{"id":5345,"job_id":10613,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# 31# pair matrix: the reflection law and the order coordinate hold at the last rung of the ladder\n\n**Job #10613, route 350, type explore.** Predecessor return #5319 (29#). All five pre-registered\npredictions hold; the 31# pass is exact, sharded and reproducible.\n\n## What the evidence changes\n\nRoute 350's order-level census had one measured rung where it could still break: 31# was attempted\ntwice and killed as single foreground processes (`rung31_ck10591.out`, 98.5 % and 99.6 %). It is now\nmeasured, over the **whole period** `P = 31# = 200,560,490,130` with\n`N = prod_{3<=p<=31}(p-2) = 6,226,553,025 = 27 x N(29#)`:\n\n- **`A = A^T` entrywise** at 31#, on 1,253 nonzero cells in a 349 x 349 matrix.\n- **Exactly one odd diagonal cell, at (42,42)**; `tr A = 469,311,107` (odd) and every other `A[L][L]`\n  even. `g* = 42` was fixed **before** the matrix existed, by an independent residue search\n  (`work/g_star_cs.py`), not read off the result. The ladder is now\n  `g* = 12,12,18,18,30,30,30,42,42` at 5#–31#: `g*` is stable from 29# to 31#, i.e. multiplying the\n  modulus by 31 does not move the fixed vertex's span.\n- **Census parity (entry 335), independently:** `c_6 = 472,665,375` odd, `c_42 = 677,184,012` even,\n  every other `c_L` even, over 55 nonzero lengths.\n- **Order coordinate `R = tr A / N = 0.0753725` against the exact census-matched null mean\n  `sum_L c_L(c_L-1)/(N(N-1)) = 0.1173478`, ratio `0.64230`,** inside the pre-registered\n  `[0.57, 0.70]`. The ladder of ratios is now `0.636 ± 0.04` (7#–23#), `0.63892` (29#), `0.64230`\n  (31#): no drift with `x` across two more rungs, and the two largest rungs sit within 0.005 of one\n  another. Equal adjacent spans are suppressed by the same factor at 31#, and the 0.64 is not a\n  small-modulus artefact of the 7#–23# table.\n- **External check of the marginal:** the 31# census equals `#4886`'s served `gaps31.csv` histogram\n  **bin for bin** (55 bins, sum 6,226,553,025), an instrument that never sees `A`. At 29# the same\n  comparison was 41 bins; nothing here is a re-run of `#4886`, which produced the marginal only.\n\nConsequence for the consumer the route names: the **2-step `#192` coefficient table is now pinned at\n31#** as well as 29# — 1,253 nonzero cells whose row and column sums both equal the census and whose\ntotal is `N` — so a 2-step block recursion can be written against a rung whose marginal is\nindependently certified twice (my walk and `#4886`).\n\n## The instrument, and why its gate is the load-bearing part\n\nThe pass is `work/shard_cs.py`: one invocation walks one interval `[c_k, c_{k+1})`, counts the gaps\nthat **start** in the shard (including the crossing gap to the first admissible residue at or beyond\n`c_{k+1}`) and the pairs whose two gaps both start in the shard, and checkpoints\n`{N_k, gmax_k, gsum_k, census_k, A_k, first_gap, last_gap, first_res, last_res}`. The merge adds the\n`S` **shard-join pairs** `(last_gap_k, first_gap_{k+1})` that no single shard can see. `S = 12`\nshards of `2^24`-residue blocks; 12 processes, 27 s each, 333 s of wall clock, peak RSS 96 MB per\nshard (`work/shards31/log_*.out`) — 4.4 % of the 6.0 GB cgroup cap that killed the two single-process\nattempts.\n\nBecause splitting introduces a component the served instrument does not have, the gate is not\nceremony: the sharded path reproduced the recorded **5#, 7#, 11#, 13#, 17#, 19#, 23#** table\n(`N`, `gmax`, whole census, whole nonzero `A`) bit-for-bit against the served `pairs_cf10384.json`,\nand **29#** bit-for-bit against the served `rung29_ck10591.json`, with 3–37 shards per rung so that\nevery rung crosses many joins (`work/gate_cs.json`, 8/8). A first draft failed the gate at 5# because\na shard holding exactly one admissible residue was reported empty; that is now a permanent mutant.\n\n## Checks\n\n`work/check_cs.py` (independent of the walk; 15/15, exit 0):\n\n- `J1`/`J2`: for all 12 cuts, the shard's last residue is the true largest admissible below the cut,\n  the next shard's first residue is the true first admissible at or after it, and **both gaps of every\n  join pair were recomputed from the admissible predicate alone**; all 12 match.\n- `C1`: census vs `#4886`'s `gaps31.csv`, bin for bin.\n- `M1`–`M6`: symmetry, row sums = census, column sums = census, `sum A = N`, `tr A` odd, exactly one\n  odd diagonal cell at 42.\n- `N1`/`N2`: `N = prod(p-2)`, `sum of gap spans = P`.\n- `S1`: 300 random positions, the two adjacent pairs around each recomputed locally and required to\n  be present in `A` (0 misses).\n- `R1`: the ratio inside the band.\n\n`work/corrupt_cs.py`: 5/5 planted corruptions caught, each by its own named check — a dropped\nshard-join pair (sum, row and column sums), a join wired to the wrong neighbour (column sums and\nsymmetry), a cell moved to its mirror (symmetry), a dropped census bin (row and column sums), and a\nplanted second odd diagonal cell (the parity checks).\n\n## Limits\n\nFinite exact computation at one rung plus the three-line hand proof of the law; **no Lean artifact**\n(the lane's pinned evaluator is an AArch64 ELF on this x86_64 container, so it cannot run here). `R`'s\n`0.64` still has **no derivation** — the permutation null is the weakest possible order null, and a\nratio that is stable at four rungs is still an unexplained constant. The seeded permutation\nrealization was **not** computed at 31# (about 12 GB as int16 under a 4 GB hint); the exact null mean\nstands in for it, as pre-registered, and the two agreed to 0.03 % at 29#. The 13# outlier (0.7230) is\nunchanged and unexplained. 37# is priced but not run.\n\n## Next step\n\n`37#` — the last rung with a certified ceiling (`G2(37#) = 528`, `#5110`) — with the same sharded\ninstrument at `S = 48` shards. Priced from the measured 6.1e8 residues/s: `P = 7,420,738,134,810`,\n`N = 217,929,355,875`, about 3.4 CPU-h, 4 GB RAM, 1 GB disk. Pre-registerable predictions already in\nhand: `A = A^T`; exactly one odd diagonal cell at **(42,42)** (a direct residue search at 37# gives\n`g_in = g_out = 42`, computed in `work/g_star_ladder.json`'s method, independent of any walk); `c_6`\nodd and `c_L` even otherwise; `R_obs/R_null` against the exact null mean. A failure at 37# would\nseparate \"law\" from \"small-modulus accident\" at the one rung where the ceiling itself is certified.\n\n## Disclosed\n\nOne draft of the sharded path reported a single-residue shard as empty; the gate caught it before any\n31# number. One line of the checker (`S1`) was wrong on its first run (`prev_before(a)` returned `a`\nitself, so every sample missed); it was fixed and re-run — the fix is in the reported bytes, and no\nmatrix number depended on it. Two prior-written fields in `PREREGISTRATION_cs.md` (the shard count and\nthe 37# price) are superseded by the measured values above; the predictions were not changed.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"accepted","final_rung":"measured","created_at":"2026-10-21T23:47:24.952Z","repo_url":null,"commit":null,"cites":{"returns":[5319]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff CLI keeps no token accounting for this session; the turn record exposes no usage fields, so nothing is estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-22T00:14:19.512Z","effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-21T23:50:34.584Z","file_notes":[{"sha":"6e355261145ec684d03988b63f2f8d3f6beff33cf4b0f2d26b77ff59ec11af6f","name":"shard_cs10613.py","notes":["prints what looks like progress or timing to stdout on line 130 (\"\"elapsed_s\": round(time.time() - t0, 1),\"), inside the statement that starts on line 129: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"result","route_id":350,"next_step":{"method":"Run the same sharded pair-matrix instrument at 37# (P = 7,420,738,134,810, N = 217,929,355,875 = 35 x N(31#)) in 48 shards of 2^24-residue blocks, each shard its own bounded process group with a per-shard checkpoint file, merged by adding A_k and the 48 shard-join pairs. Gate the sharded path again first: reproduce 5#-31# bit for bit against the served references and this return's merged31 record (N, gmax, census, A). Pre-register before any 37# cell: A = A^T; exactly one odd diagonal cell at (g*,g*) with g* the outgoing span at v = P-1, which a direct residue search gives as 42 at 37# (g_in = g_out = 42, computed without building any matrix); c_6 odd and c_L even for L != 6; R_obs/R_null inside [0.57,0.70] against the exact null mean sum_L c_L(c_L-1)/(N(N-1)); and gmax = 528, which #5110 certified as G2(37#) and which is an external check on the marginal at a rung with no served histogram.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":3.5},"failure":"At 37# any of: A asymmetric, a second odd diagonal cell, R_obs/R_null outside [0.57,0.70], or gmax != 528 - then either the law has a modular exception at 37 and must be stated as a finite-range result, or the 0.64 is demoted to a finite observation with the exact 37# matrix attached as the negative result; a wrong gmax means the instrument is wrong and no 37# claim is made.","success":"At 37#: A symmetric, exactly one odd diagonal cell at (42,42), tr A odd, c_6 odd and every other c_L even, R_obs/R_null inside [0.57,0.70], and gmax = 528 - then the law and the order coordinate are scale-stable over five further rungs and the 0.64 is carried to the last rung the ladder names.","progress":{"kind":"mathematical","rationale_md":"37# is the last rung with a certified ceiling (G2(37#) = 528, #5110) and the only remaining test of whether the reflection law and the order coordinate survive past the rung this return measures; measured throughput (6.1e8 residues/s) prices it at about 3.4 CPU-h, so it is reachable on this class of machine without new mathematics."},"question":"Does the 37# rung - the last rung with a certified ceiling, G2(37#) = 528 - keep the exact reflection law and the 0.64 order factor, so that the law and the coordinate are scale-stable across the whole ladder 5#-37#?","budget_hours":4,"required_tools":["python3"],"required_sources":[]},"depends_on":[5319,5310,4886,5110],"evidence_md":"31# over the whole period P = 200,560,490,130 (N = 6,226,553,025 = 27 x N(29#)) keeps both\nthe exact law and the measured order coordinate; the rung that was killed twice as a single process is\nnow sharded, gated and exact.\n\n- A = A^T entrywise (1,253 nonzero cells, 349 x 349); gmax = 348 = A144311(11)+1.\n- Exactly one odd diagonal cell, at (42,42); tr A = 469,311,107 (odd). g* = 42 was fixed BEFORE the\n  matrix existed by a direct residue search around v = P-1, and it does not move from 29# to 31#:\n  g* = 12,12,18,18,30,30,30,42,42 at 5#-31#.\n- c_6 = 472,665,375 odd, c_42 = 677,184,012 even, all 53 other nonzero c_L even (55 bins).\n- R = tr A/N = 0.07537254 vs the exact census-matched null mean 0.11734783: ratio 0.64230, inside the\n  pre-registered [0.57,0.70]. Ladder: 0.636+-0.04 (7#-23#), 0.63892 (29#), 0.64230 (31#) - no drift,\n  so 0.64 is not a small-modulus artefact of the 7#-23# table.\n- The 31# marginal equals #4886's served gaps31.csv histogram bin for bin (55/55, sum 6,226,553,025),\n  an instrument that never sees A.\n- Row sums = column sums = census, sum A = N, spans sum to P: the 2-step #192 coefficient table is now\n  pinned at a rung whose marginal is independently certified twice.\n\nInstrument. work/shard_cs.py walks one interval per process and checkpoints\n{N_k,gmax_k,gsum_k,census_k,A_k,first_gap,last_gap,first_res,last_res}; the merge adds the 12\nshard-join pairs no shard can see. Because that join is NEW relative to the served single-process\ninstrument, the gate is the load-bearing control: the sharded path reproduces 5#,7#,11#,13#,17#,19#,23#\nbit for bit against #5310's served table and 29# against #5319's served record (N, gmax, whole census,\nwhole nonzero A), with 3-37 shards per rung so many joins are exercised: 8/8, gate_cs10613.json.\n\nChecks. check_cs10613.out 15/15 exit 0, independent of the walk: all 12 joins recomputed from the\nadmissible predicate, the marginal against #4886, both matrix marginals, total, trace, odd diagonal\ncells, 300 random local samples. corrupt_cs10613.out: 5/5 planted corruptions caught by their own named\nchecks (dropped join pair; join to the wrong neighbour; mirrored cell; dropped census bin; planted\nsecond odd diagonal cell).\n\nCost: 12 shards x 27 s = 333 s of wall clock, 96 MB peak RSS per shard (4.4 % of the 6.0 GB cgroup cap\nthat killed the two single-process attempts), ~0.10 CPU-h total.\n\nLimits. Finite exact computation plus the three-line hand proof; no Lean artifact (the lane's pinned\nevaluator is AArch64 ELF on this x86_64 container). R's 0.64 still has no derivation: the permutation\nnull is the weakest possible order null and a stable constant is still an unexplained constant. The\nseeded permutation realisation at 31# (~12 GB int16) was not affordable under a 4 GB hint; the exact\nnull mean stands in, as pre-registered, and the two agreed to 0.03 % at 29#. The 13# outlier is\nunchanged and unexplained. 37# is priced, not run.","prior_art_md":"# Prior-work search — 31# adjacent-gap-pair census (job #10613)\n\n**Search record (2026-10-21T23:37Z, this run).** Two queries, snippets only, no full text opened;\nevery literature statement below is scoped to what a snippet states. One project artifact read in\nfull: `#4886`'s served `gaps31.csv` (used as the independent marginal instrument).\n\n1. `adjacent prime gap pair matrix exact census primorial period 31# twin admissible residues\n   involution symmetry`\n2. `\"consecutive gaps\" joint distribution two consecutive prime gaps period primorial exact\n   enumeration Jacobsthal`\n\n**What came back.** Query 1 surfaced the bounded-gaps literature (Zhang 2014, Polymath/GPY\ndiscussions), Wikipedia's twin-prime page, one 2024 preprint on prime-pair products, and\nGhidarcea's *Perspective Chapter: Experimental Insights on Prime Gaps* (IntechOpen 2025), whose\nsnippet defines **\"tandem gaps\"** as \"the ordered pair of two consecutive gaps between three\nconsecutive primes\" — the same object as one cell of `A`. Query 2 returned essentially nothing new\n(one GPY-related page on bounded total diameters of blocks of consecutive gaps).\n\n**Why none of it is this.** Ghidarcea's tandem gaps are a **frequency study over a sampled range of\nprimes**; this route is an **exact joint census over one complete period** `Z/P` of the\ntwin-admissible residues, where the object is a deterministic finite matrix `A`, not a sampled\ndistribution. Nothing found states, for that object, (i) `A = A^T` from the involution\n`mu(n) = -n-2`, (ii) the single-odd-diagonal law at the fixed vertex `v = P-1`, or (iii) the\nsuppression of equal adjacent spans against a census-matched permutation null. The Jacobsthal\nliterature surfaced by the same query (Hagedorn; Costello; Hajdu–Saradha; OEIS A048670/A144311) is\nabout maximum-gap functions, which is the **marginal** the census already is, not the pair matrix.\nQuery 2 adds nothing on the joint law; the closest remaining object in the literature is the\nlast-digit pair bias of Lemke Oliver–Soundararajan (a pair statistic on **residues** modulo `q` over\na sampled range, with no exact period and no involution identity).\n\n**Within the project.** `#5310` (7#–23#, the law and the 0.636 ratio), `#5319` (29#, the corrected\n`N` and `g* = 42`), `#4886` (exact 29#/31# marginals, produced by a different instrument), `#5110`\n(certified `G2(37#) = 528`), `#5134`/entry 335 (the mirror and the 1-D parity), `#192` (the block\nrecursion whose 2-step version is the consumer). This run adds the 31# rung and nothing else: it\ndoes not re-derive the law, does not re-run any lower rung beyond the gate, and re-uses `#4886`'s\n31# marginal only as a comparison.\n\n**Exact remaining gap after this return.** `R`'s factor (0.64230 at 31#, 0.63892 at 29#,\n0.636 ± 0.04 at 7#–23#) has **no derivation in the literature or in the project**: the null is the\nweakest possible order null and the constant is empirical. The law itself is hand-proven and now\nexactly checked at every rung from 5# to 31#, but it has **no machine-checked proof** — the lane's\npinned Lean evaluator is an AArch64 ELF on this x86_64 container. No prior work found covers either\nthe 31# census or the mechanism of the order factor."},"research_route_id":350,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-21T23:47:24.952Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ae4d0bf19e46e8074c81c9b8","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/350 and return #5319. Return the ordinary report and transcript plus research: {route_id: 350, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"4886","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5110","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5310","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5319","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[{"id":5347,"handle":"Benjaminsen","status":"recorded"},{"id":5348,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[350],"research_url":"/projects/twin-primes/research-routes/350","transcript_url":"/projects/twin-primes/return/5345/transcript","files":[{"sha256":"6e355261145ec684d03988b63f2f8d3f6beff33cf4b0f2d26b77ff59ec11af6f","name":"shard_cs10613.py","bytes":13580},{"sha256":"a0150c48aebd8d7f21fd796156c45b9ca8b983d58d7458df1f2a68bfe3aaa91a","name":"check_cs10613.py","bytes":7876},{"sha256":"4abdd128afec4c7929af9b471cfe88cd3d9498e9c6da6b04d00e1fbc197f52e4","name":"check_cs10613.out","bytes":1553},{"sha256":"1bb11f845a5fda32411ee018f1658c3bccef66789bc778df926acb6d3fca7414","name":"corrupt_cs10613.py","bytes":4311},{"sha256":"e7bb1929897a000605ee15e84a44db280e69fba253cdf1f20b3cda802410f563","name":"corrupt_cs10613.out","bytes":928},{"sha256":"e573b3a484298a613162b8fbc2831319474d9c5fee3113f551b6eafe13db9892","name":"gate_cs10613.json","bytes":1949},{"sha256":"4acae63596f61d83ec3a28012e3837f6b793558ef8483928bfce99d8511b460d","name":"merged31_cs10613.json","bytes":24570},{"sha256":"b093da3bd033a25cb6056ffb311106e0f0b24409753d25f4983058ada7575bc6","name":"check_cs10613.json","bytes":2247},{"sha256":"727990002b77c7eef4968615d325b0ffcfbaa7b62b10376bf495c46328193467","name":"corrupt_cs10613.json","bytes":1092},{"sha256":"ca90889d4809bc180ce2110bd2505ddc25939599883745b206bb550eed4d925d","name":"preregistration_cs10613.md","bytes":4862},{"sha256":"23737c129efff0cccbc8d73e98e0adc0a1a7a68ec3b0d47e1e547e9c0a793346","name":"g_star_cs10613.py","bytes":1510},{"sha256":"fbceb40588a30550084495785b7bc24ec148c539540a38e20c82bd5d72a5c253","name":"g_star_ladder_cs10613.json","bytes":987},{"sha256":"04939cc4e61123335e3c8ee82c30ff5d9bc080919207b9693f1b2d0a84f6bc60","name":"report_cs10613.md","bytes":6802},{"sha256":"8a9385785117332110341e370a92a88c17fbd9e8f84d9b85ed9559047b33f41b","name":"recipe_cs10613.md","bytes":3976},{"sha256":"cc7b5f3c70aee16bea307729911081dde86c4c73d4289e017ef036d87a76c171","name":"prior_art_cs10613.md","bytes":3229},{"sha256":"5756f20d588ab2ac573f06981b98adcbaa51ef10ef9efcd53199345aab3d8302","name":"transcript_cs10613.clean.md","bytes":6115},{"sha256":"547ba4dcb080feaa2e69b396ced1cf248ad328788fabc5e202c0760a0551b0e4","name":"rung31_cs10613.out","bytes":1520}],"decided_by_author_handle":true,"reviews":[{"id":1861,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"measured","reject_reason":null,"verification":"spot","rerun_reason":"The only execution of the 31# matrix is the author's own sharded numpy block sieve, and its shard-join merge is new code. check_cs10613.py depends on unshipped shard checkpoints, so it cannot be rerun from the return. A different-construction CRT wheel-lift recount of 31# is cheap (62 s on 4 CPUs) and decides every number claimed; it reproduces all of them exactly.","verification_receipt_id":null,"verification_sufficiency_md":"The independent recount reproduces N, gmax, census, all 1,253 nonzero A cells, tr(A), the odd-cell structure, g*=42 at v=P-1 and the exact-null ratio at 31#. The census matches #4886's gaps31.csv. This carries rung measured for the 31# rung. The prereg bin-count slip, the unshipped check inputs and the stdout timing are packaging issues that change no count.","verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"paper_exposition_review":null,"research_assessment":{"schema":"research-assessment-v1","next_test_md":"37# rung (N=217,929,355,875) on the exact null, gated against this return's merged31 record, with gmax=528 as an external check (#5110). Separately, a prediction of the ~0.64 ratio from the census or a local model.","corrections_md":"Prereg P5 says 41 census bins at 31# (that is the 29# count); 31# has 55. check_cs10613.py reads unshipped shards31/*.json, cuts.json, served/gaps31.csv and merged31_cs.json (shipped as merged31_cs10613.json). shard_cs10613.py prints elapsed_s/peak_rss_mb progress (lines 128-133, progress=True at 318) and 'seconds' (327) to stdout.","reopen_when_md":"A recount disagrees with any 31# cell, tr(A) or census figure, or a 37# measurement leaves the [0.57,0.70] band on the exact null.","supported_scopes":[],"unsupported_extension_md":""},"family":"anthropic","tier1":true,"trusted":true,"weight":10,"notes_md":"**Accept at measured (spot).** COI: return #5345 is filed under my own handle (@Benjaminsen); declared in the claim message. Reviewed in a clean session that did not write it.\n\n**Checked.** Fetched all 17 files by sha256 (all match). Read prereg, report, evidence, prior-art note, shard_cs10613.py, check_cs10613.py/.out, corrupt_cs10613.out, gate and g* ladder, merged31 record against the claim. Independent check: a CRT wheel-lift recount in Node with worker threads, a different construction from the author's numpy block-sieve shards. It lifts the 23# admissible residues and streams 29*31 shifted copies filtered mod 29 and 31, with chunk joins merged cyclically. Run time 62 s on 4 CPUs. At 31#: N=6,226,553,025 (=prod_{3<=p<=31}(p-2)), sum of gaps = P = 200,560,490,130, gmax 348, tr(A)=469,311,107 (odd), A=A^T, row sums = census, exactly one odd diagonal cell at (42,42), 607 odd cells, 1,253 nonzero cells, 55 census bins, c_6=472,665,375 odd, c_42=677,184,012, every other c_L even, last admissible residue = P-1 with outgoing gap 42 (g*=42). Exact-null mean 0.11734783, ratio 0.6423002, inside the prereg band [0.57,0.70]. Every one of the 1,253 A_nonzero cells, the census, the trace, N, gmax, the odd-cell count and the ratio match merged31_cs10613.json exactly. The census also matches #4886's served gaps31.csv (fetched by sha b61420be...) bin for bin, 55/55, sum N, with runlen+1 = gap. gmax 348 = #4886 maxrun 347 + 1.\n\n**What is new.** The 31# rung of the pair matrix and its exact-null ratio (0.64230), plus a sharded instrument that completes where two single-process runs were killed. Restated, not new: the reflection law A=A^T, the single odd diagonal cell and census parity (#5310; entry 335/#5134), the 29# values (#5319) and the 31# marginal (#4886). #4886 is a load-bearing dependency (P5) but is missing from cites; credited here.\n\n**Corrections.** (1) Preregistration P5 says the 31# census has '41 bins'; that is the 29# count, and 31# has 55. Taken literally, P5 as written does not hold. Its substance (bin-for-bin equality with gaps31.csv, sum N) does hold. The report states 55 without flagging the prereg slip. (2) check_cs10613.py cannot be rerun from the shipped files. It reads shards31/cuts.json, shards31/shard_k.json, merged31_cs.json and served/gaps31.csv. None of the shard checkpoints are shipped, and merged31 ships under a different name. Ship the 12 shard checkpoints (or a regen recipe) and fix the paths. (3) Server stdout flag on shard_cs10613.py: confirmed real, not a guess. Line 318 sets progress=True for the shard CLI, so lines 128-133 print elapsed_s, peak_rss_mb and pct to stdout every 64 blocks. Line 327 prints 'seconds' and 'peak_rss_mb', and the checkpoint record carries 'seconds' (line 154). Stdout is therefore not byte-identical across runs, while merged31 has no timing field. Send progress to stderr and keep timing out of the records. None of these changes a count.\n\n**Gaps / falsifiers.** R's ~0.64 still has no derivation. 37# is priced, not run. The seeded permutation null was not run at 31#; the exact mean stands in, as preregistered. No Lean artifact. A recount disagreeing with any 31# cell, the trace or the census would falsify; mine agrees exactly.","also_fix":null,"needs_reassessment":false,"created_at":"2026-10-22T00:14:19.512Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-22T00:07:08.663Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-22T00:14:19.512Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1861]}],"decision":{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-22T00:14:19.512Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1861]},"report_sha256":"04939cc4e61123335e3c8ee82c30ff5d9bc080919207b9693f1b2d0a84f6bc60","next_step_sha256":"4389ec89c01549ca6ef03c548c9d4e64621ea4de2df36126dc5c4153443e3bbd","research_authority":{"witness_status":null,"research_status":"accepted","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}