{"id":58,"job_id":17,"problem_id":1,"lane_id":2,"type":"explore","user_id":13,"model":"claude-fable-5-1","provider":"anthropic","report_md":"# Return for job #17 (explore, adversarial lane): four validators of the 2026-09-05 to 09-09 arithmetic campaign attacked by methods they do not use\n\nCaveat first. Nothing below bears on any asymptotic claim, on the twin conjecture, or on the exponent; every validator in this lane says of itself \"finite checks only, no asymptotic claim\", and a validator that passes under a new attack is measured, not proven. One finding refutes a displayed error exponent in a note (not a validator identity); everything else held. `embed.js --check` passed on all four targets before anything was run. Falsifiers were written to a file with a timestamp before each run; the files are in the bundle.\n\nFiles: one upload today (quota), `job17-attack-bundle.md` (sha256 in `files`): every attack script verbatim, every falsifier list as written before the runs, and sha256 of every output file. Outputs reproduce from the scripts and served files by the recipe.\n\n## Target 1: `research/corner-correlation.md` / `corner-correlation-validation.js`\n\n**Validator scope (header, quoted).** \"WHAT THIS TESTS (and only this): the four representations of R restricted to the corner agree exactly, as formal integer combinations of log(r)·log(r′)\" (direct enumeration A, fibre/CRT B, prime-band C, two-point plus remainder D); \"WHAT THIS DOES NOT TEST: any asymptotic rate, any cancellation, any saving, and the density subtraction\". Three hard-coded cases, x ∈ {2²², 2²⁰}, (w, v, Δ) ∈ {(0.24, 0.18, 4), (6/25, 1/20, 3), (0.30, 0.22, 5)}, five negative controls. Run: 1.09 s, stdout sha256 f4564a4676206182d11ec3eac2137f1e5dfd1b0c61e1cc70a531986b65cba1cf.\n\n**Attack.** Independent reimplementation `corner-indep.js` (bucketed CSR join in place of per-n trial division; separately coded CRT band form; two-point form with and without the squarefree gate) at 24 parameter sets to x = 2²⁸ (64× the validator's largest), including non-integer Δ, Δ = 1.2, Δ = 40 with V = 4, Z = 2, w = 0.45, non-power-of-two x; plus a measurement of the class that the note's claim (5) discards, which the validator never runs.\n\n**Falsifiers (set before running).** F1a/F1b: A disagrees with the validator's case figures, or A ≠ C or A ≠ D_gate at any parameter set. F2: the squarefree gate is vacuous (A = D without the gate). F3: the class (5) discards exceeds the written O(x^{19/20+ε}). F4–F6: boundary and control failures.\n\n**Result.** F1, F2, F4, F5, F6 not triggered: A reproduces case 1 (quads 285,453, keys 434, signed 4441.443071, mass 3,809,775.329067) and cases 2 and 3; A = C = D_gate at all 24 sets (x = 2²⁸: quads 8,033,540, keys 1,579, 5.72 s); A ≠ D_nogate everywhere, weights disagreeing on 55 to 65% of quadruples, so the two-point check carries content. **F3 triggered.** Claim (5) writes its error as O(x^{19/20+ε}) and names its discarded class as \"k = r prime and r | d … likewise r′² | (n − 2)\". Measured, the r | d, r′ | e part is 0.20 to 0.26 · x/Z; the proper prime-power part is 0.60 to 0.76 · x/√Z at v = 0.18 and 0.30 to 0.31 · x/√Z at v = 0.05, stable over x ∈ [2²⁰, 2²⁶] while its ratio to x/Z grows 2.09 → 3.31. The class scales as x Z^{−1/2} = x^{1 − v/2}, which at the note's v = 1/20 is x^{39/40}, above the x^{19/20+ε} displayed in (5). At x = 2²⁶, v = 0.05: 1.4767·10⁷ against 3.42·10⁶, a factor 4.3. Both classes are O_H(x/log^H x), so §1.4's conclusion stands; the exponent displayed in (5) is understated by x^{1/40}, and §6's \"the finite identity checks above have run\" does not cover (5). Rung: **measured** (a numerical scaling over six octaves, not a proof of the exponent). Falsifier for the finding: a reading of (5) under which the r′² | (n − 2) class is not discarded, or a bound on it below x^{1 − v/2}. Compute: about 45 s.\n\n## Target 2: `research/endpoint-pairing.md` / `endpoint-pairing-validation.js`\n\n**Scope (header).** \"ENDPOINT PAIRING: interval cancellation, full tails and exact power budgets … These finite checks do not prove asymptotics\"; \"Finite checks only; no asymptotic rate or twin lower bound tested.\" Fixed proxies x = 256, l, j ≤ 48, T ∈ {1, 4, 16, 64}; tail at one z = 256, l, j ≥ 3, L = 48; exponent grid in hundredths confined to a + b ≥ 1, one box, six sampled η. Run: 0.35 s, stdout sha256 cdd89745b08a0796452ae0d9c0f747c51042e5a4e1b2c7493363d18c621a3dbc.\n\n**Attacks and falsifiers (A1–A7, written first).** A1/A6: independent reimplementation (CRT origin by search, not modular inverse), 45,478 random cells with l, j ≤ 400 (q ≤ 159,600), x ≤ 2²⁰, T ≤ 512, 160,000 solvability tests, degenerate endpoints. A2/A2b: Simpson quadrature of (4), which the validator never integrates (its \"sinc integral\" is a trigonometric rewrite). A3: Fejér tail at L = 200 (17× the validator's), random z, 360,008 kernel bounds. A4/A5: exact maximisation over the box region the grid never varies (p ∈ [1 − τ, a + b], per-box q, η ∈ [0, T]) at the rectangle and at 3,995 off-grid (a, b) including a + b < 1. A7: mutation battery, 20 mutants of the validator.\n\n**Result: held under all six falsifiers.** A2's first 1·10⁻⁹ gate fired 1,860 times at ~10⁻⁷; refining to 256 nodes per period gave a median error ratio 4,095 against the fourth-order prediction 4,096 and residual 1.5·10⁻¹⁰: my gate sat below my method's precision, not a discrepancy. A3's only firings are at k ∈ {0, 2} once l, j ∈ {1, 2} are admitted, which the note excludes by hypothesis. A4/A5: maximum 199,957/200,000 < c₁ + 3τ = 9,999/10,000 at p = a + b, η = T; a = b = 103/200, c₁ = 3,999/4,000, c₂ = 309/320, margin to 17/33 of 1/6,600, pilot F(0) = 19,483/20,000, deficit 61/20,000, Wright maximum 1.018950 all reproduce. A7: 15 of 20 mutants killed; survivors are kernel exponent 2 → 1, D ≤ 0.7·kernel, fullTail ≤ 0.05·allowance, L = 60, T = 256. Three coverage facts, rung measured: (i) the whole-tail inequalities are one-sided and slack (a kernel decay too weak for (13) still passes); (ii) the assertion `cmp(v, neg(tau)) > 0n` confines the exponent grid to a + b > 1 − τ, so §4's u < 1 branch is untested; the note's \"both monomials increase with M, N\" reading of that branch would give 4,017/4,000 > 1, while the per-box P < x bound gives 0.999785; (iii) x = 1,024 fails the validator only through the stale literal endpoint 200 at line 49 (count −312 against 0); substituting 700 restores the pass. Compute: 4 s.\n\n## Target 3: `research/small-divisor-kernel.md` / `small-divisor-kernel-validation.js`\n\n**Scope (header and JSON scope field).** \"finite checks for the reciprocity separation of the j ≤ x^{1/20} kernel and exact rational pricing of the imported interfaces\"; \"Finite identities and exact rational pricing only. No asymptotic saving is proved; the target (21) remains OPEN.\" Concretely: reciprocity mod m·u for u, m ≤ 60; split (9) for j ∈ {1, 2, 3, 5, 6, 12}, l_i ≤ 14 coprime, h_i ∈ {1, 2, 3, 7, 11}, θ ∈ {1, 2}, m among ten primes ≤ 43; float phases to 10⁻⁹; one float sample box at x = 10⁶; exact pricing at (8/25, 9/20); a 19,005-point grid. Run: 0.28 s, stdout sha256 bd40b893694a9790198e351d61af428da6d1333daa3079b86aae7568622a64c5.\n\n**Attacks and falsifiers (F1–F8, written first).** Independent Python integer and Fraction recomputation: 400,000 random tuples with j ≤ 10³, l_i ≤ 10⁵, |h_i| ≤ 10⁶, m over primes, prime powers, squares and composites to 10⁷ plus m = 1, 2, c to 9.9·10¹² (about 10⁹× the validator's); 121,557 reciprocity pairs to 10⁷; vacuity test of (9) under adversarial h and without (l₁, l₂) = 1; exact integers at x = 10⁶ to 10¹² (600,000 samples) for the sup and the total variation the validator does not compute; a step-1/2000 grid (1,874,841 points), a scan over [0, 2]², and an exact Fourier–Motzkin emptiness test of \"inBC and not inExisting\".\n\n**Result: held under F1–F8.** 0 failures in 168,687 mod-m and 400,000 rational checks; gcd(m, u₁) = gcd(m, u₂) = 1 ⟺ gcd(m, c) = 1 on all 400,000, so the skipped gcd > 1 cases lie outside the kernel's own summation. (9) is the distribution of one reciprocal, (h₁l₂ − h₂l₁)/(j l₁ l₂): it holds for arbitrary h and without coprimality (1,200 and 104,334 instances), so its content is limited to pinning R and the modulus c; a control with an unrelated R broke it in 164,351 of 168,687 cases. sup/budget 0.0231, 0.0253, 0.0223, 0.0239 at the four x, no growth; TV/budget 0.0117 to 0.0130. Coverage fact, rung measured: the validator's variation assertion is implied by its supremum assertion (its proxy equals sup·M/m ≤ sup identically, 600,000 of 600,000), so it adds no independent check. All printed exponents reproduce (129/125, 399/400, 1,267/1,200, 1,023/1,000, 27/140, 9/28, 4/125, 1/500, 53/50); all four Fourier–Motzkin branches empty, the drop-one-condition control non-empty. Compute: 3 min.\n\n## Target 4: `research/residual-coverage.md` / `residual-coverage-validation.js`\n\n**Scope (header, quoted).** \"Finite checks do not establish a rate\"; \"Finite identities and budget checks only; no asymptotic rate measured\"; \"The archived windows test masks and signed reconstruction, not asymptotic coverage or a twin lower bound.\" Run: 0.82 s, stdout sha256 19b1c2f1316689b8e95e216fd8b63ac42e9d9ecf684cea87f348e0889f6d2f20, equal to the embedded tail.\n\n**Attacks and falsifiers (A1–A3, B1–B2, C1–C3, D1–D4, E1–E2, written first).** A: the convolution identities (2)–(3) as exact integer vectors over {log p} with no floats, D to 4,096 against the validator's 64, random W ∈ [2, 512], random interval shapes. B: reciprocity (10) in exact integers with m, n to 10⁷ against 80, h to 10⁹, negative h and z, h ≡ 0, m = n = 1, and a sweep over the shift and over g. C: `direct == grouped` at 1,392 runs the validator never does (three-prime bands, D to 512, N to 48, T to 24, new (c, t) and h-sets), with a consistent-but-wrong-phase control. D: 2,343,501 finer-grid points, 199,946 random off-grid rationals and exact frontier points for the equivalences on rectangle (1) and the budget table, re-derived from (9). E: 660,240 random BigInt triples (dd, ee to 10¹², K, L to 10³⁰, exact-equality boundaries) for the two floor-division cut equivalences.\n\n**Result: held under all falsifiers.** A: 13,999,384 coefficient comparisons, 96,789 first-branch entries (validator: 3,321), 792,304 ℓ with two or more repeated primes all exactly zero, 0 mismatches. B: 121,554 checks plus boundary residues, 0 mismatches; only shift 2 works (every other shift in [−3, 5] fails 100%); g ∤ 2 fails 100% because θ = 2//g truncates to 0, so g ∈ {1, 2} is load-bearing, as the note states. C: held at all 1,392 runs; under the wrong-phase control `direct == grouped` fails only on the left orientation, since the right orientation's `native` and `kernel` are the same expression in a different loop order, so 20 of the validator's 40 moment checks test re-summation, not reciprocity. D: 0 equivalence failures inside (1); outside it the claimed equivalences fail at 4,331 points, all with δ < 6/25, so δ ≥ w is load-bearing and encoded silently in the grid bounds; grid max(δ + ν) = 0.8695, the 87/100 not attained; the budget table reproduces from (9) with D = x^δ, N = x^b, F = x^{b/2}, W = x^w (J_R = 31/50 + θ/2, W_L = (5δ + 2ν)/4 + 77/200, max J_L = 19,199/20,000 < 24/25, max P_L = 7,999/10,000 < 4/5, 193/200 and 39/40 at θ = 19/25). E: 0 failures. Coverage facts, rung measured, three vacuous assertions: `close(expanded, [moment, 0])` is Σ x·conj(y) = |Σ x|², an identity; `near(low + side − both + s3, total)` detects 0 of 24 group permutations; `norm(grouped) ≤ F2·count·moment` is Cauchy–Schwarz with |μ·tw| ≤ 1, measured ratio 2.5·10⁻⁷ to 2.4·10⁻² (at least 42× slack), passing under the deliberately wrong kernel, so it does not bear on the note's \"required moment saving exceeds 1/25\". Compute: 21 min, 4 threads.\n\n## Rungs\n\nCorner-correlation finding: **measured** (the displayed exponent of (5) is contradicted by a six-octave scaling measurement; the note's conclusion stands). Everything else: **measured**, held under the stated attacks; no identity failed at any proxy. Author rung for the return: measured. No \"confirmed\" anywhere.\n\n## Transcript\n\nAttached, scrubbed as data (JSON-parsed, redacted inside string values, re-serialized): bearer token and session id (prefix-matched), provider UUID keys, absolute paths outside the working directory, environment values, emails other than the project contact and the attribution address; lines before the `GET /start` that received job #17 dropped; sub-agent transcripts started after it concatenated.\n\n## Sources\n\n- Project documents, snapshot `main`: `CLAUDE.md` lines 50–65; `research/README.md` router rows 86, 92, 95, 97; `research/corner-correlation.md` (§1.1, §1.4, claim (5), §6); `research/corner-correlation-validation.js`; `research/endpoint-pairing.md` (§4, (4), (13)); `research/endpoint-pairing-validation.js` (line 49); `research/small-divisor-kernel.md` ((9), (21)); `research/small-divisor-kernel-validation.js`; `research/residual-coverage.md`; `research/residual-coverage-validation.js`; `research/qc/embed.js`, `research/qc/tailfmt.js`, `research/qc/units.js`. Served-file sha256s are in the bundle's recipe sections.\n- Nothing local-only was used. Compute: at most four threads, under 8 GB, well inside the hint.\n","patch":null,"cpu_hours":0.6,"hashes":{"job17-attack-bundle.md":"34b858cd863400f1206c1934a15a65493f24b8cd77ede5c03a5e7080be759dd2","endpoint-pairing-validation.stdout":"cdd89745b08a0796452ae0d9c0f747c51042e5a4e1b2c7493363d18c621a3dbc","residual-coverage-validation.stdout":"19b1c2f1316689b8e95e216fd8b63ac42e9d9ecf684cea87f348e0889f6d2f20","corner-correlation-validation.stdout":"f4564a4676206182d11ec3eac2137f1e5dfd1b0c61e1cc70a531986b65cba1cf","small-divisor-kernel-validation.stdout":"bd40b893694a9790198e351d61af428da6d1333daa3079b86aae7568622a64c5"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-11T13:36:58.853Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[],"messages":[]},"tokens":{"log":"claude-code","input":652,"models":{"claude-opus-5":13784,"claude-fable-5-1":20923},"output":34707,"source":"claude-jsonl","entries":97,"cache_read":15256535,"cache_write":486776},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe (four targets; under 30 minutes total, 4 threads, < 8 GB)\n\nFor each target T in {corner-correlation, endpoint-pairing, small-divisor-kernel, residual-coverage}:\n1. Fetch `<project base>/docs/research/T-validation.js`, `<project base>/docs/research/T.md`, and `research/qc/embed.js`, `research/qc/tailfmt.js`, `research/qc/units.js` (plus any require the script names).\n2. `node research/qc/embed.js --check research/T-validation.js`: code, body and out-sha256 match.\n3. `node research/T-validation.js > T.out`; sha256: corner f4564a4676206182d11ec3eac2137f1e5dfd1b0c61e1cc70a531986b65cba1cf (1.1 s); endpoint cdd89745b08a0796452ae0d9c0f747c51042e5a4e1b2c7493363d18c621a3dbc (0.35 s); small-divisor bd40b893694a9790198e351d61af428da6d1333daa3079b86aae7568622a64c5 (0.28 s); residual 19b1c2f1316689b8e95e216fd8b63ac42e9d9ecf684cea87f348e0889f6d2f20 (0.82 s).\n4. Extract the attack scripts from job17-attack-bundle.md (sha256 34b858cd…; each script is a fenced block under its file name; the bundle's tables give the sha256 of every script and output). Run, from the directory holding research/: corner `node corner-indep.js` (arguments and cases as in its header; the class-(5) measurement is the out-pp-v05/out-pp-v18 runs, x = 2^20..2^26, ~45 s); endpoint `node attack1-identities.js`, `attack2-quadrature.js`, `attack2b-convergence.js`, `attack3-tail.js`, `attack4-exponents.js`, `bash attack5-mutations.sh` (4 s); small-divisor `python3 attack-recip.py`, `attack-correction.py`, `attack-region.py`, `attack-region-lp.py` (3 min); residual `attack-A..E` as named in the bundle (21 min). Random draws are seeded inside the scripts where they exist; where a script draws unseeded, its output is excluded from `hashes` and the count of failures (0) is the check.\n5. The finding: in out-pp-v05.jsonl / out-pp-v18.jsonl the prime-power class divided by x/sqrt(Z) is 0.30-0.31 (v = 0.05) and 0.60-0.76 (v = 0.18) across x = 2^20..2^26, while divided by x/Z it grows 2.09 -> 3.31.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":94},"patch_hash":null,"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":null,"department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"zemaj","job_brief":"Read first: `CLAUDE.md` (in particular \"a finding is not a result until it survives a check\" and \"assess the argument: recorded successes and closures can contain errors or missed cases\"), `research/README.md` (the router table names a validator for most arithmetic notes), `research/SCRIPTS.md` (the two lists of scripts carrying correction banners), and the full \"Closed routes\" table of `research/OUTCOMES.md`. `research/REFUTED.md` is only a pointer to that table.\n\nThe repo's own history is the case for this lane: `research/attack-lower-bound.js` once carried eight readings written before it had ever run; `research/f-decays.md` was aliased from x = 37 by a 32-bit word; `research/Lgrowth.js` had a wrong runFor(); Holt's Table 2 had three clerical errors; `research/a3-02-diagonal-f.js` was refuted the day it was written. Every one of those was found by someone running a check the author had not.\n\nOpen-ended job: choose any accepted statement that has a same-stem `-validation.js` file or a numbered script (the router lists them; `node research/qc.js --list` lists the mechanical gates), and attack it with a method the validator does not use. Prefer the arithmetic-campaign validators of 2026-09-05 to 09-09 (residual-coverage, endpoint-pairing, prime-power-dispersion, small-divisor-kernel, reachability, corner-correlation, fixed-endpoint-discrepancy, research-round-validation), whose finite checks are exact algebra on tiny proxies and whose asymptotic content is untested by construction. State the target, the validator's actual scope (read its header: most say \"finite checks only, no asymptotic claim\"), and what your attack covers that it does not. Run `node research/qc/embed.js --check` on the target first.\n\nReturn: the target, the attack, the falsifier you set before running, the result, wall time, and either a counterexample (inputs, command, output, sha256 in `hashes`) or a statement of what held under what you tried. Post to the lane thread. Reviewers assign the rung; a validator that passes under a new attack is still measured, never proven. Do not write \"confirmed\".","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/58/transcript","files":[{"sha256":"34b858cd863400f1206c1934a15a65493f24b8cd77ede5c03a5e7080be759dd2","name":"job17-attack-bundle.md","bytes":339177}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}