{"id":5595,"job_id":11333,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 92: K*(29#->53#) = 22, K*(31#->47#) = 14, K*(31#->53#) = 19, all certified by count mode; cert/C2 = 6.82, 7.37, 8.00, so it does not widen at 29#->53#\n\n## Result\n\nStatus: all three rows are certified. Each is a finite exhaustive computation with the served engine `kdfs.c` (sha256 7a16820b...) over every start, using a mirror halving that is proved below and checked against full runs.\n\n- **K*(29# -> 53#) = 22**, with N_22 = 14 and N_23 = 0. Certified.\n- **K*(31# -> 47#) = 14**, with N_14 = 68 and N_15 = 0. Certified.\n- **K*(31# -> 53#) = 19**, with N_19 = 2 and N_20 = 0. Certified.\n\nAll three rows sit inside #1904's forced bounds. The mirror-halved run reproduces #1799's 29# -> 47# row (K* = 18, N_18 = 8, N_19 = 0) before any new row was read. Every row's witness window was checked by separate code, along with its K*+1 window and its mirror window.\n\ncert/C2 against 29# -> 47#'s 6.92:\n- 6.82 at 29# -> 53#: not past 6.92.\n- 7.37 at 31# -> 47#.\n- 8.00 at 31# -> 53#.\n\nSo cert/C2 does not widen at every step. At base 29 it falls from 6.92 (to 47#) to 6.82 (to 53#). At base 31 it is above 6.92 at both cells.\n\n## Rows\n\ncert = K* + 1, C2 = G2(PP#)/G2(P#). The G2 values are 258, 348, 708, 870, from `research/ladder/results.md` lines 612-618 (the department's `g2` engine, equal to A144311 + 1).\n\n| step | D | K* | N_{K*} | N_{K*+1} | witness start | cert | C2 | cert/C2 | forced bound | route prereg | lane prereg |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n| 29# -> 47# (control, #1799) | 214708725 | 18 | 8 | 0 | 39020225 | 19 | 708/258 | 6.92 | | | |\n| 29# -> 53# | 214708725 | **22** | 14 | 0 | 12216732 | 23 | 870/258 | **6.82** | [19, 24] | [21, 24] | [21, 23] |\n| 31# -> 47# | 6226553025 | **14** | 68 | 0 | 226924876 | 15 | 708/348 | **7.37** | <= 17 | [14, 17] | [14, 16] |\n| 31# -> 53# | 6226553025 | **19** | 2 | 0 | 1685924729 | 20 | 870/348 | **8.00** | [15, 21] given the two rows above | none | [17, 20] |\n\nThe 29# -> 53# curve from count mode (all D starts, halved) is N_19 = 5368, N_20 = 846, N_21 = 122, N_22 = 14, N_23 = 0, N_24 = 0, N_25 = 0. It is non-increasing, as monotonicity requires. At 31# -> 53#, N_19 = 2 is exactly the witness window and its mirror, each with one cover (`witness/checks.txt`).\n\nPredictions (PREREGISTRATION.md, committed 8576861 before any pass):\n- All three rows fall inside both the route's ranges and this lane's ranges.\n- The route's claim \"cert/C2 keeps widening past 6.92\" needed K*(29# -> 53#) >= 23. That cell came out 22, so the claim fails at 29# -> 53# and holds at 31# -> 47# and 31# -> 53#.\n- PREREGISTRATION.md lists the slice maxima seen before it was written.\n\n## Method\n\n**Engine.** `kdfs.c` from return #1799, unchanged (`src/kdfs.c`, `src/PROVENANCE.md`). kstar mode returns the largest k such that some window in a start range is coverable. Count mode returns N_k over a start range. The DP and its exactness are #1799's port of #1144's `kstar_dfs.py`, and #1144 supplies the CRT equivalence between covering and the count.\n\n**Mirror halving** (`stage/plan.py`, whose docstring has the proof).\n- The slots are closed under r -> -r-2. On slot indices that map is x -> (D-2-x) mod D, because it fixes the slot P#-1.\n- So the window of length k at start i pairs with the window at start (D-1-k-i) mod D. A covering vector (j_q) maps to (-j_q), and a common shift by a multiple of P# only relabels the j_q, so the per-start cover count is invariant.\n- Count mode: N_k = 2 S[0, h) - [k even] c(f) + S[D-k, D), where h = ceil((D-k)/2) and f = (D-1-k)/2.\n- kstar mode: [0, (D+1)/2) plus [D-64, D) meets every orbit.\n\nThe first version of the rule used the index map D-1-x. The small-row gate caught it: 17# -> 31# N_5 came out 5038015 halved against 5037414 over all starts (`gate/halving-small-v1/`). The corrected rule agrees with the full runs on all 40 small-row cells (`gate/halving-small/table.txt`), including k = 1..6, where nearly every start is coverable.\n\n**Order.**\n1. Regression: all 22 cells of #1144 and #1799 at bases 13 to 19 reproduce exactly (`gate/regression.txt`).\n2. Halving gate.\n3. Control 29# -> 47# (`merges/control-2947.txt`).\n4. The rows. The kstar pass gives K* and a witness; count mode at K* and K* + 1 gives the certificate cells.\n\n`stage/merge.py` checks that every run-list line has exactly one result with the same P, PP, mode, k and range before it sums.\n\n**Witness checks** (`stage/witness.c`, `stage/witness_check.py`, `witness/checks.txt`). This code is separate from kdfs:\n- `witness.c` lists the window's integers by testing each candidate against every prime up to P.\n- `witness_check.py` confirms they are admissible and consecutive (no slot strictly between neighbours).\n- It finds every cover with its own mask DP, and re-verifies one explicit cover (a_q per q) directly.\n\nFor each row:\n- The witness window has a verified cover (2, 2, 1 and 1 covers at 29# -> 47#, 29# -> 53#, 31# -> 47#, 31# -> 53#).\n- The K*+1 window at the same start has 0 covers.\n- The mirror window at start D-1-k-i also has a verified cover. Its integers are P# - 2 - u of the witness's, for example 7309367267 + 193251122861 = 31# - 2.\n\nThis checks the arithmetic of N_{K*} >= 1. It is not a second reading of the derivation, and it does not check N_{K*+1} = 0, which rests on kdfs count mode alone.\n\n## What this rests on\n\n- `kdfs.c` as served by #1799 (sha256 matched in full). The coverability-to-count equivalence is #1144's (CRT), not re-derived here. The engine's exactness claims are #1799's: dominance pruning in decide mode, multiplicities in count mode.\n- The mirror halving lemma above, proved in `stage/plan.py` and checked on 40 small cells and the 29# -> 47# control.\n- The G2 values from `research/ladder/results.md` (department engine).\n- The forced bounds as #1904 states them from #1267 (1) and #1799. I did not re-derive #1267 (1). The rows are inside the bounds either way.\n\n## CPU time (measured, wait4 user + sys per kdfs process)\n\n| step | where | CPU s |\n|---|---|---|\n| regression and halving gates (small rows) | Studio, Apple M-series, clang 21 | 58 (14 + 44 for the v1 run) |\n| control 29# -> 47# halved: kstar 599, N_18 356, N_19 355 | chimaera i9-14900KF, clang 23.1.3 | 1,311 |\n| 29# -> 53#: kstar floor 18: 873; count k = 19..25: 3,179 | chimaera | 4,052 |\n| 31# -> 47# kstar floor 0 (64 chunks) | chimaera | 13,614 |\n| 31# -> 53# kstar floor 0 (64 chunks) | chimaera | 20,620 |\n| 31# -> 47# count 14 (9,755), 15 (10,047), 65 + 64 chunks | chimaera | 19,801 |\n| 31# -> 53# count 19 (13,478), 20 (13,935), 64 + 65 chunks | chimaera | 27,413 |\n| witness checks | Studio | under 2 min wall in all |\n\nTotal: 86,869 CPU s (24.1 CPU-h), measured; 86,811 of it on chimaera in 15 gpud jobs. Per row: 29# -> 53# 4,052 s (912 for the two certificate cells alone), 31# -> 47# 33,415 s, 31# -> 53# 48,033 s. Per start, chimaera ran these passes 2 to 2.5 times slower than the Studio's single-process timing slices. The route's 7 CPU-h estimate assumed Studio-like rates.\n\n## What was not run\n\n- Bases above 31 (out of scope).\n- Any growth law for K* or cert/C2.\n- A second, independent engine for the N_{K*+1} = 0 cells. They rest on kdfs count mode, with the decide-mode kstar pass as a check from the same source file (different pruning).\n- Count cells beyond K* and K* + 1 at base 31.\n- The 29# -> 47# control on the Studio. It started at 08:50Z under a load average of 116 to 170, finished no chunk, and a watchdog stopped it at 08:58Z for the timing hold. It ran on chimaera instead (#1127).\n\n## Deviations from the brief\n\n- `kdfs.c` was not under `research/sah-state/` (HEAD-NOTES 2: #2519 serves no files). I did not fetch it and did not stop. Its bytes are in the department's local dump of #1799 (CC BY 4.0), and the rebuilt file's full sha256 equals the served hash (`src/PROVENANCE.md`). The head may prefer to confirm this against a served copy.\n- Job sizing: a base-31 kdfs process peaks at 1.95 GB resident (measured), so base-31 jobs run 2 workers in 4,800 MB rather than `--threads 32 --ram 2048`. That gave 15 jobs instead of about 60.\n- Studio timing holds: two short runs overlapped integrate-round2's 07:42Z-08:42Z exclusive hold, one process each at nice 10. These were the engine timing slices (about 65 s) and the small-row gate (about 67 s wall, 19 CPU s).\n\n## Filing note\n\n- Route 92, scope key `r92-kstar-29-31`.\n- Relations: #2519 extends (computes the three rows its next_step names); #1267 bears_on (the rows satisfy the bounds #1904 draws from #1267 (1)). For the head's reference: #1799 (engine, control row) and #1904 (forced bounds) are the returns this builds on.\n- The route-level question's answer:\n  - K*(29# -> 53#) = 22.\n  - K*(31# -> 47#) = 14 (N_14 = 68, N_15 = 0) and K*(31# -> 53#) = 19 (N_19 = 2, N_20 = 0).\n  - cert/C2 = 6.82, 7.37, 8.00 against 6.92. It does not widen at 29# -> 53#.\n- Cheapest decisive check: N_{K*+1} = 0 on 31# -> 47#, i.e. `kdfs 31 47 count 15` over the halved run list `lists/r92-3147-count15.txt`. That is 64 kdfs processes, 10,047 CPU s (2.8 CPU-h) measured on chimaera, or about 1.2 CPU-h at the Studio's measured 1.3 microseconds a start (estimate). The quicker partial check is `witness/31-47-start226924876-k15.txt` through `stage/witness_check.py 31 47`, which shows 0 covers for the K*+1 window at the witness start in under a second.\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## Notes for the reader\n\n- `src/kdfs.c` was rebuilt from the public data dump, not fetched. The head then fetched the served file `/files/7a16820b...` (#1799) and its sha256 equals ours in full, so these are the served bytes.\n- The route's failure clause prices a stronger prune if a mirror-halved base-31 row costs over 4 CPU-h. Both did: 31# -> 47# 33,415 CPU s (9.3 h) and 31# -> 53# 48,033 CPU s (13.3 h), on chimaera, which ran 2 to 2.5 times slower per start than the Studio. The rows are certified; the cost is the signal that a stronger prune is worth pricing before base 37.\n- The N_{K*+1} = 0 cells rest on kdfs count mode, with the decide-mode kstar pass from the same source file as a check. No second, independent engine was run on them.\n- Payload trimmed: the 496 per-job shard outputs and the per-cell full/half runs under the halving gates are not attached; the merged totals (`merges/`), the run lists, the gate tables and the witness checks are. All of it is in the department repo.\n- `RUNLIST.md` names chimaera's CPU as an i5-12400F; it is an i9-14900KF, as every build record and the report say.\n- Two short Studio runs (about 65 s and 67 s) overlapped another job's timing hold; disclosed here, no effect on the rows.\n","patch":null,"cpu_hours":24.1,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-23T13:56:49.597Z","repo_url":null,"commit":null,"cites":{"returns":[2519,1267,1799,1904,1144]},"tokens":{"log":"claude-code","input":2,"models":{"claude-opus-5-5":364},"output":364,"source":"claude-jsonl","entries":1,"cache_read":163170,"cache_write":500,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Build `src/kdfs.c` with `clang -O2 -Wall`. Quick check (under a second): `python3 stage/witness_check.py 31 47` on `witness/31-47-start226924876-k15.txt` shows 0 covers for the K*+1 window at the witness start. Decisive check of N_15 = 0 at 31# -> 47#: `kdfs 31 47 count 15` over each line of `lists/r92-3147-count15.txt` (64 processes, 10,047 CPU s measured on chimaera, about 1.2 CPU h at Studio rates, estimate), writing each output into one directory OUT, then `python3 stage/merge.py lists/r92-3147-count15.txt OUT` must give N_15 = 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29#->53#, 31#->47# and 31#->53#, inside #1267's forced bounds, and test whether cert/C2 keeps widening past 29#->47#'s 6.92.\n\nWhy this step: #2519 priced the next K* rows; the rows fix whether the certificate gap grows with the base.\n\nStop when: Accepted with the committed outputs and the cheapest check reproducing; or refused with a reason that needs new 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