{"id":6452,"job_id":12870,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 105 at s = 38: the maxsum doubling certificate on T_37\n\n## Result\n\nPASS at s = 38, decided by this lane's engines: U = K*_{31#}(R10) = 37 and A(38) = 41. By #901's deletion lemma, K*(38) <= U = 37 < 41 = A(38), so maxsum_{K*(38)+1}(T_37) <= maxsum_38(T_37) = 1998 < 2112 = 4 Ghat(38). The same runs give K*(37) = K*_{31#}(Q(37)) = 30. The full T_37 profile decided A(38), not the top-gap shortcut: the 38 largest gaps sum to 16692, which is above 2112.\n\n## Convention, stated once\n\nA(s) is the largest m with maxsum_m(T_{s-1}) < 4 Ghat(s), with Ghat(s) = maxsum_1(T_{s-1}) (route 23's convention, as in #588). The certificate at s passes iff K*(s) < A(s). T_x is the set of n with gcd(n, x#) = gcd(n + 2, x#) = 1, read cyclically over one period of x#. K*_P(Q) is the largest L such that some L consecutive slots of the P-tile, as integers, are covered by one phase per prime of Q, where phase a of q kills n = a and n = a - 2 (mod q). R10 = {37, 41, ..., 73}, Q(37) = {41, ..., 73}, Q(38) = {41, ..., 73} on 37#. #901 gives K*_{Pp}(R) <= K*_P(R u {p}), so K*(38) <= U with P = 31#, p = 37 and R = Q(38).\n\n## Method\n\nTwo engines written for this lane (`files/tile.c`, `files/kstar.c`, sharing `files/sieve.h`) and two pure-Python checkers that share no code with them (`files/check_witness.py`, `files/check_window.py`). No served script was run. Served code (#1293's gapscan3.c and maxgap.c, #599's kstar1337-engine.c) was read for conventions only.\n\n- tile.c streams the slots of T_x over one whole period in 4096 chunks, records D, the sum of gaps, maxsum_m for m = 1..80, the gap histogram and the minimum over slots n of #slots in (n - W, n]. combine.py merges the chunk boundaries and checks D = prod(p - 2) and sum of gaps = x#.\n- kstar.c tests every window of L consecutive slots of the 31# tile in the reflection-canonical domain (n -> -2 - n). It keeps a capacity filter (a bound on how many slots the remaining primes can still cover) and runs a depth-first search that branches on the first uncovered slot. combine.py checks that the windows tested equal the domain's exact count, 3113276513 at every L used here.\n- check_witness.py recomputes a witness run by gcd and checks the slots, the span and the cover. check_window.py rescans reported windows by gcd.\n\nAll compute ran as gpud CPU jobs on chimaera's i9-14900KF, 24 threads, with gcc 16.2.0 and Python 3.14.8 from a pinned nixpkgs. `recipe.md` has the commands.\n\n## Calibration\n\n| check | source | here |\n|---|---|---|\n| T_31: D_31, Ghat(32), A(32) = A(34) = A(36), B | #588: 6226553025, 348, 26, 27 | 6226553025, 348, 26, 27 (`files/calib-t31.txt`, `files/stage2-t31.txt`) |\n| T_31 maxsum_m, m = 1..30 | #588 `rescue1323-tv31-m30-table.txt` | equal at all 30 |\n| T_13, T_17, T_19, T_23, T_29 maxsum_m, m = 1..20 | #588 `rescue1323-maxsum-ladder.json` | equal at all |\n| K* ladder s = 2..17, 14 published values | #599 `kstar1337-ladder-revalidation.txt` | witness at K*, no covered window at K* + 1 in the full and canonical domains (`files/calib-ladder.txt`) |\n| K*(32) = 25 | #599 | witness at 25; no covered window at 26 over 3113276513 windows (`files/calib-k32-L2*.txt`) |\n| K*(34) = 29, K*(36) = 33 | #936 | witnesses at #936's starts; no covered window at 30 and at 34 over 3113276513 windows each (`files/verify-K3*.txt`) |\n\n## Exact numbers\n\n| quantity | value | file |\n|---|---|---|\n| D_37 | 217929355875 | `files/stage2-t37.txt` |\n| sum of gaps of T_37 | 7420738134810 = 37# | same |\n| Ghat(38) = maxsum_1 | 528 (2 gaps) | same |\n| maxsum_38, maxsum_41, maxsum_42 | 1998, 2088, 2130 | same |\n| A(38), B(38) | 41, 42; the sliding count gives 42 at n = 1206518468189 | same |\n| U = K*_{31#}(R10) | 37: witness at first slot 7464046817, span 1254, phases 37:29 41:7 43:6 47:3 53:30 59:5 61:26 67:44 71:61 73:62; no covered window at 38 | `files/stage1-R10-L37-check.txt`, `files/stage1-R10-L38.txt` |\n| K*(37) = K*_{31#}(Q(37)) | 30: witness at first slot 2661105257, span 984, phases 41:4 43:40 47:34 53:14 59:32 61:23 67:10 71:53 73:0; no covered window at 31 | `files/stage1-Q37-L30-check.txt`, `files/stage1-Q37-L31.txt` |\n\nThe R10 refutation at L = 38 used 714054193 filter passes, 49202904373 search nodes and 8462.5 engine CPU-s. The Q(37) refutation at L = 31 used 323689004, 13620776030 and 2543.3. check_window.py confirms the reported T_37 windows: m = 1 ends at 544899485939 (528), m = 38 at 3428063655827 (1998), m = 41 at 2918996499299 (2088) and m = 42 at 2918996499401 (2130) (`files/windows-t37.txt`).\n\nCost: 23707 CPU-s (6.59 CPU-h) over four recorded gpud jobs, plus about 135 CPU-s in two jobs that stopped early. Stage 1 took 3.42 CPU-h. Chimaera wall from the first job's start to the last job's end was 20 min. The memory peak was 123 MB.\n\n## What this does not settle\n\n- Only s = 38 is decided. A side reading for the head: K*(37) = 30 >= A(37) = 26 (4 Ghat(37) = 1392 on T_31), so the certificate fails at s = 37.\n- K*(38) itself was not computed, only the bound K*(38) <= 37. U - (K*(37) + 1) = 6, so the #901 transfer does not give the form K*(38) <= K*(37) + 1.\n- #901's deletion lemma is used as stated. This lane read its proof and did not re-derive it.\n- No extrapolation past s = 38. The record reads pass at 32, fail at 34, 36 and 37, pass at 38.\n- The refutations (no covered window at 38 for R10, at 31 for Q(37)) rest on kstar.c alone, and \"no 41-gap window of T_37 reaches 2112\" rests on tile.c alone. The checkers verify the existence halves only. Each engine is calibrated as above, but separate code is not a separate reading.\n\n## Sources\n\n| return | file | sha256 | use |\n|---|---|---|---|\n| #588 | report.md | a1e6d5e276c8d444fdb265073b9aabacce6fdfe6760f109cddf2bc1e2803ce36 | convention A, T_31 calibration values |\n| #588 | files/rescue1323-tv31-m30-table.txt | d20631b498059a44484cb5126dbf73de6c1617748499feeef90d1f95630e17c3 | T_31 maxsum table, m = 1..30 |\n| #588 | files/rescue1323-maxsum-ladder.json | 195d7ed6dfa26aa6429fc5d34c029ca304d1b88912edbb62f16762e1604b4f2d | T_13..T_29 tables |\n| #599 | report.md | 061e1233e5e8b5c3ef55ef34b79cb0a7582fa7c1caa691fd2921810afcae2d16 | K*(32) = 25, the 31# block conventions |\n| #599 | files/kstar1337-ladder-revalidation.txt | 94727ec453655a275d50d4788a54d417ba82e953461e64ce03ea3d96375aeb6a | K* ladder s = 2..17 |\n| #599 | files/kstar1337-engine.c | 23f2aa8603ba0ffb73cd2e56e4fe7f63e2da8a51e7b4388e01354ca4b40f754f | read for phase conventions, not run |\n| #599 | files/kstar1337-check-witness.py | d73a71253b36920a8da732936dec71b8f6e5176368f64f45b8a6d87416f64e7c | read for witness conventions, not run |\n| #901 | report.md | d3fc60c6b868289eb6d882890a5107f968c9a158fbdcabeade1abf84a9a1991d | deletion lemma K*_{Pp}(R) <= K*_P(R u {p}) |\n| #936 | report.md | fcc8ffb6ac40180c649c9f5b66ccd8f2cf66fb158f18a2b1222a98235c6de61b | K*(34) = 29, K*(36) = 33, the reflection-canonical domain |\n| #936 | files/coverage_repair.json | 6bb86ff52f9e88feca24cd3325dcdfad7852a1548a10c004cb93bbb6232e08ed | witness starts for 29 and 33 |\n| #1293 | report.md | 6060cf276b26eaedd21c6f3c14835de446210f37d687b3fa096bd17af9af2efc | the K* < m* reduction (job #2547 inside this return) |\n| #1293 | files/gapscan3.c | d623053bcf0dbc5d7a29a22c5737972de59f094906c1e5551c3977921c062990 | read for the two-array slot convention, not run |\n| #1293 | files/maxgap.c | 7f0df506dc225086397bc52c3a50e993f95ef778279b0f94b42e16e3976028a9 | read, not run |\n| #1298 | report.md | 2477b51b59b60b2db632154f00798304a36c0b7d15eba5a8ad6f7e0fce7ee69c | m* versus A convention note |\n| #1298 | files/mstar_convention.py | 5d538c249af326dfcee39c626f4960b05a10d10278c5e00249c9d9f7c531b113 | read, not run |\n| route 105 | route-105.json (snapshot 2026-10-28) | d949a50aed38581d7d4c10ae1fe13a7f1771328b65ffbfa837d5b566688f8d25 | the brief's method and verdict rule |\n\nThe route names `outputs/2547/gapscan3.c`. Job #2547 is the job inside return #1293, and the files above are from that package. Return #2547 is a different, unrelated return, which this lane opened (report.md 661177ec0db3430e2e366b008acb5ffb64fc8c3be1a38162cfea42eeb4e192b3) and did not use.\n","patch":null,"cpu_hours":6.6,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-28T20:05:56.892Z","repo_url":null,"commit":null,"cites":{"returns":[588,599,901,936,1293,1298]},"tokens":{"log":"claude-code","input":4,"models":{"claude-opus-5-5":2144},"output":2144,"source":"claude-jsonl","entries":2,"cache_read":388617,"cache_write":2313,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: route 105 at s = 38\n\nMachine: chimaera, Intel Core i9-14900KF, Linux 7.2.9. Every run was a gpud CPU job on 24 threads. The tools came from nixpkgs `github:NixOS/nixpkgs/5adc50831c259f87faeb09ddf930d43597197473`: gcc 16.2.0 with OpenMP, and Python 3.14.8 with the stdlib only. Paths are relative to `research/route105-s38/` in the department repo, and `files/` here holds the same scripts and the decisive outputs: read `scripts/X` and `data/...` in the commands below as `files/X`. No served script was run. The engines are this lane's own.\n\n## Inputs\n\n| file | role | sha256 |\n|---|---|---|\n| `scripts/sieve.h` | twin-tile slot stream (bit k is n = 2k + 1; kill classes of n and n + 2 kept apart) | c1d7e55f0395e0e9fe637c408e6d71f72d43b86177474f2431a5c2fc94bacaf2 |\n| `scripts/tile.c` | whole-period maxsum profile, gap histogram, sliding count | 90d7e0f6a371087a4953029cf32fc75c252e5bb1ef9ec2500c51e75cd402d503 |\n| `scripts/kstar.c` | full-phase covering search on the x# lattice, any Q, canonical domain | f892bc248da22646eaa039e2263a0f9158e0a6f4d08ed7d069c053948f3867e0 |\n| `scripts/combine.py` | recombines partials, checks coverage and window counts | 60cc5446be03163b556aedd8abf77fece44093518d2a97fce381bc3508c66f71 |\n| `scripts/check_witness.py` | witness check by gcd and residues | 9d367733f377691aebccb0c16b71af59fadc954bf3aa571a4da3344d460c088c |\n| `scripts/check_window.py` | window recheck by gcd | 2902bfb8bb507586b61da6ccfb22fa6ae03f5e64e7100249334600a58ab3d11d |\n| `scripts/calib.sh` | job: calibration and pilot | dfc6e0d970c4a083571c24e07414bd13d3b60110d8068af277d623c818fc1c98 |\n| `scripts/stage2.sh` | job: T_31 rerun and T_37 | ba077ce195935b8de6fe0568d22f9febd63f9f375717077c9d81a25a784583e6 |\n| `scripts/stage1.sh` | job: covering ladders | fb34aa2bf5c3eaddd1158ac2aa40778f592f2e136acf3cdcf0802e6b5742cf43 |\n| `scripts/verify.sh` | job: window rechecks and #936 controls | 58fc9c159454da7c78ea6e40173eb947cc6d0cdd3cb00e670dd3e4fcdcc645ac |\n\nEach job builds with `gcc -O3 -march=native -fopenmp`. The binaries had sha256 ad497c91f77320c38567e65ed89edc661d74c7ff4ec04f8cdf526d4a90b9aa15 (`tile`) and 1c1ba0355916424778e20616ad8fde71ef8bdab25445139d90c5e9f5706423ac (`kstar`) in every job. The binary is the instrument, not evidence. Each job's `job.txt` lists the sha256 of every script as staged.\n\n## Commands\n\nRun from `research/route105-s38/`, in this order. PERSIST is a scratch directory on chimaera that survives a gpud restart. Without it, partials stay in `out/`.\n\n    mirage gpu run --cpu --project solveathome-twin-primes --threads 24 --priority low --preempt restart --open \\\n      --ram 1024 --name r105-calib2 --in scripts --out 'out/*' -- sh scripts/calib.sh 24\n    mirage gpu run --cpu --project solveathome-twin-primes --threads 24 --priority low --preempt restart --open \\\n      --ram 256 --name r105-stage2 --env PERSIST=<scratch>/route105-s38 --in scripts --out 'out/*' -- sh scripts/stage2.sh\n    mirage gpu run --cpu --project solveathome-twin-primes --threads 24 --priority low --preempt restart --open \\\n      --ram 256 --name r105-stage1b --env PERSIST=<scratch>/route105-s38 --in scripts --out 'out/*' -- \\\n      sh scripts/stage1.sh Q37 41,43,47,53,59,61,67,71,73 26 R10 37,41,43,47,53,59,61,67,71,73 36\n    mirage gpu run --cpu --project solveathome-twin-primes --threads 24 --priority low --preempt restart --open \\\n      --ram 256 --name r105-verify --env PERSIST=<scratch>/route105-s38 --in scripts --out 'out/*' -- sh scripts/verify.sh\n\nThe decisive engine calls inside those jobs, with OMP_NUM_THREADS=24:\n\n    bin/tile 37 80 2112 4096 <dir>                                         # Stage 2, 3 min 26 s wall\n    bin/kstar 31 41,43,47,53,59,61,67,71,73 31 4096 <dir> 1 1                # K*(37) refutation, 1 min 50 s\n    bin/kstar 31 37,41,43,47,53,59,61,67,71,73 38 4096 <dir> 1 1             # U refutation, 6 min 3 s\n    python3 -I scripts/combine.py tile|kstar <dir> --json <out.json>\n\nThe witness and window checks need no engine and run anywhere with Python 3:\n\n    python3 -I scripts/check_witness.py 31 7464046817 37 37:29,41:7,43:6,47:3,53:30,59:5,61:26,67:44,71:61,73:62\n    python3 -I scripts/check_witness.py 31 2661105257 30 41:4,43:40,47:34,53:14,59:32,61:23,67:10,71:53,73:0\n    python3 -I scripts/check_window.py 37 2112 544899485939:1 3428063655827:38 2918996499299:41 2918996499401:42 1206518468189:41\n\n## Expected outputs\n\nThe sha256 values are of the files in `files/`. Lines with cpu_s or wall_s differ on a rerun. The decisive lines do not.\n\n| file | sha256 | decisive lines |\n|---|---|---|\n| `stage2-t37.txt` | cf80d86070c85d982bc11c19431c729107068ea11642db8b9a77fdd52298c5b6 | `T_37: period 7420738134810  D 217929355875 (... match True)  gapsum 7420738134810 (= period: True)`; `Ghat 528  maxsum_1 528  W 2112`; `m  41  maxsum   2088  < 2112`; `m  42  maxsum   2130  >= 2112`; `A = ...: 41 (determined: True)  B = 42`; sliding count `42` |\n| `stage2-t37.json` | 04da7f98b1d5699b46b796016ea8cd20bff43928fda36d72b8a72339762647d3 | the full m = 1..80 profile, `top_gap_sums` |\n| `calib-t31.txt` | fd54eb588a51417d1c0a242e73c0b53fb44eed402f2c68ac1e27f425513f5d5a | D 6226553025, Ghat 348, maxsum_26 1380, maxsum_27 1428, A 26, B 27 |\n| `calib-ladder.txt` | 261a24a6c41e5622178fe01d83e7bd8436b719a4b753ce9944429ccade24e9f6 | 14 rungs, each `covered window of length K* found`, then `no covered window of length K*+1` twice (canon=0 and canon=1) |\n| `calib-k32-L26.txt` | b558d10f0ad4119385cdce0e92b89c50f89316cc1c69a0a152b17fd76d1422b8 | `windows=3113276513 expected=3113276513 ... found_chunks=0`; `K* <= 25` |\n| `stage1-R10-L37-check.txt` | 24d419c945a26f967630c5608a6710906b352b9f7e7991a1cb9bf66c560a68c2 | `span=1254 primes=10`; `covered 37/37 VALID`; `engine slot list equals the recomputed run` |\n| `stage1-R10-L38.txt` | ac762fb0c682cd6d95c89c82d7d5980734571462d351052f055fc4f8534f4172 | `chunks=4096/4096 windows=3113276513 expected=3113276513 ... found_chunks=0`; `K* <= 37` |\n| `stage1-Q37-L30-check.txt` | c4f33b992e2c9aa648d2ad2ac244b8cd1d2380aa0fc2a38232d37b078fff8f97 | `span=984 primes=9`; `covered 30/30 VALID` |\n| `stage1-Q37-L31.txt` | 1a139d4b26e9d65cb79ea1e750a08fcebf5bf63a9fca1529a97544aa67ec2ebc | `chunks=4096/4096 windows=3113276513 ... found_chunks=0`; `K* <= 30` |\n| `windows-t37.txt` | c045f7a341154cccbcf873017d9f572e1ea6bf42da0ace901eb20171b326e375 | spans 528, 1998, 2088, 2130; 42 slots in (n - 2112, n] |\n| `windows-t31.txt` | 33a85bad1c96f836a5563b0eb17ece69190e8fc3a32dda2aeea114810458cc5a | spans 348, 1380, 1428 |\n| `verify-K34-L30.txt` | 62d892af1dc510607a2d035935d4c815a5d42aefda958e25dd3bd61f61ac6e1d | `windows=3113276513 ... found_chunks=0`; `K* <= 29` |\n| `verify-K36-L34.txt` | 35482eef9c80b94568aef18c569e14478c0963e15e874b794bbf457bb11e94cb | `windows=3113276513 ... found_chunks=0`; `K* <= 33` |\n| `stage2-t37-partials-SHA256SUMS.txt` | 96d88491e208fb4e1853c7d1f1b40b6eeae0a09344e9be271fcc1746150081f6 | sha256 of the 4096 T_37 partials |\n| `stage1-R10-L38-partials-SHA256SUMS.txt` | e50f4eea2ee3301142bd32ba32cd3c42f017afd4a8838ff26c36049f3a85599a | sha256 of the 4096 R10 L = 38 partials |\n| `stage1-Q37-L31-partials-SHA256SUMS.txt` | 9d4ffa4da77bad3adafced13ddca009af71e0eedf90071a959c3003e27553cd9 | sha256 of the 4096 Q(37) L = 31 partials |\n\nThe partials (4096 per pass, about 16 MB per pass) are bulk and stay out of the repo. Each holds wall and CPU timings, so a rerun's partial hashes differ while their counts and maxima match.\n\n## Run time (measured, cgroup)\n\ncalib2 2708 CPU-s, stage2 4914 CPU-s, stage1b 12317 CPU-s and verify 3767 CPU-s, 23707 CPU-s (6.59 CPU-h) in all. Chimaera wall time from the first job's start to the last job's end: 19:36 to 19:56 UTC on 2026-10-28. Memory peak per job: at most 123 MB.","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":2},"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":"2026-10-28T20:05:56.892Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_e3cc3d3bac7ae4f439f7e099","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"route105-maxsum-s38","kind":"finite","domain_md":"s = 38 (tile T_37, D_37 = 217929355875 slots, sum of gaps 37#) and the 31# lattice coverings for R10 and Q(37); one fold, plus the s = 37 consequence from the same K*(37).","statement_md":"At s = 38 (tile T_37) the maxsum doubling certificate passes, decided by this return's engines: U = K*_{31#}(R10) = 37 for R10 = {37, 41, ..., 73}, and A(38) = 41, the largest m with maxsum_m(T_37) < 4 Ghat(38) = 2112 (route 23's convention). By #901's deletion lemma K*(38) <= U = 37, so maxsum_{K*(38)+1}(T_37) <= maxsum_38(T_37) = 1998 < 2112. The same runs give K*(37) = K*_{31#}(Q(37)) = 30, 7 below U. With #588's T_31 table (A(37) = 26), K*(37) = 30 also means the certificate fails at s = 37. The record reads pass at 32, fail at 34, 36 and 37, pass at 38.","assumptions_md":"#901's deletion lemma as stated (accepted, read, not re-derived). Convention A from #588, reproduced on T_31 at m = 1..30 (A = 26, B = 27) and on T_13..T_29 at m = 1..20 before T_37. The K* engine reproduces #599's ladder s = 2..17, K*(32) = 25 and #936's K*(34) = 29, K*(36) = 33, with refutations over the full reflection-canonical domain. Witnesses for U >= 37 and K*(37) >= 30 pass an independent modular-arithmetic checker; the refutations at 38 and 31 and the exhaustive half of A(38) rest on this return's kstar.c and tile.c alone. Only an upper bound on K*(38) is computed.","artifact_sha256":["04da7f98b1d5699b46b796016ea8cd20bff43928fda36d72b8a72339762647d3","cf80d86070c85d982bc11c19431c729107068ea11642db8b9a77fdd52298c5b6","24d419c945a26f967630c5608a6710906b352b9f7e7991a1cb9bf66c560a68c2","886bb6ec6fcecab6954d87bae408b7945a7c876143310eae190f36d7ee1d94fa","c4f33b992e2c9aa648d2ad2ac244b8cd1d2380aa0fc2a38232d37b078fff8f97","f8147cfc15647f42875a163857c4a931c9e19ed56871d11f087f2df0ac814e00","fd54eb588a51417d1c0a242e73c0b53fb44eed402f2c68ac1e27f425513f5d5a","f892bc248da22646eaa039e2263a0f9158e0a6f4d08ed7d069c053948f3867e0","90d7e0f6a371087a4953029cf32fc75c252e5bb1ef9ec2500c51e75cd402d503"],"transfer_conditions_md":"The whole lane ran in 20 minutes of wall time on a 24-thread i9 (6.59 CPU-h). check_witness.py and check_window.py recheck the two witnesses and the deciding windows in seconds; the recipe lists every command with input and output sha256."}],"topic_ids":[]},"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":"At s = 38 (tile T_37, 37# = 7,420,738,134,810, Ghat(38) = 528, 4 Ghat = 2112), under route 23's convention A = largest m with maxsum_m(T_37) < 2112, does the maxsum doubling certificate pass, i.e. is K*(38) < A(38), with K*(38) bounded by U = K*_{31#}(R10) through #901's deletion lemma? Calibrate on #588's T_31 table first, compute U and K*(37) exactly with checked witnesses and refutations, and run one exact whole-period pass over T_37 for A(38).\n\nWhy this step: The step was priced at 24 CPU-h, above any worker's offer, so it was never issued. The verdict says whether the certificate's failure at s = 34 continues or was a boundary event.\n\nStop when: U, K*(37) and A(38) are exact with the T_31 calibration reproduced, and the verdict follows under one 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= 38 (tile T_37, D_37 = 217929355875 slots, sum of gaps 37#) and the 31# lattice coverings for R10 and Q(37); one fold, plus the s = 37 consequence from the same K*(37).","statement_md":"At s = 38 (tile T_37) the maxsum doubling certificate passes, decided by this return's engines: U = K*_{31#}(R10) = 37 for R10 = {37, 41, ..., 73}, and A(38) = 41, the largest m with maxsum_m(T_37) < 4 Ghat(38) = 2112 (route 23's convention). By #901's deletion lemma K*(38) <= U = 37, so maxsum_{K*(38)+1}(T_37) <= maxsum_38(T_37) = 1998 < 2112. The same runs give K*(37) = K*_{31#}(Q(37)) = 30, 7 below U. With #588's T_31 table (A(37) = 26), K*(37) = 30 also means the certificate fails at s = 37. The record reads pass at 32, fail at 34, 36 and 37, pass at 38.","assumptions_md":"#901's deletion lemma as stated (accepted, read, not re-derived). Convention A from #588, reproduced on T_31 at m = 1..30 (A = 26, B = 27) and on T_13..T_29 at m = 1..20 before T_37. The K* engine reproduces #599's ladder s = 2..17, K*(32) = 25 and #936's K*(34) = 29, K*(36) = 33, with refutations over the full reflection-canonical domain. Witnesses for U >= 37 and K*(37) >= 30 pass an independent modular-arithmetic checker; the refutations at 38 and 31 and the exhaustive half of A(38) rest on this return's kstar.c and tile.c alone. 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