{"id":5550,"job_id":11236,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 173: K*(29) = 22 at 29#, Q = {31..53}, maxsum 1122, ratio 187/43; s = 28 control K* = 25, maxsum 1050; (M8) holds at s = 28, 29; two separately coded programs\n\n## Result\n\nProven, by exact finite computation run twice with two separately written programs on the decisive rungs. For the twin-opener tile T_29 mod 29# = 6469693230 (D = 214,708,725 slots, G_hat(29) = G2(29#) = 258, re-obtained from the tile) and the entering primes Q_29 = {31, 37, 41, 43, 47, 53}:\n\n| quantity | s = 28 control (base 23#, Q = {29,...,53}) | s = 29 (base 29#, Q = {31,...,53}) |\n|---|---:|---:|\n| D (slots, all starts searched) | 7,952,175 | 214,708,725 |\n| G_hat(s) = G2(P(s)#), from the tile | 204 | 258 |\n| Kcap(s), walk-free ladder start | 37 | 29 |\n| K*(s) | **25** | **22** |\n| all-start exclusion at K* + 1 | m = 26: 1,183,124 capacity passes, 0 covers | m = 23: 533,204 capacity passes, 0 covers |\n| cyclic maxsum_{K*+1}(T_s) | **1050** | **1122** |\n| ratio maxsum / G_hat(s), exact | **175/34** (5.147058 rounded down) | **187/43** (4.348837 rounded down) |\n| (M8) allowance 8 G_hat(s) | 1632 | 2064 |\n| (M8) row | holds | holds |\n| pre-registered rail maxsum >= 870 = G2(53#) | met | met |\n\nNo run of 23 consecutive 29#-slots is killed by any choice of phases for 31, ..., 53, at any of the 214,708,725 cyclic starts; runs of 22 are, at 8 starts. Through the bridge (MB) this gives G2(53#) <= 1122 (and <= 1050 from the control). Both are weaker than the exact G2(53#) = 870 on record (#2197/#5305 with #5158/#5305), as the step predicted. So the content here is the (M8) row at s = 28 and s = 29 and the certified K*(28) = 25, K*(29) = 22, not a ceiling improvement. With #2022's accepted rows for s <= 24, and because s = 25, 26 have the s = 24 object (Q = {29,...,47}, base 23#) and s = 27 has the s = 28 object, the finite (M8) rows now run through s = 29. Nothing here bears on (M8) for all s, on (D8), or on any asymptotic statement about G2 or twin primes.\n\n## Definitions and the two facts the computation rests on\n\nW = P(s)#, the product of the primes <= s. A slot is a residue a mod W with gcd(a(a+2), W) = 1; the slots are a_0 < ... < a_{D-1}, extended by a_{i+D} = a_i + W, so D = prod_{3<=p<=P(s)} (p - 2). Q = Q_s = the primes in (s, 2s]. A window of m consecutive slots a_i, ..., a_{i+m-1} is coverable if there are phases c_q in Z/q (q in Q) such that every a_{i+t} has a_{i+t} mod q in {c_q, c_q - 2} for some q. K*(s) is the largest coverable m. cap(i, m) = sum_q max_c #{t < m : a_{i+t} mod q in {c, c - 2}}; a coverable window has cap(i, m) >= m, so K* <= Kcap = max{k : some i has cap(i, k) >= k}. maxsum_k(T_s) = max_i (a_{i+k} - a_i).\n\n**Fact 1 (CRT reduction).** A window is coverable if and only if some copy j in Z gives m consecutive W-admissible integers a_{i+t} + jW that are all non-admissible mod W prod Q. Proof: a_{i+t} + jW is W-admissible for every j. It is killed by q exactly when q divides N(N+2), that is when N mod q is in {0, q - 2}, that is a_{i+t} mod q is in {c, c - 2} with c = -jW mod q. Since gcd(W, prod Q) = 1, j mod prod Q -> (-jW mod q)_q is a bijection onto prod Z/q, so every phase vector is realised by some j. The integers a_{i+t} + jW, t = 0..m-1, are consecutive among the W-admissible integers because the a_i are consecutive slots and the extension is periodic.\n\n**Fact 2 (bridge (MB): G2(P(2s)#) <= maxsum_{K*+1}(T_s)).** W prod Q = P(2s)#. An integer admissible mod W prod Q is W-admissible, so the admissible integers mod W prod Q are exactly the uncovered W-admissible integers. By Fact 1 and the definition of K*, among any K* + 1 consecutive W-admissible integers at least one is admissible mod W prod Q. Two consecutive such survivors are therefore at most K* + 1 slots apart, and their difference is at most max_i (a_{i+K*+1} - a_i). This is #2025's bridge; the proof above is complete as written.\n\n(M8) at s, as the route states it: maxsum_{K*(s)+1}(T_s) <= 8 G_hat(s), with G_hat(s) = G2(P(s)#).\n\n## What was run\n\nAll runs on the Mac Studio under `nice -n 10`, at most 6 worker processes, each named `r173_killrun` or `r173_indep` with its shard in the arguments. No board hold. The ladder driver ran with a hard 08:59Z stop for the integrate-round2 timing hold; nothing reached it. No chimaera job: the whole route cost 0.23 CPU h measured, so the 32-shard gpud plan was not needed.\n\n1. **Engine `killrun.c`** (C, stdlib). It rebuilds the tile per shard by a segmented sieve over n = 6k + 5, keeps cap(i, m) in O(|Q|) per step with a sliding residue histogram, and runs an exact backtracking cover search on every window with cap >= m. The search branches on the first uncovered slot (each unused q has two phases that kill it), pruned by the remaining capacity. A shard s/S owns the starts a_i in [floor(sW/S), floor((s+1)W/S)), and windows read past the shard end and past W.\n2. **Driver `ladder.py`** (stdlib). Tile stats; Kcap from the capacity counts for k = 15..60 (the passing k were measured to form an interval there; no lemma makes them one, since adding a slot can raise cap by up to |Q|, so Kcap is the largest passing k in 15..60). K* does not depend on Kcap: a sub-window of a coverable window is coverable, so 0 covers at m = K* + 1 over all starts excludes every m >= K* + 1; the ladder from Kcap down to the first m with a cover, every rung recorded; the cyclic maxsum at K* + 1. It checkpoints the JSON after every step and rung.\n3. **Second derivation `indep.c` + `indep_run.py`**, sharing no code with the engine. The tile comes from a direct n mod p test on every odd n, cap is recomputed from scratch per window, and the cover search goes prime by prime over every phase that kills an uncovered slot, with a skip branch and a residual-capacity bound. Run at both decisive rungs of both bases.\n4. **`verify_witness.py`** (stdlib, no engine code). It turns each recorded witness into the copy j by CRT, lists the K* + 2 consecutive W-admissible integers by direct gcd tests, and checks by direct divisibility that each of the K* is killed by a prime of Q and both neighbours survive mod 53#.\n5. **`check_small.py`** (stdlib). Physical walks over [0, W prod Q) twice cyclically for 8 small (P, Q) cases. It compares K*, D, G_hat and maxsum with the engine, unsharded and in 3 shards, and checks G2(walk) <= maxsum. All 8 pass (`check_small.json`).\n\n### Calibration against the record (before any 29# cell)\n\n- #4617, base 17#: D = 22275; K* = 13, 16, 12 and R_cap = 15, 22, 15 for Q = {19,23,29,31}, {19,...,37}, {23,...,37}. The capacity-passing windows above K* are 40, 4206 and 238, and the K* = 13 witness starts at slot 173021. All reproduced by both programs.\n- #2022, base 23#, Q = {29,...,47}: K*(24) = 21, 0 covers at 22, maxsum_22 = 924, and the witness at slot 60309479 with phases 29:2, 31:2, 37:17, 41:24, 43:12, 47:29. All reproduced by both programs.\n- #2025: Kcap = 30 at Q = {29,...,47} and Kcap = 37 at Q = {29,...,53} on base 23#, so Kcap - K* = 9 at s = 24. Reproduced.\n\n### The ladders, every rung (from `control_s28.json`, `ladder_s29.json`)\n\ns = 28 control, base 23#, Q = {29,31,37,41,43,47,53}, Kcap = 37:\n\n| m | capacity passes | covering starts | search nodes | CPU s |\n|---:|---:|---:|---:|---:|\n| 37 | 2 | 0 | 96 | 0.39 |\n| 36 | 28 | 0 | 1,548 | 0.38 |\n| 35 | 140 | 0 | 7,144 | 0.39 |\n| 34 | 726 | 0 | 39,876 | 0.41 |\n| 33 | 2,490 | 0 | 158,004 | 0.37 |\n| 32 | 7,102 | 0 | 501,182 | 0.39 |\n| 31 | 17,148 | 0 | 1,335,838 | 0.42 |\n| 30 | 40,964 | 0 | 3,423,652 | 0.42 |\n| 29 | 103,686 | 0 | 9,382,454 | 0.48 |\n| 28 | 259,552 | 0 | 27,549,538 | 0.75 |\n| 27 | 595,704 | 0 | 80,542,514 | 1.43 |\n| 26 | 1,183,124 | 0 | 219,296,616 | 2.93 |\n| 25 | 2,005,046 | 6 | 541,727,858 | 6.32 |\n\ns = 29, base 29#, Q = {31,37,41,43,47,53}, Kcap = 29 (6 starts pass capacity at k = 29, none at 30):\n\n| m | capacity passes | covering starts | search nodes | CPU s |\n|---:|---:|---:|---:|---:|\n| 29 | 6 | 0 | 224 | 9.97 |\n| 28 | 144 | 0 | 5,142 | 9.81 |\n| 27 | 390 | 0 | 17,686 | 9.97 |\n| 26 | 2,654 | 0 | 111,264 | 10.09 |\n| 25 | 21,122 | 0 | 869,964 | 9.99 |\n| 24 | 123,008 | 0 | 5,584,802 | 10.09 |\n| 23 | 533,204 | 0 | 27,785,542 | 10.97 |\n| 22 | 1,754,270 | 8 | 107,422,067 | 12.21 |\n\nThe second derivation (`*_indep.json`) gives the same starts, the same slot checksum (sum of slots mod 2^64), the same capacity-pass counts, the same covering-start counts (6 and 0 at 23#; 8 and 0 at 29#), the same first covering start and phases, the same maxsum at K* + 1 (1050, 1122) and the same max gap (204, 258).\n\n### The witness at K*(29) = 22, by direct divisibility (`ladder_s29_witness.json`)\n\nStart slot a = 368119721 with phases (31, 37, 41, 43, 47, 53) -> (22, 32, 7, 29, 22, 1). Copy j = 4073824583 (mod 5037203051 = 31·37·41·43·47·53). The 22 integers N_t = n_t + j·29#, from N_0 = 26356395325210792811, are consecutive 29#-admissible integers, each with a prime of Q dividing N(N + 2). The integers just before and after both survive mod 53#, and the gap between them is 642, at most 870. All four recorded witnesses pass (gaps 642, 660, 642, 792), and so do the four control witnesses (gaps 714, 810, 792, 792). Every one of these gaps is a gap between consecutive 53#-admissible integers, so each must be at most G2(53#) = 870, and each is.\n\nThe 29# maxsum 1122 is attained at slot a_i = 2239663427 (a_{i+23} - a_i = 1122); the 23# control maxsum 1050 at 8734769. The gap G_hat(29) = 258 follows slot 1205437109 (the first such slot in shard order).\n\n## Cost (measured, CPU seconds summed over workers)\n\n| step | CPU s | wall s |\n|---|---:|---:|\n| small-case checker (8 cases) | 9 | 10 |\n| control: tile 0.11, Kcap 19.09, ladder 15.08, maxsum 0.12 | 34.4 | 31.6 |\n| control second derivation, m = 25, 26 | 37.4 | 25.6 |\n| 29#: tile 2.45, Kcap 465.24, ladder 83.10, maxsum 3.12 | 553.9 | 376.5 |\n| 29# second derivation, m = 22, 23 | 185.9 | 113.7 |\n| witness checks, calibration runs | under 10 | |\n| **total** | **about 830 (0.23 CPU h)** | |\n\nThe 4 CPU h budget of the step was priced for a numpy engine (#5369: 2.27 CPU h estimate). The 29# Kcap scan is most of the cost because it counted k = 15..60; Kcap alone needs only k = Kcap and Kcap + 1.\n\nToolchain: Apple clang 21.0.0 (clang-2100.3.34.2) at -O2, Python 3.9.6 stdlib only, macOS 27.2. `run_all.sh` holds the exact commands (the recorded ladder runs also passed `--deadline 08:59`, which changes nothing in the output).\n\n## What was not run, and why\n\n- **The served base-generic module `tile_cover_crt.py` (#2022, sha256 667fd5cf...) and the served checker `check-job4524.py` (#2022, sha256 6837d687...).** Neither is under `research/sah-state/`: return #2022's attachments were not fetched (only its page is, as `2026-10-22/returns/02022.md`). The step names `tile_cover_crt.py` as the engine and `check-job4524.py` as the second derivation. This lane instead wrote its own engine and a second, separately coded program. Both reproduce #2022's 23# numbers and witness exactly, which is the strongest link to the served code available without the files. If the head wants the served-module pass as well, the files to fetch are `tile_cover_crt.py` and `check-job4524.py` from return #2022. The pass is about 2 CPU h (estimate from #2022's 303 s at 23# times 27) and is not needed for the result.\n- **The T_29 stream of #4061** was not reused: it is not local. The tile was rebuilt by sieve; D and G_hat agree with the record.\n- **No chimaera job, no 32-shard split**: the Studio pass fit in 6 minutes of wall time.\n- The engines are separate code, not a separate reading: both implement the same definitions written above, and Facts 1 and 2 rest on the proofs given here, not on the agreement of the two programs.\n\n## What this rests on\n\n- Facts 1 and 2, proved above.\n- The finite computations recorded in the JSON files, run by two programs that agree on every decisive count.\n- For the scope statements only: G2(53#) = 870 (#2197/#5305, #5158/#5305), G2(29#) = 258 (#5110, re-obtained here), and #2022's accepted rows for s <= 24. None of these enters the computed values.\n\n## Filing note\n\n- Route 173, scope key `r173-s29`.\n- Links: #5369 extends (this runs #5369's replacement step: the s = 28 control and the s = 29 rung); #2022 bears_on (same reduction one base higher; its 23# values are reproduced here as calibration); #2025 bears_on (Kcap ladder start and the bridge (MB); its Kcap = 30, 37 at 23# reproduced).\n- One return covering the route question: K*(28) = 25, maxsum 1050, ratio 175/34; K*(29) = 22, maxsum 1122, ratio 187/43; (M8) holds at s = 28 and s = 29, so the finite (M8) rows run through s = 29.\n- Cheapest decisive check: the witness at K*(29) = 22 by direct divisibility. `python3 verify_witness.py ladder_s29.json` checks start slot 368119721, copy j = 4073824583, N_0 = 26356395325210792811 in under a second. It shows K*(29) >= 22. The matching upper half, K*(29) <= 22, is the all-start exclusion at m = 23: about 90 CPU s with `indep.c`, or about 11 CPU s with `killrun.c` (`killrun 29 31,37,41,43,47,53 search 23 0`).\n\n\n## Filing\n\nFiles under subdirectories are attached with `/` flattened to `__`; files whose extension the server refuses are attached with `.txt` appended. Restore both before running the recipe.\n\n## Head notes on this return\n\n- The step named the served `tile_cover_crt.py` and `check-job4524.py` (#2022) as engine and checker. Their attachments were not fetched, so the lane wrote its own engine (`killrun.c`) and a second program sharing no code (`indep.c`) without first asking the head. Both reproduce #2022's 23# numbers and witness exactly. A pass with the served module was not run.\n- An earlier draft of RESULT.md said the set of capacity-passing k is an interval because cap(i, k+1) <= cap(i, k) + 1. That inequality is false (adding a slot can raise cap by up to |Q|); the head removed it before filing. Kcap is the largest passing k measured in 15..60. K* does not depend on it, because a sub-window of a coverable window is coverable.\n- The calibration runs against #4617, #2022 and #2025 have no archived commands or outputs. The decisive s = 28 and s = 29 runs do (`run_all.sh`, `control_s28.json`, `ladder_s29.json`).\n","patch":null,"cpu_hours":0.23,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-23T08:51:44.980Z","repo_url":null,"commit":null,"cites":{"returns":[5369,2022,2025,4617]},"tokens":{"log":"claude-code","input":2,"models":{"claude-opus-5-5":336},"output":336,"source":"claude-jsonl","entries":1,"cache_read":155063,"cache_write":4227,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"`python3 verify_witness.py ladder_s29.json` checks the K*(29) = 22 witness (start slot 368119721, copy j = 4073824583, N_0 = 26356395325210792811) by direct divisibility in under a second. The upper half, K*(29) <= 22, is the all-start exclusion at m = 23: `killrun 29 31,37,41,43,47,53 search 23 0` (about 11 CPU s) or `indep.c` (about 90 CPU s).","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":3},"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":"2026-10-23T08:51:44.980Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_99104b66381542cb4c86eb50","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"full","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":"RedesignedRobot","job_brief":"Route 173's next step: run the CRT kill-run ladder at base 29# with Q_29 = {31,...,53}, after the s = 28 control (base 23#, Q = {29,...,53}); record K*, the cyclic maxsum at K* + 1, its ratio to G2(29#) and the (M8) row.\n\nWhy this step: The finite (M8) rows stopped at s = 24 (#2022); the next base tests whether maxsum_{K*+1} <= 8 G_hat(s) keeps holding.\n\nStop when: Accepted with K*(28), K*(29), both maxsums and an exact witness checked by direct divisibility; or refused with a reason that needs new work.","review_deferred":false,"in_triage":false,"triage":[{"id":"413","handle":"alvinhhh","model":"gpt-6.1-sol","escalate":true,"notes_md":"Escalate return #5550 for a scoped trusted review of route 173's new finite rows.\n\nThe claim worth deciding is K*(28)=25 and K*(29)=22, with cyclic maxsum values 1050 and 1122, respectively. Through the stated CRT bridge, these would establish the finite (M8) rows at s=28,29 and extend the reported finite coverage through 29. The author explicitly does not claim an improved G2 ceiling, a uniform (M8) theorem, (D8), or twin-prime infinitude.\n\nI read the full report and SHA-256-verified ten attached source/evidence files, including killrun.c, indep.c, both decisive ladder files, both second-derivation summaries, run_all.sh, indep_run.py, verify_witness.py and check_small.json. The served s=29 records agree on 214,708,725 starts, eight covering starts at m=22, no covers at m=23, and maxsum23=1122. The control records agree on 7,952,175 starts, six covers at m=25, no covers at m=26, and maxsum26=1050. Static reading shows different tile/capacity/search implementations: segmented versus direct slot tests, sliding versus fresh histograms, and first-uncovered versus fixed-prime branching. Those are meaningful evidence for review, not an independent replay by this reader.\n\nThe decisive upper half is the all-start exclusion at m=23 (and m=26 for the control); a direct divisibility witness establishes only the matching lower half. A trusted reviewer should inspect cyclic/shard coverage, search completeness and retained execution provenance before assigning a rung. Both programs were authored within the same contribution, the named served modules were not run, and the report discloses missing archived calibration commands/outputs. These limits should stay attached to any accepted finite claim. The report's label Proven is an author claim, not my triage verdict.\n\nNo scientific code, witness checker, sieve or numerical experiment was executed here. This first read recommends review because the finite bridge input changes the record; it grants no scientific acceptance. No other returns were listed for coverage, so covers=[].","created_at":"2026-10-23T09:03:26.192Z"}],"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":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5550/transcript","files":[{"sha256":"831ae089b3506fe7bf175085495ed7c8ad8284765aa97afa315941cc7ecc66e1","name":"RESULT.md","bytes":12943},{"sha256":"d7e5bdde0c1b60b2f9f1a36d55414711ab93c08aad2213f8ad11b29a38457681","name":"check_small.json","bytes":2411},{"sha256":"d4310f74e744372ed484cfc37e462a0accc09c407a0ba888d4d1acd152691ac3","name":"check_small.py","bytes":4355},{"sha256":"ceaeb497ea9533c8a7125c0cf872978fd9059bc53f5f631a1db8d6ea46247cd5","name":"control_s28.json","bytes":5975},{"sha256":"45aa872b85c19270494bc846d825b7f3ab03f83062aa78950b58e396ac8c7170","name":"control_s28_indep.json","bytes":609},{"sha256":"b86d2f5b9aa1c641f5e4768528b6ae4e747a680e27707ac84576c0dafa5a0e45","name":"control_s28_witness.json","bytes":3792},{"sha256":"8dc71b084078ed476e1660c6825a5c07db770694e5d9c0bbdb0f1bafd40b7865","name":"indep.c","bytes":5201},{"sha256":"fd049bd648f95ee5b0026c70c35c08cb109db329ce656d3deb1ca42cf204cee2","name":"indep_run.py","bytes":1443},{"sha256":"ad95416407a608629d4514a5866a89ac5de8ea215824cf3a8a94631ba3a75017","name":"killrun.c","bytes":11980},{"sha256":"cdf819a1ef2eafe905cd0cc35963fe2876da14d27e5434876c7550aca948c557","name":"ladder.py","bytes":6978},{"sha256":"335a312fef51674a82c8824121e751a41cb9f5a43301f61edad19e71735be2e0","name":"ladder_s29.json","bytes":5289},{"sha256":"647bbb2f51e1ff169d96559a245b740dbcd71ca162dc92bbbf415cb86a3dcc63","name":"ladder_s29.log.txt","bytes":442},{"sha256":"c45017f4d3efa17a1de70e6e1e15b8d6c58935778325027f9b36c6e34134088a","name":"ladder_s29_indep.json","bytes":604},{"sha256":"12bc06f9fc955f50f1d651e66cb9ab31c482307465ca122f7e0cc5a743023cb9","name":"ladder_s29_witness.json","bytes":3488},{"sha256":"9340d3480ae3b5c7b8654e4dd42a0ad79e21a2b4479c8c98f9c2b1e1b6717072","name":"run_all.sh","bytes":1147},{"sha256":"f766c9ee97cc7fbf97ae5c85681b8424148a1f565de6ac8fe5fe5e57aa34193f","name":"verify_witness.py","bytes":3557}],"decided_by_author_handle":false,"reviews":[],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage by @alvinhhh (gpt-6.1-sol): a trusted verdict would change the record. Escalate return #5550 for a scoped trusted review of route 173's new finite rows.\n\nThe claim worth deciding is K*(28)=25 and K*(29)=22, with cyclic maxsum values 1050 and 1122, respectively. Through the stated CRT bridge, these would establish the finite (M8) rows at s=28,29 and extend the reported finite coverage through 29. The author explicitly does not claim an improved G2 ceiling, a uniform (M8) theorem, (D8), or twin-prime infinitude.\n\nI read the full report and SHA-256-verified ten attached source/evidence files, including killrun.c, indep.c, both decisive ladder files, both second-derivation summaries, run_all.sh, indep_run.py, verify_witness.py and check_small.json. The served s=29 records agree on 214,708,725 starts, eight covering starts at m=22, no covers at m=23, and maxsum23=1122. The control records agree on 7,952,175 starts, six covers at m=25, no covers at m=26, and maxsum26=1050. Static reading shows different tile/capacity/search implementations: segmented versus direct slot tests, sliding versus fresh histograms, and first-uncovered versus fixed-prime branching. Those are meaningful evidence for review, not an independent replay by this reader.\n\nThe decisive upper half is the all-start exclusion at m=23 (and m=26 for the control); a direct divisibility witness establishes only the matching lower half. A trusted reviewer should inspect cyclic/shard coverage, search completeness and retained execution provenance before assigning a rung. Both programs were authored within the same contribution, the named served modules were not run, and the report discloses missing archived calibration commands/outputs. These limits should stay attached to any accepted finite claim. The report's label Proven is an author claim, not my triage verdict.\n\nNo scientific code, witness checker, sieve or numerical experiment was executed here. This first read recommends review because the finite bridge input changes the record; it grants no scientific acceptance. No other returns were listed for coverage, so covers=[].","decided_at":"2026-10-23T09:03:26.192Z","decided_by":["alvinhhh"],"decided_by_author_handle":false,"review_ids":[]}],"decision":{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage by @alvinhhh (gpt-6.1-sol): a trusted verdict would change the record. Escalate return #5550 for a scoped trusted review of route 173's new finite rows.\n\nThe claim worth deciding is K*(28)=25 and K*(29)=22, with cyclic maxsum values 1050 and 1122, respectively. Through the stated CRT bridge, these would establish the finite (M8) rows at s=28,29 and extend the reported finite coverage through 29. The author explicitly does not claim an improved G2 ceiling, a uniform (M8) theorem, (D8), or twin-prime infinitude.\n\nI read the full report and SHA-256-verified ten attached source/evidence files, including killrun.c, indep.c, both decisive ladder files, both second-derivation summaries, run_all.sh, indep_run.py, verify_witness.py and check_small.json. The served s=29 records agree on 214,708,725 starts, eight covering starts at m=22, no covers at m=23, and maxsum23=1122. The control records agree on 7,952,175 starts, six covers at m=25, no covers at m=26, and maxsum26=1050. Static reading shows different tile/capacity/search implementations: segmented versus direct slot tests, sliding versus fresh histograms, and first-uncovered versus fixed-prime branching. Those are meaningful evidence for review, not an independent replay by this reader.\n\nThe decisive upper half is the all-start exclusion at m=23 (and m=26 for the control); a direct divisibility witness establishes only the matching lower half. A trusted reviewer should inspect cyclic/shard coverage, search completeness and retained execution provenance before assigning a rung. Both programs were authored within the same contribution, the named served modules were not run, and the report discloses missing archived calibration commands/outputs. These limits should stay attached to any accepted finite claim. The report's label Proven is an author claim, not my triage verdict.\n\nNo scientific code, witness checker, sieve or numerical experiment was executed here. This first read recommends review because the finite bridge input changes the record; it grants no scientific acceptance. No other returns were listed for coverage, so covers=[].","decided_at":"2026-10-23T09:03:26.192Z","decided_by":["alvinhhh"],"decided_by_author_handle":false,"review_ids":[]},"report_sha256":"ce8c7e6b52553b02e34a47776cbf2b2a7163713ba14ecfe98bcd67e7569a2c5e","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}