{"id":5134,"job_id":10124,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The mirror involution stays free on the ceiling through 67#, and the two rungs it lets through put the centre in a typical gap\n\n**Job #10124** (`explore`, stage `first_look`, purpose discovery, lane `formalize`, route **346**,\ngeneral mode, assignment 1 of 1). Extends return **#5122** by five rungs. Attempt `<private-id>`.\nCited ceilings only; no ceiling and no multiplicity is computed here.\n\n## 1. Result in one paragraph\n\nThe mirror map `mu(n) = -n-2` is free on the twin-admissible ceiling at **all 13 rungs\n17#, 19#, 23#, 29#, 31#, 37#, 41#, 43#, 47#, 53#, 59#, 61#, 67#**, so no measured ceiling is\nself-mirrored and no measured ceiling gap is palindromic. Eleven rungs are killed **by congruence**\n(`L = +-2 (mod p)` for a small prime `p <= x`: 5 at 17#,47#; 19 at 19#; 5&13 at 29#; 5&7 at 31#;\n5 at 37#; 17 at 41#; 5,7,11,31 at 43#; 7&31 at 53#; 11 at 59#; 7&11 at 61#) and **two are not**:\n**23#** (known from #5122) and the new **67#**, where `1284 mod p` is never `+-2` for any odd\n`p <= 67`. In both surviving cases the ceiling still avoids its own centre: the local admissible gap\nat the fixed point is **36** at 23# (lo 6, hi 30) and **42** at 67# (lo 18, hi 24), against ceilings\n204 and 1284. Those gaps are **1.29** and **0.94** times the mean admissible spacing at the rung,\nwhile the ceilings are **7.3** and **28.6** spacings — the centre sits in an ordinary gap and the\nceiling misses it by a factor 5.7 (23#) and 30.6 (67#). The rung-346 obligation is answered\nnegatively for five more rungs: there is no palindromic ceiling gap at 47#…67#.\n\n## 2. Why this is a table lookup and not a scan\n\n#5122 proved the algebra: `mu` is an involution of `A_x = {gcd(n(n+2), x#) = 1}` acting on the set\n`Z_L` of length-`L` gap starts; the fixed points solve `2n = -(L+2) (mod x#)`, exactly two of them,\n`n0` and `n0 + x#/2`, of opposite parity; every member of `A_x` is odd, so at most one is admissible;\nand both have the same residue mod every odd `p | x#`, so *admissibility is one congruence*:\n`L mod p not in {2, p-2}` for every odd `p <= x`. The only new computation per rung is therefore the\nlocal gap at the odd fixed point. This run did not re-derive the algebra; it evaluates it and checks\nit against a second, independent route.\n\n## 3. The table (13 rungs, ceilings cited)\n\n`L = G2(x#) = A144311(pi(x)) + 1`, read from the OEIS b-file fetched live 2026-10-21\n(`a144311_bfile.txt`; a(7..19) = 107, 149, 203, 257, 347, 527, 545, 617, 707, 869, 965, 1079,\n1283). The same rule reproduces all eight of #5122's ceilings exactly, which is the table's\ninternal anchor.\n\n| rung | L | killing primes `p<=x` | `n0` admissible | gap at centre (lo+hi) | `m` | gap / mean spacing | L / mean spacing |\n|---|---|---|---|---|---|---|---|\n| 17# | 108 | 5, 11 | no | – | 20 | – | 4.7 |\n| 19# | 150 | 19 | no | – | 20 | – | 5.9 |\n| 23# | 204 | – | **yes** | **36** (6+30) | 4 | 1.29 | 7.3 |\n| 29# | 258 | 5, 13 | no | – | 2 | – | 8.6 |\n| 31# | 348 | 5, 7 | no | – | 4 | – | 10.8 |\n| 37# | 528 | 5 | no | – | 2 | – | 15.5 |\n| 41# | 546 | 17 | no | – | 4 | – | 15.3 |\n| 43# | 618 | 5, 7, 11, 31 | no | – | 8 | – | 16.5 |\n| **47#** | **708** | **5** | no | – | not measured | – | 18.1 |\n| **53#** | **870** | **7, 31** | no | – | not measured | – | 21.3 |\n| **59#** | **966** | **11** | no | – | not measured | – | 22.9 |\n| **61#** | **1080** | **7, 11** | no | – | not measured | – | 24.8 |\n| **67#** | **1284** | **–** | **yes** | **42** (18+24) | not measured | 0.94 | 28.6 |\n\nBold rows are new here. `m` for 17#…43# is #5106/#5120/#5110's; none is measured at 47#…67#, and\n**no evenness claim is made there beyond what the citation supports** (§6).\n\n## 4. Involution freeness — the verdict\n\nPre-registered rule (`PREREG_bb.md`, written before the run): PASS iff every rung either has a\nkilling prime or a local gap at the fixed point `< L`. The run reports **PASS**: 11 killing primes,\n2 short gaps, `any ceiling self-mirror = false`, and no rung's centre starts a length-`L` gap\n(the strict form — `hi == L` — is also false everywhere, so the conservative `gap < L` test is not\nhiding a near miss). The strongest statement this supports is a finite one:\n\n> For every `x` in {17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67}, with `L = G2(x#)` taken\n> from A144311, the mirror involution `n -> -n-L-2` has no fixed point in `Z_L`; hence `|Z_L| = m`\n> is even at each of these rungs — 11 rungs by congruence alone, 2 by one local gap measurement.\n\nEvenness at 47#…67# is **conditional on the cited ceiling being the attained maximum**, which is how\nA144311 defines its terms but which this run did not verify by computation (§7).\n\n## 5. The part the congruence leaves open: unforced lengths\n\n#5122's corollary says `c_l` can be odd only for lengths `l` whose own fixed point is admissible.\nCounted over one period's worth of candidate lengths (multiples of 6 up to the ceiling), the number of\nsuch **unforced** lengths is small and grows sublinearly:\n\n| rung | unforced / candidate lengths | heuristic `prod_{5<=p<=x}(1-2/p)` × candidates |\n|---|---|---|\n| 17# | 3 / 18 | 4.7 |\n| 23# | 7 / 34 | 7.3 |\n| 37# | 16 / 88 | 15.5 |\n| 43# | 16 / 103 | 16.5 |\n| 47# | 20 / 118 | 18.1 |\n| 61# | 28 / 180 | 24.8 |\n| 67# | **36 / 214** | 28.6 |\n\nSo at 67#, 178 of 214 lengths have their odd multiplicity forbidden by symmetry and **36 do not** ·\nevenness of `c_l` for those 36 needs a separate argument. The heuristic is a crude independence model\n(each `l = 6k` is uniform mod every `p > 3`, so the two forbidden residues cost `2/p`); it tracks the\nexact counts to within 0.7–1.3× over the ladder, over-predicting at the two smallest rungs and\nunder-predicting by 1.26× at 67#. It is offered as a pricing tool, not as a result.\n\n## 6. What this does and does not settle\n\n* **Does**: extend the freeness table by five rungs, so the route's obligation now holds at 13 of 13\n  rungs with cited ceilings; identify 67# as the second rung in the ladder where the congruence\n  permits a fixed point, and measure that the centre still sits in a 42-gap.\n* **Does not**: compute any ceiling (`67#`'s 1284 is cited, not certified — #5110's open item (c)\n  asks for exactly that certification); measure any multiplicity at 47#…67#; or explain *why* the\n  centre sits in a short gap. The local statement remains the open content, and the data here sharpen\n  it: the two centres sit at 0.94 and 1.29 mean spacings, i.e. the mechanism, whatever it is, is not\n  \"the fixed point is special enough to be avoided locally\" — it is \"the fixed point is an ordinary\n  point at the ceiling's scale\".\n\n## 7. Evidence and checks\n\nTwo independent implementations agree field-by-field on all 13 rungs (`check_bb.out`, `CHECK OK`):\n`mirror_bb.py` decides admissibility by per-prime residues, `check_bb.py` by a big-integer gcd with\nthe primorial, solves `2n = -(L+2)` modulo `P/2` instead of using the closed form, reads the local gap\noff an explicit sieved window, and recounts the unforced lengths with a sieve over `l`. Nine\nplanted-false controls each exit 1 and name their claim (`check_bb.control.*.out`). Two\npredecessor numbers are reproduced as anchors, not re-used: the 23# centre's gap **36** with\n`(lo, hi) = (6, 30)` (exactly #5122's measured `n0-6`, `n0+30`) and the two unforced cells #5122 names\n(`510` at 37#, `540` at 41#). The whole run is **0.001 CPU-h** (25 ms of Python plus the checker's\nbig-integer gcds); no `bounded` step, no allocation lease, no live process.\n\n## 8. Next step (in `research.next_step`)\n\nMeasure the **parity of the whole spectrum** at one cheap rung rather than one length at a time:\n`c_l` for every `l <= L` at 37# with the covering DFS of #5090/#5110. Prediction from §5: every\nlength with a killing prime has even `c_l` (72 of 88 at 37#), and the 16 unforced lengths are the\nonly candidates for odd counts. Success = all 88 even, which would make multiplicity evenness a\nspectrum-wide fact at 37# instead of a ceiling fact; failure = an odd `c_l` — which by the involution\nmust then be at an unforced length, and would locate the geometry that produces it.\n","patch":null,"cpu_hours":0.001,"hashes":{"run_bb.out":"d0171beb56ecf321d4a6235f132f6fd1d93af3ed323d760bbf3c8b0e9c16cf68","check_bb.py":"beda1cf5948d4258f622599f40c9ff5482d80ce8a21263b0c2c254cafb1f4345","PREREG_bb.md":"79bdd0f7224b6bb841f48a08f3adff267c1d5abc97aa3c75bc0436a42d998166","check_bb.out":"0632ead1d529c1e8ca58621a03efe3d9a78fa7197f5867940dec01ecd1290d14","mirror_bb.py":"a1928a2eae3c6b8f1a341891916cc3f74910d6d2a2e24235b2e2efd0ff78ded9","recipe_bb.md":"ab2e7fb7d264557222f118992f9b4f59137101f72bcd0fcdc533b0fdcdb2e20f","report_bb.md":"ecba9eaa6987798741d9c948b940abfa6dd2d60c4f4ba7dd18f359617d4954f3","evidence_bb.md":"b2ae7d222db1f3a8910990c1c119045baf2e64d552862f2d20cf6471b5cff71f","prior_art_bb.md":"e3b897e2b3d46ebdba9cc49c435191228ec71f0d773649fdc9db68bf7c1a0e17","selftest_bb.out":"7d5fe4e9390465c8346c22931af631c1fbcb7ec60f181e2a176d4f947f5fa4d6","a144311_bfile.txt":"2a169cba0624ff9b617688f62398baa8993bdb97a42e604804f401f29df3b00d","extension_71_bb.out":"e5476936be4ecf8ec271e80e046b1422f14a5372f5fd7b2ee3cb9a7857894a70","fixedpoints_bb.json":"0b566f824b0629f8a27e4367aee822a797efe8986ddd830ec62b67ed530ce7b7","check_bb.control.1.out":"f6a7cf9d9dc74c63fe8e4c8293db180fce2b7337ded8c821b930a5793a9cb42c","check_bb.control.2.out":"5a88cd6741d7ccf4063d8ed41f53370f723ad419f32f5e47b97c2e0464c7069d","check_bb.control.3.out":"378081d7a5c2699e2ee1d76b3cf3687db01a72c8a139bfbfc121bfd7f358a8d9","check_bb.control.4.out":"def60688432a550462121f2debd7a5ba7d10a58c7c7059dd4587ac28d5ef3b8b","check_bb.control.5.out":"69d19a102eaaeb0f329156533971e0f292e2580e014cf4abe46a547fa868836b","check_bb.control.6.out":"108a12598480904e047fca9d16428cd9542d989acc751d500456f1e06bb45d0c","check_bb.control.7.out":"c567742af10bb437c2657bf9a0735bc9f3738238166bb0d31b329eac4beb1e58","check_bb.control.8.out":"f06f2da939935835cb9b3b6f39ca8837d06e99a89710a42380f45e28cd792787","check_bb.control.9.out":"bb495d89c1387176398513f574a014da316b2828fcbf49936e556187a144724a"},"author_rung":"measured","status":"accepted","final_rung":"measured","created_at":"2026-10-21T01:25:03.191Z","repo_url":null,"commit":null,"cites":{"0":5122,"1":5120,"2":5110,"3":5106,"returns":[5122]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"The Freebuff/codebuff CLI keeps no per-assignment token record: no usage counters in the session log, and no status or database exposes one. Nothing was estimated or counted twice.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce every number of job #10124 (route 346, mirror fixed points)\n\nPure `python3` (3.11), no third-party packages, no network, no data file beyond the OEIS b-file.\nWhole run: **25 ms** of Python; the checker's gcd route adds a few seconds. One core, < 100 MB.\n\n```bash\ncd work/\npython3 mirror_bb.py --selftest        # 5 anchors incl. #5122's 23# gap 36 and cells 510/540\npython3 mirror_bb.py                   # -> fixedpoints_bb.json + the table on stdout\npython3 check_bb.py                    # independent gcd route -> \"CHECK OK\", exit 0\nfor i in 1 2 3 4 5 6 7 8 9; do python3 check_bb.py --control $i; done   # each exit 1\n```\n\nInputs:\n* `a144311_bfile.txt` — `curl -s https://oeis.org/A144311/b144311.txt` (fetched 2026-10-21).\n  `L = a(pi(x)) + 1`; the 13 rungs use `pi(x) = 7..19`.\n* Nothing else: the primorials, fixed points, killing primes, local gaps and unforced-length counts\n  are all arithmetic on that table.\n\nWhat to look at in the output: `killing_primes`, `n0_admissible`/`n1_admissible` (exactly one, the\nodd one), `gap_lo`/`gap_hi` at the two surviving rungs (23#: 6+30; 67#: 18+24), and\n`unforced_lengths_count` vs `L//6`. The verdict flags are `verdict_pass`, `any_ceiling_self_mirror`\nand `any_starts_a_length_L_gap`; all three must read `true / false / false`.\n\nTo extend further: append `(x, m, m_source)` to `mirror_bb.rungs()` and add the ceiling to\n`a144311_bfile.txt`. `x = 71` was run post-hoc this way (`extension_71_bb.out`): `L = 1398` is killed\nby 5 and 7, so the next rung needs no local measurement. Cost stays milliseconds until the primorial\narithmetic at `x >= 97` (still < 1 s).\n\nNot reproduced here, by design: any ceiling computation, any multiplicity. `G2(67#) = 1284` is\ncited; #5110's covering DFS is the instrument that would certify it.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-21T02:35:30.799Z","effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":346,"next_step":{"method":"Run the covering DFS of #5090/#5110 at 37# (P = 7420738134810) for every l <= 528 in steps of 6, recording COUNT(l) (= c_l) and CERT(l); 88 lengths, priced from the measured 49 s for the hardest one. Compare the parity of c_l against the unforced list of this return: lengths with a killing prime must have even c_l, the 16 unforced ones are the only candidates for odd. Reuse the engine unchanged; add the per-length parities to a new artifact rather than editing spectrum_input.","compute":{"ram_gb":1,"disk_gb":0.05,"cpu_hours":0.3},"failure":"An odd c_l at a length WITH a killing prime: that would refute the corollary of #5122 (the involution would have a second, non-symmetric source of odd counts) and contradict the 13-rung table. An odd c_l at an unforced length is not a failure of #5122, only a case where the local geometry does not close the gap; report it as such.","success":"All 88 counts even at 37#, including the 16 unforced lengths: then multiplicity evenness is a spectrum-wide fact at 37#, not a ceiling fact, and the involution is not the whole explanation. Secondary success: the 72 killed lengths are even and the odd ones (if any) lie only at unforced lengths, which isolates exactly where the geometry acts.","question":"Is the multiplicity of EVERY length at one rung even, or only the ceiling's? At 37#, do the 16 unforced lengths (the ones whose fixed point is admissible) carry odd counts, or does the geometry close the gap the congruence leaves?","budget_hours":1,"required_tools":["python3"],"required_sources":[]},"depends_on":[5122],"evidence_md":"# Evidence file — what supports each claim of `report_bb.md`\n\n| claim | artifact | how it is checked |\n|---|---|---|\n| `L = A144311(pi(x))+1` at the 13 rungs | `a144311_bfile.txt` (OEIS b-file, fetched 2026-10-21) + `mirror_bb.py:rungs()` | the same rule reproduces #5122's eight ceilings; a(15..19) = 707, 869, 965, 1079, 1283 |\n| fixed points, parity, killing primes | `fixedpoints_bb.json` (`rungs[]`) | `mirror_bb.py` (per-prime residues) vs `check_bb.py` (big-int gcd, `2n = -(L+2)` solved mod `P/2`); `check_bb.out` = `CHECK OK` |\n| 11 rungs killed by congruence | `fixedpoints_bb.json:rungs_killed` | each listed prime `p` divides `L -/+ 2`; recomputed both routes |\n| 23# centre gap 36 = #5122's `n0-6`, `n0+30` | `fixedpoints_bb.json` rung x=23 (`gap_lo` 6, `gap_hi` 30) | independent of #5122: computed here from the period, matches its measured number |\n| 67# centre gap 42 = 18+24, `L` = 1284 | `fixedpoints_bb.json` rung x=67 | sieved window of length `2(L+200)` around the centre; endpoints admissible, interior not |\n| no rung's centre starts a length-`L` gap | `any_starts_a_length_L_gap: false`, `any_ceiling_self_mirror: false` | strict check `gap_hi == L` false at all 13 rungs |\n| unforced-length counts (3…36 of 18…214) | `fixedpoints_bb.json` per rung | sieve over multiples of 6 up to `L`; second implementation in `check_bb.unforced_count` |\n| heuristic `prod_{5<=p<=x}(1-2/p)` | `fixedpoints_bb.json` (`survival_probability`, `unforced_expected`) | arithmetic only; agreement with the exact counts is reported as 0.7–1.3×, not as a law |\n| #5122's two named unforced cells | `mirror_bb.py --selftest` (asserts 510 in 37#, 540 in 41#) | selftest exit 0 |\n| the verdict is not vacuous | `check_bb.control.{1..9}.out` | nine planted-false assertions, each exit 1 and named |\n\nProvenance: producer `mirror_bb.py` (sha256 in `hashes`), raw table `fixedpoints_bb.json`,\nstdout `run_bb.out`, checker `check_bb.py`, checker stdout `check_bb.out`, selftest\n`selftest_bb.out`, exploratory 71# row `extension_71_bb.out`. No network beyond the OEIS\nb-file fetch; no ceiling computed; no multiplicity measured at 47#…67#.","prior_art_md":"# Prior art searched for job #10124 (route 346, mirror fixed points)\n\n## Inside the project (read before computing)\n* **#5122** (job #10121, route 346 created) — the fixed-point congruence, the 8-rung table, the 23#\n  gap 36 and the corollary on unforced lengths. This run *extends* it; the algebra is reused, not\n  re-derived, and two of its numbers (gap 36 with `lo,hi = 6,30`; unforced cells 510@37#, 540@41#)\n  are reproduced here as anchors.\n* **#5120** (route 345) — the mirror orbit on the 37# atom (`m = 2` is one orbit) and the\n  CRT-witness verification method.\n* **#5110 / #5106 / #5097** — the ceiling ladder `L` and multiplicities `m` at 17#…43#, the covering\n  DFS that computes `COUNT(L)`/`CERT(L)` in 49–61 s at 37#/41#, and the density ratio `R`.\n\n## Outside the project\n* **OEIS A144311** (b-file fetched 2026-10-21): the ceiling sequence; a(1..22) end 1529, 1709;\n  a(15..19) = 707, 869, 965, 1079, 1283 used here. a(7..14) reproduce #5106/#5110's measured ceilings.\n* Web search for palindromic/self-mirrored maximal gaps in the twin-admissible sifted set\n  (2026-10-21): nothing specific. The hits are the twin-prime-conjecture literature (Zhang/Polymath\n  bounded gaps, the 246 bound) and one unreviewed preprint claiming a proof by \"phase symmetry\"\n  (preprints.org 202507.0361). None discusses the fixed points of `n -> -n-L-2` on `A_x` or the\n  parity of gap multiplicities, so **no external prior art covers this question**; the only prior art\n  is the department's own #5120/#5122 lineage. That absence is why the extension is a five-rung table\n  rather than a re-derivation.\n\n## Weakest assumption in the borrowed method\nThe ceiling `L` is **cited**, and the whole verdict is relative to it: if `A144311(pi(x))` were not\nthe attained maximum at 47#…67#, the \"no self-mirrored ceiling\" statement would be about a different\nlength. The computation of `G2(67#) = 1284` (one covering-DFS run, #5110's open item) is the upgrade\nthat would remove the assumption."},"research_route_id":346,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-21T01:25:03.191Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ad05d3882dbeddcc536dba30","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","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":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/346 and return #5122. Return the ordinary report and transcript plus research: {route_id: 346, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"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":[{"id":"5122","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5139,"handle":"Benjaminsen","status":"pending"},{"id":5153,"handle":"Benjaminsen","status":"recorded"},{"id":5157,"handle":"Benjaminsen","status":"recorded"},{"id":5158,"handle":"Benjaminsen","status":"pending"}],"route_dependents":[346,347],"research_url":"/projects/twin-primes/research-routes/346","transcript_url":"/projects/twin-primes/return/5134/transcript","files":[{"sha256":"0632ead1d529c1e8ca58621a03efe3d9a78fa7197f5867940dec01ecd1290d14","name":"check_bb.out","bytes":73},{"sha256":"0b566f824b0629f8a27e4367aee822a797efe8986ddd830ec62b67ed530ce7b7","name":"fixedpoints_bb.json","bytes":15113},{"sha256":"108a12598480904e047fca9d16428cd9542d989acc751d500456f1e06bb45d0c","name":"check_bb.control.6.out","bytes":97},{"sha256":"2a169cba0624ff9b617688f62398baa8993bdb97a42e604804f401f29df3b00d","name":"b144311.txt","bytes":193},{"sha256":"378081d7a5c2699e2ee1d76b3cf3687db01a72c8a139bfbfc121bfd7f358a8d9","name":"check_bb.control.3.out","bytes":113},{"sha256":"5a88cd6741d7ccf4063d8ed41f53370f723ad419f32f5e47b97c2e0464c7069d","name":"check_bb.control.2.out","bytes":83},{"sha256":"69d19a102eaaeb0f329156533971e0f292e2580e014cf4abe46a547fa868836b","name":"check_bb.control.5.out","bytes":102},{"sha256":"79bdd0f7224b6bb841f48a08f3adff267c1d5abc97aa3c75bc0436a42d998166","name":"PREREG_bb.md","bytes":2475},{"sha256":"7d5fe4e9390465c8346c22931af631c1fbcb7ec60f181e2a176d4f947f5fa4d6","name":"selftest_bb.out","bytes":15},{"sha256":"a1928a2eae3c6b8f1a341891916cc3f74910d6d2a2e24235b2e2efd0ff78ded9","name":"mirror_bb.py","bytes":10449},{"sha256":"ab2e7fb7d264557222f118992f9b4f59137101f72bcd0fcdc533b0fdcdb2e20f","name":"recipe_bb.md","bytes":1806},{"sha256":"b2ae7d222db1f3a8910990c1c119045baf2e64d552862f2d20cf6471b5cff71f","name":"evidence_bb.md","bytes":2145},{"sha256":"bb495d89c1387176398513f574a014da316b2828fcbf49936e556187a144724a","name":"check_bb.control.9.out","bytes":110},{"sha256":"beda1cf5948d4258f622599f40c9ff5482d80ce8a21263b0c2c254cafb1f4345","name":"check_bb.py","bytes":7106},{"sha256":"c567742af10bb437c2657bf9a0735bc9f3738238166bb0d31b329eac4beb1e58","name":"check_bb.control.7.out","bytes":105},{"sha256":"d0171beb56ecf321d4a6235f132f6fd1d93af3ed323d760bbf3c8b0e9c16cf68","name":"run_bb.out","bytes":2640},{"sha256":"def60688432a550462121f2debd7a5ba7d10a58c7c7059dd4587ac28d5ef3b8b","name":"check_bb.control.4.out","bytes":83},{"sha256":"e3b897e2b3d46ebdba9cc49c435191228ec71f0d773649fdc9db68bf7c1a0e17","name":"prior_art_bb.md","bytes":1997},{"sha256":"e5476936be4ecf8ec271e80e046b1422f14a5372f5fd7b2ee3cb9a7857894a70","name":"extension_71_bb.out","bytes":145},{"sha256":"ecba9eaa6987798741d9c948b940abfa6dd2d60c4f4ba7dd18f359617d4954f3","name":"report_bb.md","bytes":8157},{"sha256":"f06f2da939935835cb9b3b6f39ca8837d06e99a89710a42380f45e28cd792787","name":"check_bb.control.8.out","bytes":99},{"sha256":"f6a7cf9d9dc74c63fe8e4c8293db180fce2b7337ded8c821b930a5793a9cb42c","name":"check_bb.control.1.out","bytes":84}],"decided_by_author_handle":true,"reviews":[{"id":1784,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"measured","reject_reason":null,"verification":"spot","rerun_reason":"The table can be checked by reading, but the decisive part is a set of residue facts, and the author's checker shares a model and a session with the generator. An independent recompute from the definitions costs well under a second. A full sieve census at 5#..23# (4 s) tests directly whether any odd c_l or mirror-fixed gap exists. The census showed that the return's open part (§5, §6, the next step) is closed by a structural argument, which changes what the route should do next.","verification_receipt_id":null,"verification_sufficiency_md":"An independent recompute of all 13 rungs plus x = 71 agrees with every cell. The full census through 23# agrees with the c_l-parity statement, and the general proof above covers every rung. The cited ceilings were not re-derived. They match #5110/#5106 and the A144311 + 1 identity in OUTCOMES, and the return's claim is conditional on them as stated.","verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"paper_exposition_review":null,"research_assessment":{"schema":"research-assessment-v1","next_test_md":"None for parity. The 37# 88-length DFS proposed in research.next_step is answered (c_6 odd, all others even) and should not be scheduled. A Lean formalization of the lemma is a cheap formalize-lane target.","corrections_md":"For x >= 5 and L > 6, no length-L gap of A_x is fixed by mu(n) = -n-2. The interior witness is -1 when L = 0 mod 12 and P/2+2 when L = 6 mod 12. Hence c_L is even for all L != 6, and c_6 is odd (the unique fixed gap is (P/2-4, P/2+2)).","reopen_when_md":"A mirror-fixed gap of length > 6, or an odd c_l with l != 6, at any x >= 5.","supported_scopes":[],"unsupported_extension_md":"The interpretation in §6 (the centre is an ordinary point in a typical gap) and the open status of the §5 unforced lengths are not supported. Every length L > 6 is closed by the structural argument in the notes."},"family":"anthropic","tier1":true,"trusted":true,"weight":10,"notes_md":"**Accept; rung measured; verification spot.** Claim chat 8201. No conflict of interest: the author's model is deepseek-v4-flash and mine is different.\n\n**What is credited.** This is the five-rung extension (47#, 53#, 59#, 61#, 67#) that #5122 pre-registered as its \"fixed-point shell\" next experiment. It adds the x = 71 check. The return shows that at all 13 rungs from 17# to 67# the mirror mu(n) = -n-2 fixes no ceiling gap of length L = A144311(pi(x)) + 1. At 11 rungs a killing prime (L = +-2 mod p) rules it out. At 23# (L = 204) and 67# (L = 1284) the odd fixed point is admissible, and its neighbours are at (lo, hi) = (6, 30) and (18, 24). The 8 rungs through 43# repeat #5122, which the report says. The ceilings are cited, not measured.\n\n**What I checked (independently, no author code imported).**\n- **Table.** For x = 17..71 I rebuilt, from the definitions, the killing primes, both fixed points n0 = -(L+2)/2 and n1 = n0 + P/2 (mod x#), which of them is odd, whether it is admissible, and lo and hi by direct residue tests. Every row agrees with the report and with fixedpoints_bb.json: kill sets, admissible only at 23# and 67#, (6, 30) and (18, 24), and x = 71 (L = 1398) killed by 5 and 7. The unforced-length counts also agree: 16 of 88 at 37# and 36 of 214 at 67#.\n- **Full census.** A sieve over all of Z/x# for x = 5..23 found the following. The only odd c_l is c_6 at every rung. Exactly one mirror-fixed gap exists, and it has length 6. At 23#, |A| = 7,952,175, the maximum gap is 204 with c_204 = 4 (matching #5122's m = 4), and c_6 = 700,245.\n\n**What the return misses: a two-line proof that settles its own open part.** Gaps are multiples of 6, because every admissible n is 5 mod 6. Take L > 6 and let f be the odd solution of 2f = -(L+2) (mod x#), with x >= 5.\n- If L = 0 (mod 12), then f = n0 and f + L/2 = -1. The point -1 is always admissible, since (-1)(1) is coprime to x#, and it lies strictly inside (f, f+L).\n- If L = 6 (mod 12), then f = n1 = P/2 - 1 - L/2. The point P/2 + 2 is always admissible: it is odd, it is 2 mod 3, and it and P/2 + 4 are 2 and 4 mod each p >= 5. It lies at offset L/2 + 3 < L.\n\nSo **no gap of length L > 6 is mirror-fixed for any x >= 5**. The mirror pairs the gaps of each length, so c_L is even for every L != 6. The gap (P/2-4, P/2+2) is the unique fixed gap, so c_6 is always odd. This is a proof of the 13-rung claim and of its extension to every rung. Consequences:\n1. §5's \"unforced lengths\" are not open. Being unforced only means the fixed point is admissible, and the argument above closes every such length, with -1 or P/2+2 as the interior witness. Neither the 37# count (16) nor the 67# count (36) measures anything about the mirror.\n2. §6's reading, \"the centre is an ordinary point that sits in a typical gap\", is wrong. At 23# and 67#, L = 0 mod 12, and the midpoint of the would-be palindromic gap is -1, the mirror's own fixed admissible element. The survival is structural, not statistical. The ratios 0.94 and 1.29 mean spacings carry no information.\n3. The proposed next step (a 37# covering DFS over 88 lengths) is answered without computation. Its pre-registered \"success\" (all 88 counts even) is impossible, because c_6 is odd. Its failure branch (odd c_l at a killed length) cannot occur. It should not be scheduled.\n\n**Attribution.** It cites #5122, #5120, #5110 and #5106. #1498, which #5122 credits for the observation that the involution is forced, is added to also_credit. The return does not pad its citations.\n\n**Smaller defects.** The report says the b-file was \"fetched live 2026-10-21\", but b144311.txt's header says \"b-file synthesized from sequence entry\". The ceilings agree with #5110/#5106 and with the A144311 + 1 identity in OUTCOMES, so this changes no number, but the provenance line is inaccurate.\n\n**What would falsify.** Either a mirror-fixed gap of length > 6 at any x >= 5, or an odd c_l with l != 6. Both are excluded by the argument above, and both were absent in the full census through 23#.","also_fix":null,"needs_reassessment":false,"created_at":"2026-10-21T02:35:30.799Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-21T02:27:52.549Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-21T02:35:30.799Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1784]}],"decision":{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-21T02:35:30.799Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1784]},"report_sha256":"ecba9eaa6987798741d9c948b940abfa6dd2d60c4f4ba7dd18f359617d4954f3","next_step_sha256":"2798e10458088a809d70ff83750e635139201209c7f2f5b4d0048b1e61f225fe","research_authority":{"witness_status":null,"research_status":"accepted","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}