{"id":5525,"job_id":11194,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 205: on the eight recorded rungs 7#..31#, c ln x fails at 31# only; c(ln x)^2, c generic_max^0.584 and c gap_len^0.466 all survive; 37# falsifiers are core_run >= 22, 23, 25\n\n## Result\n\nMeasured on the recorded ladder x = 7, 11, 13, 17, 19, 23, 29, 31, with nothing recomputed. The refuted clause reproduces: core_run/ln x reaches 4.9505 at 31#, over its bar 1.25 * 3.7359 = 4.6698 (+32.5% over the x <= 23 maximum), so c * ln x fails at 31# and only there. Of the three other preregistered forms, all three survive the 25% rule on all eight rungs under both fits: c * (ln x)^2 (least constant 1.4416), c * generic_max^0.584 (1.6382) and c * gap_len^0.466 (1.1113). Their two fits agree, and each passes the permuted-pairing control as written (12 of 40320 arrangements). So the next_step's success clause (\"exactly one form\") is not met, and neither is its failure clause: the outcome is three survivors. The single new rung that tells them apart is 37#. core_run(37#) >= 22 falsifies the (ln x)^2 form, >= 23 also the generic_max form (generic_max(37) = A048670(12) - 3 = 63, confirmed from the OEIS b-file), and >= 25 all three. Any value <= 21 falsifies none. The control is weaker than it looks: the recorded core_run sequence is non-decreasing, so any increasing regressor gives the true pairing the least residual. The control therefore shows a monotone growth signal and does not separate the forms. Nothing here is a bound on core_run, on G2 or on anything past 31#.\n\n## The argument\n\nInputs (`inputs.json`, committed with the preregistration before any fit):\n\n| x | core_run | gap_len | generic_max | source |\n| - | - | - | - | - |\n| 7 | 5 | 29 | 7 | #2462 |\n| 11 | 5 | 41 | 11 | #2462 |\n| 13 | 7 | 65 | 19 | #2462 |\n| 17 | 9 | 107 | 23 | #2462 |\n| 19 | 11 | 149 | 31 | #2462 |\n| 23 | 11 | 203 | 37 | #2473 (and #4559) |\n| 29 | 11 | 257 | 43 | #4559, from `results_cg.json` (sha256 52d6821f...) |\n| 31 | 17 | 347 | 55 | #4559, from `results_cg_x31.json` (sha256 2d15608b...) |\n\ngap_len + 1 equals the department's proved G2(x#) at all eight rungs (`research/ladder/results.md`, rows n = 4..11), an independent check of that column.\n\nProcedure (`PREREGISTRATION.md`): OLS of ln core_run on ln f, fitted on S6 = {x <= 23} and on S8 = all eight; ratio r = core_run/f^b; bar B = 1.25 * max over S6 of r; a form survives when every rung passes under both fits.\n\n| form | fit | b | bar B | least c on 8 rungs | r(29)/B | r(31)/B | verdict |\n| - | - | - | - | - | - | - | - |\n| F1 c ln x | both | 1 | 4.6698 | 4.9505 | 0.6995 | 1.0601 | refuted at 31# |\n| F2 c (ln x)^2 | both | 2 | 1.6506 | 1.4416 | 0.5878 | 0.8734 | survives |\n| F3 c g^b | S6 | 0.5505 | 2.1411 | 1.8722 | 0.6479 | 0.8744 | survives |\n| F3 c g^b | S8 | 0.5838 | 2.0067 | 1.6382 | 0.6099 | 0.8164 | survives |\n| F4 c L^b | S6 | 0.4707 | 1.3042 | 1.0831 | 0.6189 | 0.8305 | survives |\n| F4 c L^b | S8 | 0.4663 | 1.3332 | 1.1113 | 0.6204 | 0.8336 | survives |\n\n(g = generic_max, L = gap_len.) Envelope at 37# from the two fits: F3 18.320 vs 18.403, F4 20.695 vs 20.658, so the fits agree well inside 25%. All comparisons are far from floating-point ties (`fit_r205.out` prints every ratio).\n\nFalsifying core_run(37#), the least integer over the form's bar at the S8 exponent (the S6 exponent in brackets):\n\n| form | bar value at 37# | falsifier N | caps |\n| - | - | - | - |\n| F2 | 21.52 | **22** [22] | below 63 and 527 |\n| F3 | 22.54 | **23** [21] | below 63 and 527; needs generic_max(37) |\n| F4 | 24.78 | **25** [25] | below 63 and 527 |\n\nF3's falsifier as a function of generic_max(37): 22 for 56..60, 23 for 61..65, 24 for 66..70. The fit used 63 as a recalled value. The OEIS b-file the head fetched (`research/sah-state/oeis/b048670.txt`) confirms A048670(12) = 66, so 63 is correct and the falsifier 23 stands. The caps are proved bounds on the observable: core_run <= generic_max (a run of L both-killed positions m..m+L-1 needs m..m+L+1 all sharing a factor with x#, so L + 2 <= j(x#) - 1) and core_run <= gap_len = 527 (G2(37#) = 528). Every falsifier lies below both caps, so each surviving form can be falsified at 37#. Discrimination: 22 falsifies F2 only; 23 or 24 falsifies F2 and F3; 25 or more falsifies all three. The bars are one-sided, so a small core_run(37#) falsifies nothing.\n\nPermuted-pairing control (exhaustive over the 8! = 40320 arrangements of the core_run values):\n\n| form | p_survive | p_as_good |\n| - | - | - |\n| F1 | 0.9786 | 0 (F1 is refuted on the true pairing) |\n| F2 | 1.0000 | 12/40320 = 0.0003 |\n| F3 | 0.7530 | 12/40320 = 0.0003 |\n| F4 | 0.7506 | 12/40320 = 0.0003 |\n\nReading, attacked as a referee would:\n\n- 12 = 2! * 3! is the number of arrangements equal to the recorded sequence (two 5s, three 11s). The recorded sequence 5, 5, 7, 9, 11, 11, 11, 17 is non-decreasing and every regressor (ln x, generic_max, gap_len) is increasing in x. For F2's fixed exponent the residual is least exactly when the sum of y_i u_i is largest, which by the rearrangement inequality is the sorted pairing. For F3 and F4 the measured count is also 12. So p_as_good <= 0.05 holds for each survivor, as preregistered, but it measures one thing: that core_run is monotone in x (a 12/40320 event under random pairing). It cannot tell F2, F3 and F4 apart.\n- p_survive = 1 for F2: the 25% bar never fails for F2 under any arrangement of these eight values, so F2's survival of the bar carries no information. F3 and F4 survive 75% of arrangements, so their bars have some power but not much. The bar test, as the next_step wrote it, is weak on this ladder. The fit residuals (log-scale RSS on S8: F2 0.1926, F3 0.0958, F4 0.0711) rank F4 first, but the preregistration made no decision on RSS between forms, and none is made here.\n- F3 and F4 each fit two parameters to eight points. Their exponents are stable between S6 and S8 (0.551 to 0.584, 0.471 to 0.466), and the S6 fits, chosen without 29# and 31#, pass at both new rungs. That is a small out-of-sample check, two points per form, not evidence of a law.\n- All eight recorded core_run values are odd. No parity constraint is proved or used here. If one held, the least attainable falsifiers would be 23, 23 and 25.\n\n## What this rests on\n\n- The x <= 19 rows of #2462 and the 23# row of #2473, as quoted in their reports. The 23#, 29# and 31# rows are also read here from the bytes of #2451's `results_cg.json` and `results_cg_x31.json` (`verify_json_r205.py`, 22 of 22 checks). The sha256 match 52d6821f... and 2d15608b.... The symbol whose count equals tB is 2 at every rung. len(types) = gap_len and G2 = gap_len + 1. one_class_g - 3 = generic_max. The longest run of 2 is 11, 11, 17, as #4559 reports.\n- gap_len checked against the department's proved ladder. generic_max + 3 checked against the OEIS A048670 b-file at all eight rungs and at 37 (A048670(4..12) = 10, 14, 22, 26, 34, 40, 46, 58, 66), the OEIS contributors' values (Hagedorn and others), not re-derived.\n- Arithmetic in IEEE doubles; the margins in `fit_r205.out` are at least 0.06 relative, far from any rounding question.\n\n## CPU and what was not run\n\nMeasured (`time.process_time` inside the script, and `/usr/bin/time` user time): stages 0 to 3 under 0.001 s; stage 4 (the 40320-arrangement control) 3.830 s; total 3.831 s CPU (user 3.86 s, wall 6.1 s under `nice -n 10` on the Studio). `verify_json_r205.py`: 0.06 s user. Route total about 3.9 s CPU; the preregistration estimated 0.05 CPU h.\n\nNot run:\n- Any wheel scan, the 37# rung itself, and any of the statistics the next_step forbids recomputing (g, the generic maximum, t0 = t2, the density tracking, route 208).\n\n## Files\n\n`PREREGISTRATION.md` and `inputs.json` (commit 606e09e, before the fit); `fit_r205.py`, `fit_r205.out`, `fit_r205.json` (commit 6af4c52); `verify_json_r205.py`, `verify_json_r205.out` (byte-level read, after the head supplied the files).\n\n## Filing note\n\n- Route 205. Scope key `r205-growth-form`. Links: #4559 extends (this applies #4559's next_step to its own recorded ladder); bears_on #2462 and #2473 (inputs).\n- Rung: measured, finite. The claim: on x = 7..31, c ln x is refuted at 31#; c (ln x)^2, c generic_max^0.584 and c gap_len^0.466 all survive the 25% rule; the 37# falsifiers are 22, 23 and 25 (generic_max(37) = 63 from A048670(12) = 66); the 29# and 31# inputs are re-read from #2451's bytes by sha256. The permuted-pairing control passes but only shows monotone growth.\n- Cheapest decisive check: `nice -n 10 python3 -I research/small-fits/r205/fit_r205.py` (4 CPU s, stdlib) and `nice -n 10 python3 -I research/small-fits/r205/verify_json_r205.py` (0.1 s), then `git diff` on both `.out` files.\n\n","patch":null,"cpu_hours":0.002,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-23T08:08:35.495Z","repo_url":null,"commit":null,"cites":{"returns":[4559,2451,2462,2473]},"tokens":{"log":"claude-code","input":4,"models":{"claude-opus-5-5":550},"output":550,"source":"claude-jsonl","entries":2,"cache_read":326959,"cache_write":6518,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Cheapest decisive check: `python3 -I fit_r205.py` (4 CPU s, stdlib) regenerates fit_r205.out; `verify_json_r205.py` re-reads #2451's two JSONs by sha256 and A048670(12) from the OEIS b-file (paths in the script are repo-relative).","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":"2026-10-23T08:08:35.495Z","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 205's next step: on the same eight recorded rungs (x = 7..31), pre-register candidate growth forms for core_run(x), fit each, apply the refuted clause's 25% rule to each, and say which forms survive.\n\nWhy this step: #4559 refuted c ln x at 31#; the route needs to know which growth form the both-killed core run follows before bounding runs.\n\nStop when: The return is accepted with the forms pre-registered before fitting, the 29# and 31# inputs read from #2451's bytes by sha256, and the survivors named with a deciding rung; 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/5525/transcript","files":[{"sha256":"f4a6117f43c7410b7ac36eb739527d3ba77b2048e28db5757a75060136917ed9","name":"PREREGISTRATION.md","bytes":5927},{"sha256":"ede7d2a8a4e14cfd5fbf57d96f618b07e5de9a4a4724eef1d93c1d6cb70cb7ec","name":"RESULT.md","bytes":8518},{"sha256":"7254f541f13ef127040624f006f3879680e3ff870238a343b650153b4586ca0a","name":"fit_r205.json","bytes":5781},{"sha256":"d181bf8bf870683f542ae888759f6b56c69ca15fc001ce703ed4b09e8af97b47","name":"fit_r205.out","bytes":5042},{"sha256":"9d8c946ea50f6b3d124716c91907aef4f7f21585a83e2cb40d7b6d96adee9f67","name":"fit_r205.py","bytes":10974},{"sha256":"29c70fbdb2af8dfd3d0375bf6a7bd3212077c8aee1e256dbf885cf7d7114d5fa","name":"inputs.json","bytes":2158},{"sha256":"653f67670a5807714ee50a0680c944ce8376cfbe03a0381e8b1cd11b81082af3","name":"verify_json_r205.out","bytes":1370},{"sha256":"b450d5612cbebdabb3fe0d89745ace38a000542400fc34e60520e1071609aa83","name":"verify_json_r205.py","bytes":3300}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"bf5634bffc9ab15eff4d0310409290ebc363217fceca51882bb8ae85d75bb739","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}