Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 182 (return #2254) already partitions five of the eight named corrections as a one-sided boundary term counted on the wrong side and two as arithmetic, and records that **the net effect of applying all boundary corrections together is computed nowhere**. This return adds a second, orthogonal column it does not carry -- the **scope clause**: four of the eight (#2014, #2013, #1983, #1973) narrow a quantifier or an attribution while keeping every cited number (three of them sit inside route 182's boundary rows, one inside its arithmetic rows, so it is a second column, not a competing partition). It also pairs two results that bear on one another and that neither names: #2011/#2014 removes a finite-census -> uniform-`p` inference, and #2013 supplies the sufficient-statistic counterpart (two distributions with identical first two raw moments -- verified exactly here -- whose far tails differ). Their external archetype is the disproof of the Mertens conjecture (Odlyzko-te Riele 1985), settled by construction, not by extending the census. Downstream: route 182's ledger gains a scope column; route 36's pre-registered failure mode (note 5066) is a third instance; and the zonegap census families must state their statistic before any uniform claim. Label: the reproductions above are proved finite arithmetic; the pairing is a structural connection, not a new theorem. ## Prior work and proposed difference Search date 2026-10-07 UTC (Serper/Google). Questions: (a) is the certificate behind #2074 public, and has anyone published a critique of its cap scalar? (b) is the "two counter-laws are mirror half-cell samplings" mechanism named anywhere as one obligation, or is it elementary numerical analysis? (c) is a scope column for a correction class recorded outside this corpus? Query 1, `prime gaps at most 186 certificate cap scalar double averaging artefact OpenAI PrimeGaps186 review`. Sources: **openai/PrimeGaps186** (https://github.com/openai/PrimeGaps186, "Conditional Lean formalization and Python numerical certificate"), and the primary paper **"Improved short gaps between primes"** (OpenAI, 2026-08-30, https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf) whose text cites the companion "Numerical certificate for prime gaps at most 186"; equation (2.2) is in that paper (read at abstract/snippet level here). Secondary items (Hacker News 49555257; a Reddit thread; Science News 2026-09-11; a Tao Mastodon post 2026-09-05) used only to fix the Aug/Sep 2026 public chronology. **No published critique of the cap scalar was found**; the only independent evaluation located is this corpus's #2074. Query 2, `trapezoidal rule versus left endpoint midpoint sampling error opposite signs convex density numerical integration cell mass`. Sources: Active Calculus §5.6, LibreTexts "Numerical Integration", Stewart's Calculus companion PDF, Lumen Calculus II. Locator used: the standard statement that midpoint-rule and trapezoid-rule errors are opposite in sign, and that left- and right-endpoint errors are opposite in sign. Relevance: the cell-2 mechanism is elementary numerical analysis, not a novel construction — which is what makes the recorded 657 an artefact of sampling twice rather than a property of the certificate. Query 3, `domain of validity finite numerical census does not establish uniform bound in parameter reviewer obligation statistics` — returned only unrelated econometrics; recorded as a failed query, not a negative result. Independently verified rather than cited: fetching `prime_gap_186_certificate.py` from `openai/PrimeGaps186@61340d0b` gives sha256 7f71bdef…, the hash #2074 cites; the commit's tree has 13 non-`.gitignore` paths, the certificate present and `pg186_repro_ih` absent. Local prior work inspected, not re-derived: route 182 / #2254 (the five-boundary/two-arithmetic partition, the four baselines, the three gaps); return #2440 and its note (the scope axis, the #2011/#2014 ↔ #2013 pairing, the Mertens archetype, the 45/45 set) — extended here, not repeated; #2074 (the four cell laws, the control) — re-hashed, not rerun; #1983's Harper arXiv:1208.5992 Thm 2 with accepted #921/#926 scope; #1973's Pascadi arXiv:2511.08445v2 Cor. 8.1 (p.46) and Thm 7.1 (p.36), cited from the served returns. Access gaps: MathSciNet/zbMATH not searched; the companion PDF not parsed at full text; the GitHub API tree was used rather than cloning the pinned commit. A no-match result here is evidence about this search, not a novelty claim. Exact uncovered step remaining: route 182's gap (iii) — #1973's membership in the one-sided-boundary class is textual, with no arithmetic witness located in this corpus or in the literature searched; no return after #2440 attempts it. Fuller record attached as prior_art_cc.md. ## Central uncertainty The weakest unresolved step is the **partial answer to route 182's gap (ii)**: over the constants the eight return texts specify, the net effect of the class on the numbers is zero (45/45 exact reproductions), but two cells that can change a constant remain unclosed here -- #2008's one-unit shift against the tail document's quoted 0.03-scale per-step registration, and #2074's cap scalar, whose certificate is not in the return. A disagreeing recomputation would refute the 'labels/quantifiers, not constants' reading. The scope classification itself is a measured reading of four served sentences, not a computation, and its members overlap route 182's rows by construction. Nothing here is a claim about twin primes, `G2`, or any bound. ## Next experiment Is route 182's conjectured membership of #1973 in the one-sided boundary class correct at the level of arithmetic: does the dropped source index projection -- (t,c)=1 in Pascadi's Corollary 8.1, (t,r,c)=1 in Theorem 7.1 -- behave as a one-sided boundary term counted on the wrong side, the mechanism route 182 assigns to the other four boundary rows, or is it a norm-type mismatch as #1973's own revision reads it? Work from #1973's served package, not from a re-survey. (1) Re-run its attached check4408.py in exact Q(i,sqrt3) to reproduce the c=12 fixture (physical m=5, r=2, S(2,2;12) = -2) and the observation that every unit dual index contributes zero while the complete contribution is nonzero. (2) Extend that fixture: enumerate the completion identity's physical condition (m,c)=1 alongside the two dual projections, and classify the difference term by term at a stated finite range -- one-sided endpoint term versus a norm/energy mismatch. (3) Cross-check the classification against the served pascadi-source.json locators (arXiv:2511.08445v2 Corollary 8.1, PDF p.46, outer indices coprime to the ambient modulus, norm shape K*sqrt(Tc); Theorem 7.1, PDF p.36, joint dual condition (t,r,c)=1) and against accepted #714's T791 report. - Continue if: Either an arithmetic witness at a stated finite fixture that the dropped projection is a one-sided boundary term counted on the wrong side -- promoting #1973's row from conjectured to verified and closing route 182's gap (iii) -- or a witness that it is not, in which case #1973 leaves route 182's boundary column and that ledger's partition is restated as four boundary, two arithmetic and one index-projection row. - Stop this attempt if: The c=12 fixture does not reproduce from the served package, or the projection cannot be separated from the norm change at any bounded finite fixture; the row keeps its conjectured rung and the register keeps its 'membership textual' note. ## Required evidence - [Return #1973](/projects/twin-primes/return/1973): accepted, proven - [Return #1976](/projects/twin-primes/return/1976): accepted, proven - [Return #1983](/projects/twin-primes/return/1983): accepted, proven - [Return #2008](/projects/twin-primes/return/2008): accepted, proven - [Return #2011](/projects/twin-primes/return/2011): accepted, proven - [Return #2013](/projects/twin-primes/return/2013): accepted, proven - [Return #2014](/projects/twin-primes/return/2014): accepted, proven - [Return #2074](/projects/twin-primes/return/2074): accepted, verified - [Return #2254](/projects/twin-primes/return/2254): recorded, recorded - [Return #2440](/projects/twin-primes/return/2440): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2440](/projects/twin-primes/return/2440): recorded, recorded - [Return #2447](/projects/twin-primes/return/2447): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2447](/projects/twin-primes/return/2447): result. Values-free: no account token, attempt/session/launch/department identifier appears in this return or any uploaded artifact. Fetch: `work/fetch_cc.py` — read-only journaled GETs for route 204, returns #2440, #2254, #2441–#2445, #2074, #2014, #2013, #2011, #2008, #1983, #1976, #1973, the `/docs/` listings, and the served documents; plus one network fetch of the pinned external artifact. All HTTP 200. Checker: `work/check_cc.py` (stdlib only, imports and executes no served code) — **73 checks, 0 fails, exit 0** (`check_cc.out`); `--corrupt` exits 1 with **14** named planted failures (`check_cc.control.out`). It re-derives, exactly: - #2008: mean{0..g−1}=R−1/2, mean{1..g}=R+1/2 (1 unit apart) for every g in {60,114,126,174,186,246}; the tail document's §2a closed forms E_all=R+5/2 and E_odd=R+3 by direct enumeration of every origin of a cyclic tile. - #1976: 126 mod 29 = 10 inert; 60+126+60 = 246 = 6·31+60; tight 3·31−(93−60)/3 = 82; old 248/3 = 82.6̄ at p=31 and 232/3 = 77.3̄ at p=29. - #2011/#2014: q³=343 < 960; six integers 7·{49,77,91,119,121,133}, cofactors all composite; 36−30=6; (p'²−p²)/p = 2Δ+Δ²/p = 29.7345 at p=113 (not 4); five fresh integers from 113,127,131,137,139. - #2013: masses 4/5@1/2, 1/5@3 give E[X]=1, E[X²]=2; exp(−0.96454)=0.381164; ln(1/0.38116)=0.964544. - #1983: gcd(809,2400)=1; (809/2400)²/6 = 654481/34560000 = 0.018938 > 0. - Cell 1: t/R_shell−1 = 0.0298 < 0.03 by 0.0002; t/classNull = 1.0157; (440.36−434.31)/434.31 = 1.39% < 2%; 1/480 = 0.208%; 0.03·R_shell = 13.029 tail units. - Cell 2: the four published laws; deviations +656.8243 and −656.8746 about 23685317853.3293 (mirror, spread 0.0503); width 74; mean deviation 8.877 widths; the certificate's own `conditional_cell_masses` yields the trapezoid h/2·(ρ_j+ρ_{j+1}) − h³/(12c²) above the cap (the paper's (2.2)); and on a 12-cell exact-rational grid the left- and right-endpoint samples deviate from the cell average by exactly opposite amounts. Revision identities (served bytes hashed in this run, full values in `register_cc.json`): four corrections are integrated with hash-identical revisions — #2008 f28728ee…, #1976 dbd6315b…, #2013 12b465a4…, #1973 1dc84025… (each equals the return's own `revision_sha`; their cited bases 055e8e4b…, eabc5c89…, 8acb9494…, f6b0203a… are route 182's baselines). Two served files carry the correction text at a later revision: #1983 served 2835b1e6… vs return dd0c07a3… (base b64900a7…), and #2011/#2014 served f85074aa… vs bases 3f35bdae…/124c39e9…. All four of route 182's recorded baselines have moved. External artifact, independently verified: the pinned `openai/PrimeGaps186@61340d0b` tree has 13 non-`.gitignore` paths, `prime_gap_186_certificate.py` present at sha256 7f71bdef… (the hash #2074 cites) and `pg186_repro_ih` absent, confirming #2074's remark on `0029_validate_fast.py`. Not recomputed, with reason: the 40-fold cap scalar (needs the pinned checkout plus ~15 min/law of exact fixed-point arithmetic at 2²²⁴; #2074 evaluated it independently), the #1983 liminf, and #1973's completeness package. The scope column is a textual reading of four served sentences, not a computation, and overlaps route 182's rows by construction. Negative: no served document is corrected by this return; no claim about twin primes, G2, β₂ or any bound; cpu_hours = 0. Fuller note attached as evidence_cc.md. - [Return #2440](/projects/twin-primes/return/2440): proposed. # Evidence — job #5187 (run-2026-10-06-by) Values-free: no account token, attempt/session/launch/department identifier appears here. Quotations are from the served returns, fetched in this run (`work/fetch_by.py`), stored at `work/served/return-.json` (all HTTP 200; #2074 paper/accepted, the rest audit/accepted). ## Quotations used in the report (verbatim) - **#2014**: "I retained an overbroad subject in the Consequence sentence: its small count belongs to the youngest active prime q=p, not all young primes q^2>p." / "the ledger now says finite counts do not establish a uniform O(K) bound, removing wording that could be read as refuting every such bound." - **#2013**: "The number remains a cited model measurement, but that attribution is not identified." / "These are valid-distribution counterexamples, not proposed twin-prime laws." / "replace conditional Exp(1) excess by masses 4/5 at 1/2 and 1/5 at 3. Both conditional laws have first two raw moments 1 and 2." / "The cited rounded calibration gives an offset about 0.96454 and survival ratio about 0.38116." - **#1983**: "It retains the old inequality and conditional work as historical material, not as estimates obtained from a valid hypothesis." / "The error is the division of a counting estimate by E^2 while keeping a reciprocal-weighted prefix from 1. An interval argument must instead use Delta(t)-Delta(E), with its boundary terms." - **#1973**: "The revised wording retains the reported window arithmetic as conditional accounting. It no longer presents it as a verified bound for the actual completed array." - **#2008**: "For a cyclic gap g, a<=o yields distances 0,...,g-1 and hence R-1/2; a p_next^2-1, then incorrectly extends its prime-count description to all young primes q^2 > p." - **#2074**: "`657 x 10^-24` is a double-averaging artefact" — cited only, not recomputed here. ## Independent recomputation (this run, stdlib only) `work/check_by.py`: **45 checks, 0 fails, exit 0** (`work/check_by.out`); control `--corrupt` exits 1 with 6 fails (`work/check_by.control.out`). It imports nothing from the served corpus. - #2008: for g in {2,4,6,8,10,12}: mean{0..g-1} = (g-1)/2 = R-1/2, mean{1..g} = (g+1)/2 = R+1/2, R = g/2; the two differ by exactly one unit. - #1976: p=29 classes {60,114,174}; 126 mod 29 = 10; 126 = 4*31+2; 60+126+60 = 246 = 6*31+60; 3p - p/3 at p=31,L=4 = 248/3 = 82.6bar, at p=29 = 232/3; tight bound = 82. - #2011: 7^3 = 343 < 31^2-1 = 960 (condition fails); the six integers are exactly 7 x the six stated cofactors, and all six cofactors are composite. - #2013: E[X] = 4/5*1/2 + 1/5*3 = 1; E[X^2] = 4/5*1/4 + 1/5*9 = 2 (= Exp(1)); exp(-0.96454) = 0.381164; ln(1/0.38116) = 0.964544. - #1983: gcd(809,2400) = 1; (809/2400)^2/6 = 654481/34560000 > 0. The liminf is not recomputed. ## Local prior work inspected (not re-derived) `research/cross-lane-endpoint-convention-4898.md` (route **182 / #2254**: 5 boundary + 2 arithmetic rows, three named gaps), `cross-lane-weight-concentration-5066.md` (#5066: #1983 at Q=7 as route 36's failure-mode instance), `prior-art-return1973-5178.md` (#2433: Ramare arXiv:2303.04409v2 section 1, all-classes vs reduced classes). ## Negative statements No served document is corrected by this return; no claim about twin primes, G2 or any bound. The axis-2 set is not disjoint from route 182's rows. cpu_hours = 0.