{"id":2064,"job_id":null,"problem_id":1,"lane_id":null,"type":"audit","user_id":17,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Audit: research/history/staging/attack-0830-rec-cheapest.md, ledger status PARTIAL -> ANSWERED and verdict refreshed (job #4603, explore return #2063)\n\n**One ledger block and one new rider, in a 458-line note.** The body is unchanged. Base: the served sha c6996fcb… (v2, #1681).\n\n1. **Status.** `status: PARTIAL` becomes `ANSWERED`. The question's three parts are answered on the record:\n   - routes (§2: none closes);\n   - mixed strategy (§3: z^0);\n   - proof or truth gap (§6, plus `redteam-0904-floor-growth-2.md`, which refutes the proof-gap reading, and `redteam-0904-item0.md`, which confirms the truth-gap reading).\n\n   OUTCOMES already lists the route CLOSED as a truth gap. `research/qc/questions.js`: \"the status belongs to the QUESTION, not the note's review grade\". #1681 edited only the verdict and left the status untouched.\n2. **Verdict.** \"At every computable z it is a proof gap\" becomes \"no computation reaches the failing level\", with both conventions (10^31.68 at (2.698721, 4.2165), 10^15.30 at (3.0, β₂)). Two numbers move to the served successors (`blind-0830-omega-floor.md`): Ω is exact to z = 113 (1980), and the certified family reaches 9.351e32 at z = 1e9 against z^{u₀} = 8.87e37. The verdict now carries the 09-04 scope (Rosser vector-sieve lattice, every admissible level split, 1643 splits, minimum margin 0.644). Every other clause (PROVEN exact half, DERIVED growth red-teamed twice, s ≤ 3, 16s/9 = 4.7088, onset 1.875e6, no exponent moved) is kept.\n3. **Rider** dated 2026-09-28, recording the change and its sources.\n\nQUESTIONS.md is generated from this block and needs only regeneration. The id and question lines are unchanged, as are `todo: 0` and the TODO Ledger guard.\n","patch":"--- a/research/history/staging/attack-0830-rec-cheapest.md\n+++ b/research/history/staging/attack-0830-rec-cheapest.md\n@@ -2,12 +2,21 @@\n \n <!-- ledger\n id: Q-rec-cheapest-0830\n-status: PARTIAL\n+status: ANSWERED\n todo: 0\n question: At the cheapest legal point of REC(s, u0), s -> 1+sqrt(e) and u0 -> beta_2, where the saving asked over the triangle inequality is only z^{1.031}, does any of the five dead routes close, does a mixed mean-square-plus-cap strategy close, and is the failure a proof gap or a truth gap?\n-verdict: No route closes and the failing step is yesterday's statement, so PARTIAL on the routes; but the step is located as a TRUTH gap, not a proof gap, asymptotically: the certificate's own value at a CRT-planted doubly-smooth window is cc(r) = -(A1A2 + A1B2 + B1A2) with every term a count of exit chains of the Rosser supports (PROVEN, checked exhaustively at z <= 73), the window bound T <= cc(r) + H - 1 is exact, and the exit-chain count is DERIVED (red-teamed twice) to grow like z^{16s/9 - o(1)} at four primes and z^{2s - o(1)} in the limit, with hypothesis s <= 3 and conditional on that DERIVED growth, REC(s, u0) is false for every u0 < 16s/9 = 4.7088 at s -> 1+sqrt(e), the whole legal band (2, beta_2] included, and RML(alpha) with it; at every computable z it is a proof gap (Omega(z) exact to z = 73 reads 251 against z^{u0} = 7.188e7 (1.05e6 is HM(u0), not z^{u0}; redteam-0830-floor-sign.md), and a certified lower bound 8.86e17 at z = 5e5 sits a factor 1.2e6 under z^{u0}); the September 4 rider reports the exact construction onset z = 1.875e6 at s = 2.698721, not the early analytic 1.055e6; no exponent moved.\n+verdict: ANSWERED at the rungs stated, all three parts. Routes: none of the five routes of attack-0829n-rml-proof.md sec.4 closes at the cheapest point; the failing step is the same reduced arrow. Mixed strategy: a mean-square-plus-cap mixture is worth z^0 at every split point, because with no moment above the second the mean square adds nothing at one position. Gap: a TRUTH gap, not a proof gap. The certificate's value at a CRT-planted doubly-smooth window is cc(r) = -(A1A2 + A1B2 + B1A2), with every term a count of exit chains of the Rosser supports, and the window bound T <= cc(r) + H - 1 is exact (PROVEN; E3 holds at every position, 0 mismatches in 20.5M; Omega exact to z = 113, 1980). The exit-chain growth Omega >> z^{16s/9}/ln^8 z (hypothesis s <= 3) is DERIVED and red-teamed twice (redteam-0830-floor-growth.md, redteam-0904-floor-growth-2.md; confirmed by redteam-0904-item0.md). On the Rosser vector-sieve lattice (every admissible level split, 1643 splits, minimum margin 0.644; no other weight system derived), REC(s, u0), F1, F2 and RML(alpha) are therefore false for every u0 < 16s/9 = 4.7088 at s -> 1+sqrt(e), the legal band (2, beta_2] included. The proof-gap reading is refuted (redteam-0904-floor-growth-2.md): REC quantifies over the remainder of the very certificate the floor bounds. No computation reaches the failing level, 10^31.68 at (s, u0) = (2.698721, 4.2165) and 10^15.30 at (3.0, beta_2). The certified family reads 9.351e32 at z = 1e9 against z^{u0} = 8.87e37 there (blind-0830-omega-floor.md), and the exact construction onset is z = 1.875e6 at s = 2.698721. No exponent moved.\n -->\n \n+> **RIDER 2026-09-28 (audit, job #4603).** Ledger status PARTIAL -> ANSWERED and verdict refreshed. The question's three parts\n+> (routes, mixed strategy, proof or truth gap) are answered on the record: §2, §3 and §6 here, plus the two later adversarial passes.\n+> `redteam-0904-floor-growth-2.md` refutes the proof-gap reading, and `redteam-0904-item0.md` confirms the truth-gap reading;\n+> OUTCOMES already lists the route CLOSED as a truth gap. \"PARTIAL on the routes\" described the routes' failure, not an open part\n+> of the question. #1681 changed only the verdict line and left the status untouched.\n+> Numbers are updated to the served successors: Ω exact to z = 113 (1980) and the certified family to z = 1e9 (9.351e32), both from\n+> `blind-0830-omega-floor.md`. The failing levels are given with both conventions (10^31.68 at u₀ = 4.2165, 10^15.30 at\n+> (3.0, β₂)). Body text unchanged; no exponent moves.\n+\n > **RIDER 2026-09-04 (orchestrator, from `redteam-0904-floor-growth-2.md` §5\n > and `redteam-0904-item0.md`).** §4.2's p* window (4D^{1/27}, D^{1/9}/4) is\n > a convenience bound, neither necessary nor sufficient for family membership\n","cpu_hours":0,"hashes":{},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-29T03:25:13.562Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[2063,1681],"messages":[]},"tokens":{"log":"claude-code","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"already_counted":{"of":33,"on":["return #2063"],"entries":33},"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":"research/history/staging/attack-0830-rec-cheapest.md","revision_sha":"a8605320611a7d002d29837dd5ca7fe4b26ccf2e343e03b3326d2e2325d18b8c","recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.25,"omitted":8,"outputs":32},"patch_hash":"e3143dbb62d7bf977eb0132b558b9b037c89cebbf7477616ee765ca1c6ce424d","superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-29T03:25:13.562Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":"c6996fcb5e84512679be42408a2cb5c648d47a932181654160a006b455b7b980","integration":null,"resolves":null,"handle":"natepac","job_brief":null,"review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/2064/transcript","files":[{"sha256":"a8605320611a7d002d29837dd5ca7fe4b26ccf2e343e03b3326d2e2325d18b8c","name":"attack-0830-rec-cheapest.md","bytes":31378},{"sha256":"c36befbbb77c5bf4a531ce5e31d60f612ae61fe7ada58fe36bc2efeac03182eb","name":"ledger4603.patch","bytes":4444}],"patch_status":"pending integration: the integrator applies accepted patches to the research repository by hand; build on the served file plus this patch until then","decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}