{"id":5520,"job_id":11186,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 174: U_5 prunes every maxsum maximiser at s = 24 and 25 (U_5 = 20 < K* = 21; CM 924 -> 912), not at s = 22 (4 of 12 open maximisers keep U_5 = 20 = K*)\n\n## Result\n\nFinite computation, measured, exact enumeration of the bounded-arity union relaxation U_5 in the served definitions and mask convention of #5497. Every value was produced twice in C, once in complement-intersection form and once in direct union form, and once more by an independent stdlib Python check that uses a different algorithm (115/115 PASS).\n\n- **s = 24 and s = 25 (base 23#, Q = [29,31,37,41,43,47], K* = 21): U_5 prunes every maxsum maximiser.** U_5(i, 21) = 20 < 21 at both open maximisers 635201 and 7316952 (a mirror pair). The other two maximisers, 311372 and 7640781, were already pruned by U_1 = 20 and have U_5 = 19. The first pruning arity at these rungs is 5, a proper arity (|Q| = 6). The implied drop against A_profile[K*-1], the convention in route 174's success clause, is **CM(24) = CM(25): 924 -> 912** (CM/Ghat with Ghat(2s) = 708: 1.3051 -> 1.2881). The drop does not rest on the maximisers alone. Every 21-slot window at base 23# with span above 912 (12 windows: four at 924, eight at 918) has U_5 <= 20.\n- **s = 22 (base 19#, Q = [23,29,31,37,41,43], K* = 20): U_5 does not prune.** At the 12 open maximisers, U_5 = [20, 20, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20]. 8 of the 12 drop below K*, but 4 starts keep U_5 = 20 = K*: 5182, 37634, 341020 and 373472, two mirror pairs. Among arities m <= 5, no arity prunes every maximiser at s = 22, and CM(22) = 750 stays as it is under U_5. U_6, the exact coverage, was not enumerated.\n\nSo route 174's m = 5 step meets its success clause (\"U_5 < K* at every open maximiser of at least one of the three rungs\") at s = 24 and s = 25. At s = 22 it settles nothing: 4 of the 12 open maximisers keep U_5 = K*, and the failure clause, which needs U_5 >= K* at every open maximiser of all three rungs, does not hold either. The figure 912 at s = 24, 25 matches the bound #5497 deduced from the served Step-2 identity (its section 3, \"deduction, not re-verified\"). This return reaches the same figure through enumeration of a proper relaxation, without the identity.\n\n## The computation\n\nDefinitions (from the served `depth_buffy29.py` of #5497, sha256 5a706a36..., which restates #2025 and #4263):\na is the sorted list of n in [0, W) with gcd(n(n+2), W) = 1, W = base#, D = |a|; pos_j = a[(i+j) mod D] (+W if wrapped), 0 <= j < k; B_q(r) is the bitmask of j with pos_j = r (mod q); cv_q(r) = B_q(r) | B_q((r-2) mod q); Cm_q(r) = full & ~cv_q(r); span(i,k) = a[(i+k) mod D] (+W if wrapped) - a[(i-1) mod D] (-W if i = 0);\n\nU_m(i,k) = max over S in Q with |S| = m of  [ k - min over r_q (q in S) of popcount(AND_{q in S} Cm_q(r_q)) ] + sum over q not in S of max_r popcount(cv_q(r)).\n\n`u5.c` builds the tile, scans span(., K*) over all D starts for the maximum and the full maximiser list, builds masks only at the requested starts, and enumerates every joint residue assignment of every m-subset. `--union` evaluates each subset term as max popcount(OR cv_q(r_q)) instead; the two forms share only the mask construction. `check_u5.py` (stdlib) rebuilds the tiles with a bytearray sieve and finds each subset term by branch and bound over the inclusion-maximal distinct masks of each prime. OR is monotone, so the maximum is attained at maximal masks.\n\n### Gates and control (read before any U_5 value)\n\n| check | expected (served) | got |\n|---|---|---|\n| base 13#, s = 16, k = 17, starts [19,638,829,1448]: U_1..U_4 | [18,16,16,18] x3, [17,16,16,17] (`served_u34_result.json`, the unmodified served job8173 instrument run by #5497; `gate_buffy29.json`) | equal |\n| same starts, assignment counts m = 1..4 | 119, 5590, 129530, 1480489 | equal |\n| same starts, U_5 (= exact, |Q| = 5) | `exact_m5` = [15,14,14,15] (`gate_buffy29.json`) | equal |\n| s = 22: MS, maximisers (16), U_1 at all 16, open list (12), U_2/U_3/U_4 at the open starts | #5497 `analysis_buffy29.json` | equal |\n| s = 24: MS, maximisers (4), U_1, open list [635201, 7316952], U_2 = U_3 = U_4 = [21,21] | same | equal |\n| assignment counts m = 1..4 at s = 22 / s = 24 | 204, 17193, 765960, 19017099 / 228, 21537, 1078680, 30207579 | equal |\n| max span at k = K*-1 | A_profile[K*-1] = 720 (s = 22), 912 (s = 24, 25) | equal |\n| mirror i <-> D-1-K*-i on the maximiser sets | closed | closed |\n\nU_4 at s = 22 reproduced #5497 exactly: [20,21,21,20,20,21,21,20,20,21,21,20].\n\n### U_5 at every maxsum maximiser\n\nAssignments per start at m = 5: 249,379,116 (s = 22) and 448,408,452 (s = 24, 25), the figures #5497 priced.\n\ns = 22, K* = 20, MS = 750 (U_1 .. U_5; binding 5-subset for the starts that stay open):\n\n| start | U_1 | U_2 | U_3 | U_4 | U_5 | status under U_5 |\n|---|---|---|---|---|---|---|\n| 5182 | 20 | 20 | 20 | 20 | **20** | open (S omits 29: 16 + 4) |\n| 37634 | 21 | 21 | 21 | 21 | **20** | open (S omits 23: 15 + 5) |\n| 39825 | 21 | 21 | 21 | 21 | 19 | pruned |\n| 49725 | 20 | 20 | 20 | 20 | 19 | pruned |\n| 49726 | 20 | 20 | 20 | 20 | 19 | pruned |\n| 61623 | 22 | 22 | 22 | 21 | 19 | pruned |\n| 110391 | 17 | 17 | 17 | 17 | 17 | pruned by U_1 |\n| 163449 | 18 | 18 | 18 | 18 | 18 | pruned by U_1 |\n| 215205 | 18 | 18 | 18 | 18 | 18 | pruned by U_1 |\n| 268263 | 17 | 17 | 17 | 17 | 17 | pruned by U_1 |\n| 317031 | 22 | 22 | 22 | 21 | 19 | pruned |\n| 328928 | 20 | 20 | 20 | 20 | 19 | pruned |\n| 328929 | 20 | 20 | 20 | 20 | 19 | pruned |\n| 338829 | 21 | 21 | 21 | 21 | 19 | pruned |\n| 341020 | 21 | 21 | 21 | 21 | **20** | open (mirror of 37634) |\n| 373472 | 20 | 20 | 20 | 20 | **20** | open (mirror of 5182) |\n\ns = 24 and s = 25 (identical inputs: Q_24 = Q_25, K* = 21, same tile), MS = 924:\n\n| start | span | U_1 | U_2 | U_3 | U_4 | U_5 |\n|---|---|---|---|---|---|---|\n| 311372 | 924 | 20 | 20 | 20 | 20 | 19 |\n| 635201 | 924 | 21 | 21 | 21 | 21 | **20** |\n| 7316952 | 924 | 21 | 21 | 21 | 21 | **20** |\n| 7640781 | 924 | 20 | 20 | 20 | 20 | 19 |\n\nAt 635201 two 5-subsets attain U_5 = 20: S omitting 47 (union term 16, plus 4 from q = 47) and S omitting 31 (15 + 5). Neither reaches 21.\n\nAll 21-slot windows at base 23# with span > 912 (`--above 912`): spans 918 (8 windows) and 924 (4). The eight span-918 windows (311366, 635200, 747754, 3308212, 4643941, 7204399, 7316953, 7640787) have U_1 <= 20 already and U_5 in {18, 19, 20}.\n\n### The CM drop\n\nRoute 174's success clause measures the drop \"against A_profile[K*-1]\", as #5497 did at s = 19, 20. At s = 24, 25 that gives 924 -> 912. The drop removes 12 of the 216-slot excess of CM over Ghat(48) = Ghat(50) = 708, or 5.6 % (measured from the served values).\n\nWhat I checked behind that figure: no 21-slot window with span above 912 has U_5 >= 21, and the largest span over windows of 20 slots is 912 (recomputed; it equals the served A_profile[20]). I did not read the served definition of CM in `bridge_profile.json` (#2025). If the envelope also takes windows of other lengths into account, the 912 figure rests on the route's convention, not on a check of mine.\n\n## Cost (measured)\n\nProcess CPU time (`CLOCK_PROCESS_CPUTIME_ID` and `/usr/bin/time -p` user). Machine: Studio, Apple M3 Ultra, Apple clang 21.0.0 `-O2`, arm64. All runs used `nice -n 10`. They ran while integrate-round2 held an exclusive studio-cpu timing hold (load average 26 to 30); every run here was under a minute (HEAD-NOTES item 8), and the figures may carry some load noise.\n\n| step | CPU |\n|---|---|\n| tile + full span scan, base 19# | 0.046 s |\n| tile + full span scan, base 23# | 1.37 to 1.38 s |\n| U_4 per start (19.0M / 30.2M assignments) | 0.003 s / 0.004 s |\n| U_5 per start, s = 22 (249.4M assignments) | 0.034 to 0.036 s |\n| U_5 per start, s = 24, 25 (448.4M assignments), one start first (635201) | 0.063 s |\n| control run U_1..U_4: s = 22 / s = 24 (whole process) | 0.09 s / 1.38 s |\n| six labelled all-maximiser processes, m = 1..5 (s22c, s22u, s24c, s24u, s25c, s25u) | 0.65, 0.65, 1.63, 1.63, 1.63, 1.63 s |\n| four labelled `--above 912` processes (a24c, a24u, a25c, a25u) | 2.22 to 2.25 s each |\n| `check_u5.py` (all rungs) | 10.78 s |\n| full reproduction `run_u5.sh` (sequential, compile and check included) | 33.72 s user, 51.85 s wall |\n\nTotal for the record run: 19.7 s of C CPU and 11.4 s of Python CPU. With the reproduction rerun included, about 65 s CPU in all (measured sum). #5497 had estimated 5 to 15 minutes for s = 22 in its Python instrument; the C enumeration took 0.42 s of U_5 CPU for the 12 open starts.\n\nLabelled worker pids are in `out/pids.txt`; all exited normally and none was killed.\n\n## Not run\n\n- U_6 (exact coverage) at any start, and any exact cover search (out of scope). So nothing here decides whether exact coverage prunes s = 22. #5497's identity deduction bounds it by 19 there; that bound was not re-checked.\n- U_5 at s = 22 windows other than the 16 maximisers: not needed, because four maximisers stay open.\n- The served `job4535_union_probe.py` (#2025, sha256 prefix 20a3ab76) and `job8173_union_depth34.py` (#4263, prefix dedd6f82) were not read. They are not under `research/sah-state/` (HEAD-NOTES item 2). The definitions come from #5497's `depth_buffy29.py`, which restates them. The base-13 gate matches the output of the unmodified served job8173 instrument (`served_u34_result.json`), which pins the mask convention. If a reviewer wants the convention checked against the source itself, H13 has to be fetched.\n- The served Python instrument (`depth_buffy29.py`, `run_buffy29.py`) was not executed; its outputs were compared instead. The Studio python has no numpy.\n- `bridge_profile.json` (#2025) was not read; see \"The CM drop\".\n- No quiet-Studio timing. The exclusive hold was in place, and the whole job is under a minute of CPU.\n\n## What this rests on\n\n- Served definitions and values: #5497 attachments `depth_buffy29.py`, `analysis_buffy29.json`, `gate_buffy29.json`, `served_u34_result.json` under `research/sah-state/2026-10-23/attachments/05497/` (sha256 checked against `SHA256SUMS.txt`: all OK). Route 174's m = 5 step text is in `research/sah-state/2026-10-23/returns/05497.json` (`research.next_step`).\n- `u5.c` sha256 `1e2ff22447253bf5c609df839bdfef66f5ba891fd86a7f3ca64c0e7346a48538`; `check_u5.py` sha256 `54b3eddc7f2070072ab50637d54700599bced225c5a57f368f2adb51363aae22`; `run_u5.sh` sha256 `5f9621ffbb436c4de84c8c8fbcf334730a6bf282519afe8036f3c9c17a34ec7e`.\n- Planted-fault check: lowering one stored U_5 value by 1 in a copy of `u5_s22c.jsonl` makes `check_u5.py` fail on exactly that start.\n- Exact command: `sh run_u5.sh` from `research/union-m5` (binaries to `$BIN`, default a fresh temp dir; outputs to `$OUT`, default `out`). A full rerun of the final `u5.c` reproduced every stored value in `out/` (0 mismatches over 14 output files; only the CPU fields differ).\n\n## Filing note\n\n- Route: 174. Scope key: `r174-u5`.\n- Links: #5497 **extends** (U_5 at #5497's open maximisers, with #5497's definitions, counts and starts); #5493 **bears_on** (the step check that released the U_2..U_4 step and listed the open starts).\n- Answer to the route's m = 5 question: U_5 prunes every maxsum maximiser at s = 24 and s = 25 (first pruning arity 5 < |Q| = 6; CM 924 -> 912 against A_profile[20]). It does not prune at s = 22 (4 of 12 open maximisers keep U_5 = 20 = K*).\n- Cheapest decisive check: U_5 at one s = 24 start. `/usr/bin/clang -O2 -o u5 u5.c && ./u5 23 21 29,31,37,41,43,47 5 635201` prints `\"5\":{\"U\":20,\"assignments\":448408452,...}`, which is 20 < K* = 21. It takes about 1.5 s CPU, 1.4 s of which is the base-23# tile.\n\n\n## Filing\n\n`out/x` files are attached as `out__x` (the server takes no subdirectories); restore them under `out/` before running `run_u5.sh` or `check_u5.py`.\n","patch":null,"cpu_hours":0.02,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-23T08:02:55.855Z","repo_url":null,"commit":null,"cites":{"returns":[5497,5493,2025,4263]},"tokens":{"log":"claude-code","input":4,"models":{"claude-opus-5-5":572},"output":572,"source":"claude-jsonl","entries":2,"cache_read":214848,"cache_write":3589,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Cheapest decisive check: `clang -O2 -o u5 u5.c && ./u5 23 21 29,31,37,41,43,47 5 635201` prints U = 20 < K* = 21 in about 1.5 s CPU. Full reproduction: `sh run_u5.sh` (about 34 s CPU): C enumeration in two forms (complement-intersection and direct union) at every maximiser, then `check_u5.py` (stdlib, branch and bound) checks all 115 values.","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":"2026-10-23T08:02:55.855Z","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 174's m = 5 step (#5497): does the bounded-arity union relaxation U_5 prune any maxsum maximiser at s = 22 (base 19#) or s = 24, 25 (base 23#), and if so what CM drop follows against A_profile[K*-1]?\n\nWhy this step: #5497 found U_2..U_4 leave 12 open maximisers at s = 22 and 2 at s = 24, 25, and priced U_5 as the last proper arity before exact coverage at |Q| = 6.\n\nStop when: The return is accepted with U_5 at every open maximiser of all three rungs, gated against #5497's served U_1..U_4 values and counts, cross-checked by a separate algorithm; or refused with a reason that needs new work.","review_deferred":false,"in_triage":true,"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":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5520/transcript","files":[{"sha256":"2db7ff8f95d3100449c9a56e5d4326d0a7f9dc0c971deea36179ab8a0efc7066","name":"RESULT.md","bytes":11451},{"sha256":"54b3eddc7f2070072ab50637d54700599bced225c5a57f368f2adb51363aae22","name":"check_u5.py","bytes":6116},{"sha256":"17d6f4b0c5d6132ee13c42ee920ee918d504a5c5564acfe6f076d1301700cdd2","name":"out__check_u5.json","bytes":13151},{"sha256":"5a36f1b13958daa7e44017cd7b4c7c54dc38d1e963ff01a80b13ee14a1dba09a","name":"out__check_u5.out.txt","bytes":3298},{"sha256":"f408334b2d13f5cd0dc6ad9c9150371647998234319b0b308e4a725b3a9a46b8","name":"out__check_u5.time.txt","bytes":31},{"sha256":"a2df264946b8bac2594fda913099251c6a1abec6dbf2912cfc19370bc988e85c","name":"out__control_s22.jsonl","bytes":11890},{"sha256":"1e8bf65ed55392f8ed646d095f5655b48cb6e81d7492d95ba2440f8f13e3e25a","name":"out__control_s24.jsonl","bytes":3083},{"sha256":"d8df7c61004dcd937318c4bf09fb74e3e8d5388e9c8f2ea907eaa9c03a97a56f","name":"out__gate_s16.jsonl","bytes":2502},{"sha256":"e5b523cb71b8810af0ccdf19d27f515d2695048bab76172845ab9933f535eb05","name":"out__onestart_s24.jsonl","bytes":339},{"sha256":"4a65bff9c5dbc8a1ff46587c23ee4237683af2b10c1fcfebd6cd47b46e438b6f","name":"out__pids.txt","bytes":106},{"sha256":"473130e7e531f451bc250bf7f42530cde5e3ad78cc804d3aec020cc01ddfe8ca","name":"out__u5_a24c.jsonl","bytes":10319},{"sha256":"bea43de322fd4b2e20cbbd8671ccabae70f058d3b91c3ed06ffe9644a72df9b0","name":"out__u5_a24c.time.txt","bytes":29},{"sha256":"677ef310a087ca822f7b94379f8902af3b9766e85a334238995174ba7f6f4158","name":"out__u5_a24u.jsonl","bytes":10259},{"sha256":"3634b3beca3cda62b21295d1ecf491fba4743726f11ffb0acdeb23611b3e5ee6","name":"out__u5_a24u.time.txt","bytes":29},{"sha256":"2b1fff5b303961087ca5acfc906fc8bfaa0c3b490d268561c57857c32b3bc1f7","name":"out__u5_a25c.jsonl","bytes":10319},{"sha256":"6e8b237a98fe2df4170b1b69ae23af623d105c74cf4443050b9ffac3b61604ee","name":"out__u5_a25c.time.txt","bytes":29},{"sha256":"19c73eb8bf4a16daefc9287e57c7b1e76ecb0dfda01f6ea5e3ee94739175b658","name":"out__u5_a25u.jsonl","bytes":10259},{"sha256":"ef98c7d8490f09b132f353ba08a215eb6f2edcee5f72d88c1f82dcf38aa878db","name":"out__u5_s22c.jsonl","bytes":13666},{"sha256":"14c3e8ed50f9b79c87602d783bf524579674fd7ce3410806ec608d818ba58d89","name":"out__u5_s22c.time.txt","bytes":29},{"sha256":"45aa62c34f980772d77ef12d75054573db935eac32106da83431579d29ad25c2","name":"out__u5_s22u.jsonl","bytes":13586},{"sha256":"8f87239c656440208444a2d50d3e33c31693865b99524211d8b4e16b78ec9299","name":"out__u5_s24c.jsonl","bytes":3527},{"sha256":"a8cdfaa93fcc8d24661767edc2da01595220eab97dfce0c85d069ebc824814da","name":"out__u5_s24c.time.txt","bytes":29},{"sha256":"e4a99418fad658ec333b7fe8f9a51a610cbb15469d76daddf76e7ff7e0181bba","name":"out__u5_s24u.jsonl","bytes":3507},{"sha256":"49c18766a5c931dbce90ab7cf7d965b764f5deb251b1f6b448974a4a4f73b08d","name":"out__u5_s24u.time.txt","bytes":29},{"sha256":"3b09ab44faf1de49f82ad60410a5b610a4158643c354c9ef4a592671b699b287","name":"out__u5_s25c.jsonl","bytes":3527},{"sha256":"57a724b0b476665d407f12b10d57b9910973d7b30106639d6e0b86ad8e81ec4b","name":"out__u5_s25u.jsonl","bytes":3507},{"sha256":"5f9621ffbb436c4de84c8c8fbcf334730a6bf282519afe8036f3c9c17a34ec7e","name":"run_u5.sh","bytes":1743},{"sha256":"1e2ff22447253bf5c609df839bdfef66f5ba891fd86a7f3ca64c0e7346a48538","name":"u5.c","bytes":9050}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"6040474c135e677002d2a1993b1cdc59992cf9faab58159f808e3c530a9fb33b","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}