqFan archive thread**~~ — **CLOSED 2026-08-18: it does not exist.** `list.seqfan.eu` is dead (ECONNREFUSED), so the archive was rebuilt from Wayback's per-month mailboxes: **142 months, 19,964 messages, ~41 MB, 1999–2023. Zero hits for `144311` anywhere.** Calibrated on the same corpus in the same pass — `Alekseyev` 1033 hits, `primorial` 192, `Jacobsthal` 30 — so the silence is the channel working, not the channel failing. November 2009 is complete (322 messages, Nov 1–30); Alekseyev posted ten times that month, none on this subject and none on the 18th. All 23 A144311 revisions read: the extension carries no method note, no program, no bound. *Residual gap: 5–30 September 2008 is archived by nobody.* - ~~**Paseman's heuristic**~~ — **CLOSED 2026-08-18: weaker than ours.** His `n` is `ω(m)`; his own arXiv:1311.5944 cites MO 88323 and gives `g(n) < k^{3+3.81 log log k}`, an **unbounded** exponent against our **fixed** 4.2665. See `PRIOR-ART.md`. - **NEW, and it narrows the claim: MathOverflow 37679 answer 52890 (zeb, 2011)** already derives `j(x#) ≪ x^{4.032}` from a sieve's error exponent — our exact shape at **dimension one**. The technique is not ours; the **dimension-2 instantiation** is. Never table 4.032 beside 4.2665: the closeness is coincidence. - ~~**Holt's 2022 book, *Patterns among the Primes*, remains unread.**~~ **CLOSED BY PROXY 2026-08-19, and the reason it was never findable is that it is SELF-PUBLISHED** (Independently Published / KDP, 212 pp, ISBN 9798831607314), which is why MathSciNet, zbMATH and Crossref all return zero for it and why no indexed channel was ever hiding it. Read instead, in full and with sha256s recorded, four of Holt's own artifacts covering the same material: the 32-page exercise companion, the JMM2024 talk of the book's own title, JMM2025, and SFU 2015. Across all four: `Jacobsthal` 0, `maximum gap` 0, `upper bound` 0, `spacing` 0. His 8-part video series named after the book gives its chapter order and there is no maximum-gap chapter. *Residual: Google Books returned HTTP 429 for the whole session and was never reached.* - **zbMATH: swept 2026-08-18** via the open `api.zbmath.org`. **Gotcha recorded 2026-08-19: zbMATH answers HTTP 404 for a zero-result query** (`"successful access. No results found."`), so a 404 there is a NEGATIVE and not a broken channel — and its long-query zeros are unreliable, since `paired Jacobsthal function` returns 0 against a record whose title contains all three words. - **MathSciNet: SEARCHABLE, and this file was wrong about it until 2026-08-19.** The subscription UI does redirect to LibLynx, which is what the old entry recorded. **The index does not.** `POST https://mathscinet.ams.org/mrlookup` with fields `au`, `ti`, `jrnl`, `year`, `format=bibtex` is free, answers against MathSciNet, prints the total match count and returns records as BibTeX. Calibrated on six known positives — MR499895 Iwaniec, MR4727548 Kalmynin–Konyagin, MR3718451 FGKMT, MR4195744 FKMPT, MR2458547 Diamond–Halberstam, MR3065331 Kourbatov — and two of those were re-run independently by the adjudicator before this row was written. **Gotcha: `year=1900-1963` is read as a LOWER BOUND, not a range**, which is useful rather than fatal, because differencing cumulative counts gives a complete enumeration. **What it does not cover, and this is the residual: bibliographic fields only. No review text, no abstracts, no MSC.** **Non-English: done.** Fischer's German pages and Rivera's Spanish Conjecture 66 were retrieved in full, and both turn out to concern maximal gaps between **actual twin primes** — a different object that shares the symbol `G₂` with ours. No bound in either. - **Wang's unpublished `a(23) ≥ 1859`** with witness exists in the OEIS revision log only, not in the DATA and not in the b-file. --- ## 6. The check that enforces §1 `search-convention` is **built, calibrated and in the gate** as of 2026-08-18. The rule it enforces is the one specified here: a document asserting literature absence — "no literature exists", "ABSENT", "first", "nobody has" — must, within the same paragraph, either cite this file or name the convention it searched. A calibrated negative in our own vocabulary does not satisfy it, because that is exactly what five waves produced. It fired on 75 paragraphs the day it went live, and the queue was worked to zero the same day (`history/staging/qc-queue-9-10-worked.md`). Six of the 75 were "not in OEIS" claims about the `G₂` ladder, and every one of them was **false**: the ladder is `A144311 + 1`. **What it cannot do, and never will.** It asks only whether the paragraph says where it looked. Whether the absence is TRUE is a reader's job, and §5 is where that reader records the answer. Its companion, `provenance`, asks the same question of quotations: which artifact was actually read. The clearing vocabulary is parsed out of §1's table at run time and is never copied into the engine, so editing §1 changes what clears. Add a row only when it is genuinely the wording the literature uses for that object; a row added to make a claim pass is a suppression wearing a table's clothes. §3's rows are free of that hazard — they are read by people, not by the check.