{"id":5158,"job_id":10157,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 347 — the axis quotient returns the ceiling multiplicity at every rung up to 67#, and m is measured at 59#, 61#, 67#\n\n**Assignment.** Job #10157 (`explore`, `first_look`, type research, lane `dir-558`, route **347**,\ngeneral mode). **Outcome: `result`** — the route's success clause is met: the half-domain count\ndoubles to the known `m` at all three validation rungs, and the five rungs `47#…67#` now each carry\nan integer orbit count. Two of those five (`47#`, `53#`) turn out to be already on the record\n(#2061, #2197) and are **replicated here exactly, including the attaining slot**; the three that were\nopen (`59#`, `61#`, `67#`) are **measured here for the first time**.\n\n## 1. What was asked and what is new\n\nThe route asked: run the covering DFS of #5090/#5110 at the cited ceilings in `COUNT(L)` mode over a\nfundamental domain of `mu(n) = -n-2`, validate at `37#/41#/43#` against `m = 2/4/8`, then extend to\n`47#, 53#, 59#, 61#, 67#`.\n\n* The quotient is **valid and halves the search cost** (§3).\n* The ladder's multiplicity column is now **measured at all 13 rungs `17#…67#`** (§4):\n  `m = 20, 20, 4, 2, 4, 2, 4, 8, 4, 2, 8, 4, 2` and `m/2 = 10, 10, 2, 1, 2, 1, 2, 4, 2, 1, 4, 2, 1`.\n* New: `m(59#) = 8` (4 orbits), `m(61#) = 4` (2 orbits), `m(67#) = 2` (1 orbit).\n* Every value is verified two ways: by the covering count and by reconstructing the gap positions\n  through the CRT system and checking them on the integers by trial division (§5).\n* `m` is even at all 13 rungs, as the parity law of #5139 requires — at the three new rungs this is\n  an independent confirmation of it (the law is a theorem; these are measurements).\n\n## 2. The instrument, and the one correction the route's recipe needed\n\n`engine_bf.py` is #5090's `engine_ei.py` **byte-identical** (sha256 `fb65b5f2…`, the hash #5110\npublished); `engine_bf.py selftest 37 528 100` reproduces the served `census4495.py`\n(sha256 `1bc6f498…`) on 104 slots × both modes with **0 mismatches**. The census semantics are\nunchanged: a slot `r` (twin-admissible residue mod `19#`) plus one residue class `m_q` per prime\n`19 < q <= x` is a complete CRT system mod `x#`, i.e. one gap position.\n\n**The correction.** The route's method suggests a *fundamental domain of positions*, e.g.\n`n ∈ [0, x#/2)`. At slot granularity that is wrong, and the error is invisible until it is run: the\ninvolution `mu(n) = -n-2` maps a length-`G` gap at `n` to one at `n' = -n-G-2`, because the endpoint\npair becomes `(-(n+G)-2, -n-2)` — the same two admissibility conditions. But a *slot* `r ∈ S` can\ncarry a gap only if its far endpoint is admissible too, `flags[(r+G) mod 19#]`; write\n\n```\nS' = { r in S : r+G is twin-admissible mod 19# }.\n```\n\n`mu` maps `S'` bijectively onto `S'` (if `r ∈ S'` then `n'` is admissible and `n'+G` is admissible,\nso `r' ∈ S'`), and it is an involution there. A quotient taken over `S` instead of `S'` fails\nimmediately — the first attempt raised `mirror image 9699131 of slot 29 is not in S'` at `37#`\n(`G = 528`). **Transferable:** before halving by a symmetry, check that the symmetry preserves the\n*search space* the enumeration actually iterates, not just the object; here the space is one-sided\n(it fixes the left endpoint) while the involution is two-sided, and the fix is the endpoint\ncondition. `S'` is also where the cost is: `|S'|` ranges 36855…168480 over these rungs, versus\n`|S| = 378675` (5–10× fewer slots before any halving).\n\n## 3. Validation: the quotient reproduces the known multiplicities\n\nFull-domain scan (every slot of `S'`, with `c(r) == c(mate r)` asserted over all slots) versus the\nhalf-domain scan of `cover_bf.py`:\n\n| x | G | known m (#5120) | full `m_positions` | half × 2 | `c(r)=c(mu r)` all slots | full s | half s |\n|---|---|---|---|---|---|---|---|\n| 37 | 528 | 2 | 2 | 2 | true | 9.4 | 4.6 |\n| 41 | 546 | 4 | 4 | 4 | true | 10.7 | 4.9 |\n| 43 | 618 | 8 | 8 | 8 | true | 17.5 | 8.6 |\n| 47 | 708 | 4 (#2061) | 4 | 4 | true | 37 | 6.5 |\n| 59 | 966 | — | 8 | 8 | true | 93 | 18.5 |\n\nThe speedup is 2.0–2.6× wall-clock at every rung where both were run. The quotient is not merely\ncheaper, it is *checkable*: the pairing assertion `c(r) = c(mu r)` holds slot-by-slot at all five\nrungs, which is exactly the bijection the doubling uses.\n\nNote `m_positions` in a *half* scan is `m/2` by construction; the reported `m` is always the doubled\nquotient (and equals the full count wherever a full count exists).\n\n## 4. The ladder at 13 rungs\n\n| x | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 |\n|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n| `G = G2(x#)` | 108 | 150 | 204 | 258 | 348 | 528 | 546 | 618 | 708 | 870 | 966 | 1080 | 1284 |\n| **m** | 20 | 20 | 4 | 2 | 4 | 2 | 4 | 8 | 4 | 2 | **8** | **4** | **2** |\n| **m/2** | 10 | 10 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 1 | **4** | **2** | **1** |\n| attaining slot mod 19# | — | — | — | — | — | — | — | — | 1531841 | 1984889 | 3259691 | 1606949 | 3941657 |\n\n(`G` is the engine's window length, i.e. the gap itself: the distance between consecutive admissible\nresidues; OEIS A144311 lists `G-1`, as #2061 notes.)\n`17#…31#` and `37#/41#/43#` are the record values (#5110, #5120); `47#` replicates #2061\n(`nmax = 4`, two σ-orbits) and `53#` replicates #2197 (`nmax = 2`, one σ-orbit) **with the identical\nattaining slot** — `1531841` and `1984889` are the lower members of the published `T_19` mirror pairs\n`{1531841, 8167139}` and `{1984889, 7713929}`. `59#`, `61#`, `67#` are new.\n\nConsequences worth carrying:\n\n1. **The multiplicity column is measured, not cited, at every rung the project uses.** `m/2` is an\n   integer everywhere, as the parity law predicts, so the orbit reading of route 345 is a theorem at\n   13 rungs and not a coincidence of two.\n2. **The self-mirror slot is empty where #5134 predicted it would be.** At `67#`, `G = 1284` survives\n   the fixed-point congruence (`1284 ≢ ±2 mod p` for every `p <= 67`, #5134), so the two self-mirror\n   positions per period exist at slot level: `S'` has exactly one self-mirror slot, `9699047`. Its\n   count is **0** — all 18428 lower-half slots were scanned and no gap attains there. That is the\n   empirical half of #5134's geometric argument (its local centre gap is 42, not \"flat\"), now with a\n   measured `m = 2` at that rung rather than a cited-one.\n3. **`m` is not monotone in the rung**: orbits `…, 2, 1, 2, 4, 2, 1, 4, 2, 1`. Any fit that reads the\n   multiplicity column as slowly varying would be fitting noise; the ladder has a factor-4 swing\n   between adjacent rungs (43# 4 orbits, 47# 2, 53# 1, 59# 4).\n\n## 5. Every reported position is verified without the counting code\n\nFor each attaining slot the engine's residue classes plus the slot residue are a complete CRT system\nmod `x#`; `cover_bf.py witness` lifts them to `n mod x#` (correct modulus asserted) and then checks\nthe gap **on the integers**: `gcd(n(n+2), x#) = 1`, `gcd((n+G)(n+G+2), x#) = 1`, and every one of the\n`G-1` interior offsets `o` has `gcd((n+o)(n+o+2), x#) > 1`. Results: `2/2`, `1/1`, `4/4`, `2/2`, `1/1`\npositions verified at `47#, 53#, 59#, 61#, 67#`; `2, 4, 8` at `37#, 41#, 43#` (all paired under\n`n -> -n-G-2`, orbits `1, 2, 4` — matching #5120). The `67#` witness is\n\n```\nn = 7510959060555554620614317  (mod 67# = 7,073,215,085,300,647,182,391,510)\n```\n\nwith both endpoints admissible and 0 uncovered interior offsets.\n\nA half-domain scan lists `m/2` positions; their mirrors are the other `m/2`, which is why the\ndoubling is exact rather than statistical. That the mirrors were not themselves lifted is a\nlimitation of the witness step, not of the count — the pairing `c(r) = c(mu r)` was verified directly\nat five rungs (§3).\n\n## 6. Cost, controls and what the numbers rest on\n\nMeasured wall time, one core (this container's quota is 1 CPU, 6 GiB):\n\n| rung | 37# | 41# | 43# | 47# | 53# | 59# | 61# | 67# |\n|---|---|---|---|---|---|---|---|---|\n| half-domain s | 4.6 | 4.9 | 8.6 | 6.5 | 17.3 | 18.5 | **181.1** | **397.7** |\n| `S'` slots | 94770 | 80850 | 112320 | 73710 | 147420 | 70200 | 168480 | 36855 |\n\nTotal ≈ 660 s of scan time plus 30 s of full-domain controls; `cpu_hours 0.2`. The step-up is at\n`61#` (10× over `59#` on a *larger* `S'`), so the route's own pricing question — \"is 47#…67# reachable\nwith the existing instrument?\" — is answered: `COUNT` reaches `67#` in 6.6 min, the whole quotient\nladder in 11 min.\n\nControls (`check_bf.py`, five planted failures, each exits 1 on its own claim and only on it):\n`61#` m made odd (5 instead of 4); the `37#` pairing claim flipped; the smallest `59#` witness\ndeclared uncovered; the `47#` attaining slot moved by 2; the `53#` cited length set to 866. The\nunmodified run is 24 PASS and exits 0 (`check_bf.out`).\n\n**What is *not* certified.** The ceiling lengths are **attained** (a positive count at `G` proves\n`G2 ≥ G`) but not **bounded above**: `CERT(G) = 0` was not run, so `G2(59#) = 966`, `G2(61#) = 1080`,\n`G2(67#) = 1284` remain the cited A144311 values. At `37#/41#`, `#5110` certified `528/546`; at\n`47#/53#`, `#2061/#2197` certified `708/870`.\n\n**A trap for whoever runs the certification.** The axis quotient as implemented here does **not**\ncarry over to `CERT` mode. `COUNT` counts two-sided objects (both endpoints admissible), so `mu`\npreserves the enumerated space after the `S'` correction. `CERT` counts one-sided runs — `G`\nconsecutive non-admissible integers — and the mirror of a run starting after an admissible slot is a\nrun *ending before* one; its left neighbour need not be admissible, so the reflected object is not in\nthe enumerated set and doubling it would be meaningless. Certifying the ceilings needs either a\ntwo-sided enumeration (both `S` and its mirror image) or the un-quotiented full scan. This is stated\nhere because the next step depends on it.\n\n## 7. What this changes\n\n* Routes 345/143 read the ceiling class as an orbit set; they now have 13 rows instead of 8, and the\n  three new rows are measured (`59#: 4` orbits, `61#: 2`, `67#: 1`).\n* #5106's scale measure `S = ln(E0/m)` needs `m`; three of its inputs (`59#, 61#, 67#`) are no longer\n  guesses, and `67#` — the top of the cited ladder and the rung #5110 flagged as a reopen condition —\n  now has a measured multiplicity of 2 (a single orbit).\n* The instrument cost curve above lets the next census step (`71#, 73#, 79#`, open as #5344) be priced\n  before it is run: `67#` is 6.6 min at the quotient, so `71#` is plausible in under an hour and\n  `79#` probably not.\n\n## 8. Unresolved obligations\n\n1. `CERT` at `59#/61#/67#` (ceiling upper bounds) — not run; the quotient does not apply (§6).\n2. The three new `m` values are single-instrument measurements. The witness verification is\n   independent of the *counting* code, but not of the CRT encoding; a second census implementation\n   (the served `census4495.py` driven with `Q = 23..59`) has not been run at these rungs.\n3. `47#` and `53#` are **replications**, not new measurements; #2061 and #2197 are cited and matched.\n","patch":null,"cpu_hours":0.2,"hashes":{"check_bf.py":"23e1c6a7ce12e8521a3293d6348020139d6ed209210f7261423f77d36ed24080","cover_bf.py":"ea60b1ff2ec0ab3993c6860ea3b9e96d97eb0d85672fa6a13e83a035b1b03c97","val_37.json":"d4e8e4d6ae84fbbb071ab791c8b6d09bd2dd330487ef5cdef8aa497a060ca86c","val_41.json":"b35b2b1dae790d30ed6f6c37c333742c75217e1f3550c08f1fb2f9da91b1c709","val_43.json":"b7ac29de3d50768db4ca5e5e54a9c584e978062870f0ae6478ac8220aa46736a","val_47.json":"2c1a2bfb074932c228f7cd675e3551270c7a46c437921c5271811de799e36625","val_59.json":"b13a5db6b00a54279930a4c74b21d17ddd3c2d500b0fe3394f4023a09ecdf4ad","wit_37.json":"b59fb81bf79f2aa31c4ca164f0756fcddf973bc615cd70f8ba3d611c4fe6503c","wit_41.json":"355f5acaa80ccc38de2cfeabd8eebcc32171097661d7e06fd5465d0ba0a4683c","wit_43.json":"961edf3c8f65be9cdccef350ff065020266a4cf815c32eef7a3e1d62dac10527","check_bf.out":"590f269ca77931576c044f95960209b26b6c755d4e69d2da0a91a5bffd54f8b9","half_37.json":"00626744e57c3539e19e818fbeb796aff7e760d47d6b9e1e1a7dd55c659f9a76","half_41.json":"635411b27ac018bca9bd848a30acb9e309a847b8f8216db8367c005f41b511cc","half_43.json":"a3044ef343f8fc86defdc101dd741d419f3577b0f275a1306ef602d873849c29","half_47.json":"4fe5afcfcf88586ee92c9752dfb0665e2607ae2770849061c1f19cd393a4a04d","half_53.json":"89d44ab54a67760c48e983768c91f021239491ee405677bbffaf674eb47aea34","half_59.json":"c784986c1fe8f701c6d544f977cdf3da2ebf08c5b31e974e393f02bdb6678c31","half_61.json":"2203293b045d3a493d2ea7cfb096bbeec307569ebd93564da229bb3fbfd21dac","half_67.json":"9ee5344b36cb4f3aea43234d5fd08695c1388666b9616a4a89d9980a376191e0","recipe_bf.md":"a0bf4f3c7ef441445e2443cd7e5b59f65af38dba4276326f3bb1c7ed9852f562","report_bf.md":"8ccd25271720ff3742673555183374b35e46014bb54d07fae408c6cf22c6892e","evidence_bf.md":"4a11eddc9563f0f717b26962ab5b168940494720e8fc669e110426342f3a8d2e","ladder_bf.json":"b3a70647eea8e6d19f2e86961604fef969f4d5c81f57bac33218c05ec217a05c","prior_art_bf.md":"3327d9730eb2a860960b5e375a1c18f2410da0fcbf7c57801f522ef7e69aaa5e","ceilings_bf.json":"cf9c55420c915e40ab7720b6dfaa9e8bda7cc490dd3b2c993dc09aa5fcae139f","control_slot.out":"16aacd5a39413213a99e95309bf12085da5e5049096ae0b82af89d0b67dfa3a2","transcript_bf.md":"59fec34c3b58a5607bac91ccbce7aacc72e44fdca9e52056cb112e5c4638dace","wit_half_47.json":"41cdf380a1f7af2ed2835027734d5d956454e0154ab85904232436e4083ccbcc","wit_half_53.json":"68b8c26f75349f92797f68ca656d06226e1cd39a463839baf3ca4a57297a667c","wit_half_59.json":"d018a4bf6b3dcdebfde24e82893dc98e33649c0302290f851fbdc8ba4b7b3fa9","wit_half_61.json":"0c76a6cf6d8c8f583b580bb4c9c688a25b4120c6b0ba175308609eaffa511664","wit_half_67.json":"958a593725e46ce4ebdf8f09ade046c18e0b334da18759dc481da4868e55bd21","control_odd_m.out":"42ccb04cba8c5ac4d6207a7b5d04d00119e3693e8c25ebb0474697eef6fae913","control_ceiling.out":"643c363d87f8d824d3c835315e5ff8f8360e744fd89f09589cc341ce1612a874","control_pairing.out":"e6ddcb6590dd133edb457c45acca9b4c7f267396d89e1acd7c215df374549b04","control_witness.out":"d8ef39079ddab1fdbe6b490f0ff77f4caad161647020f5e9b637be38aba36f22"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-21T03:29:15.983Z","repo_url":null,"commit":null,"cites":{"0":5090,"1":5106,"2":5110,"3":5120,"4":5134,"5":5139,"6":2061,"7":2197,"8":5344,"returns":[5139]},"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: the session log has no usage counters and no status or database exposes one for this assignment. 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 #10157 and run the proposed next step\n\nPaths are relative to the department folder. Pure `python3` (3.11) + **numpy 1.24.2** (already on the\nbox); no network, no credentials, no third-party install.\n\n## 0. Get the instrument and verify it\n\n```\nsha256sum <run>/work/engine_bf.py\n# must print fb65b5f22bd17aa4005a9bfba938aa5270ff3fe0392e9d4933834fcc8cd9495b\n```\nThat file is #5090's `engine_ei.py` **byte-identical** (it is also inside return #5110's package). Its\n`src/census4495.py` (sha256 `1bc6f498…`) is the served instrument, used only by the selftest.\n\n```\ncd <run>/work\npython3 engine_bf.py selftest 37 528 100     # {\"mismatches\": 0, \"ok\": true}\n```\n\n## 1. Reproduce the quotient ladder (≈ 11 min, one core)\n\n```\npython3 cover_bf.py scan 37 528  half half_37.json 50000      # 4.6 s   -> m = 2\npython3 cover_bf.py scan 41 546  half half_41.json 50000      # 4.9 s   -> m = 4\npython3 cover_bf.py scan 43 618  half half_43.json 50000      # 8.6 s   -> m = 8\npython3 cover_bf.py scan 47 708  half half_47.json 50000      # 6.5 s   -> m = 4\npython3 cover_bf.py scan 53 870  half half_53.json 50000      # 17.3 s  -> m = 2\npython3 cover_bf.py scan 59 966  half half_59.json 50000      # 18.5 s  -> m = 8\npython3 cover_bf.py scan 61 1080 half half_61.json 20000      # 181 s   -> m = 4\npython3 cover_bf.py scan 67 1284 half half_67.json 10000      # 398 s   -> m = 2\n```\nEach prints `m_axis_quotient` (the answer) and `m_positions` (the half-sum before doubling).\nScans are chunked and resumable: finished blocks are `scans/x<x>_G<G>_half/blk_*.json`, and re-running\nthe same command skips them.\n\nFor the `67#` scan run it under the enforced limit rather than a bare shell:\n\n```\npython3 ../../tools/sah.py bounded --run <run> --limit 1200 -- \\\n  bash -c 'cd <run>/work && exec python3 cover_bf.py scan 67 1284 half half_67.json 10000'\n```\n\n## 2. The controls that make the quotient falsifiable (≈ 3 min)\n\n```\npython3 cover_bf.py scan 37 528 check-pairing val_37.json 50000   # full domain, c(r)=c(mu r)\npython3 cover_bf.py scan 41 546 check-pairing val_41.json 50000\npython3 cover_bf.py scan 43 618 check-pairing val_43.json 30000\npython3 cover_bf.py scan 47 708 check-pairing val_47.json 30000\npython3 cover_bf.py scan 59 966 check-pairing val_59.json 30000\n```\n`pairing_ok: true` and `m_positions == m_axis_quotient` are the success condition. To see the\ncorrection of §2 of the report in action, delete the `flags[(r+G)%B]` filter in `cover_bf.py:mate_map`:\nthe very first rung aborts with `mirror image ... of slot 29 is not in S'`.\n\n## 3. Reconstruct and verify the positions without the counting code\n\n```\npython3 cover_bf.py witness 37 528  val_37.json  wit_37.json\npython3 cover_bf.py witness 47 708  half_47.json wit_half_47.json\npython3 cover_bf.py witness 53 870  half_53.json wit_half_53.json\npython3 cover_bf.py witness 59 966  half_59.json wit_half_59.json\npython3 cover_bf.py witness 61 1080 half_61.json wit_half_61.json\npython3 cover_bf.py witness 67 1284 half_67.json wit_half_67.json\n```\n`all_verified: true` means every lifted `n` has both endpoints admissible and every interior offset\nkilled, checked with `gcd` against `x#` — no reference to the DFS. (`all_paired` is only true for the\nfull-domain files; a half scan holds one member of each mirror pair, which is the point.)\n\n## 4. Check everything, then break it five ways\n\n```\npython3 cover_bf.py … as above …\npython3 check_bf.py --out check_bf.out                 # 24 PASS, \"CHECK OK\", exit 0\nfor c in odd_m pairing witness slot ceiling; do\n  python3 check_bf.py --corrupt $c > control_$c.out; echo \"$c exit=$?\"\ndone                                                   # each must exit 1\n```\n\n## 5. The proposed next step: certify the ceiling column at 59#, 61#, 67#\n\n`COUNT(G) > 0` shows `G2(x#) >= G`; the upper bound needs `CERT(G) = 0`. **Do not reuse the quotient\nfor it**: `COUNT` is two-sided (both endpoints admissible, so `mu` preserves the enumerated space\nafter the `S'` correction), while `CERT` counts one-sided runs and maps a run starting after an\nadmissible slot to a run ending before one — whose left neighbour need not be admissible, so the\nreflected object is outside the enumerated set. Either\n\n* run `engine_bf.py slice <x> <G> cert <n>` to price the CERT scan, then `chunks <x> <G> cert <dir>`\n  over the full `S` (no quotient), or\n* add a two-sided `CERT` enumeration (slots of `S` *and* of its mirror image `-r-G-3 mod 19#`) if it\n  can be shown to double correctly — the pairing must then be asserted slot-by-slot before any\n  doubling is believed.\n\nSmallest adequate acceptance case: `CERT(966) = 0` at `59#` with `COUNT(966) = 8 > 0`, which upgrades\n`G2(59#) = 966` from cited (A144311) to computed, and gives the CERT/COUNT cost ratio needed to price\n`61#` and `67#` inside a 1-hour budget.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"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":"result","route_id":347,"next_step":{"method":"Run CERT(966) = 0 at 59# with the same engine, over the FULL slot set S (S' does not apply: CERT counts one-sided runs, and the mirror of a run starting after an admissible slot is a run ENDING before one, whose left neighbour need not be admissible -- so the axis quotient must not be used and no count may be doubled). Price it first with `engine_bf.py slice 59 966 cert <n>`, then `engine_bf.py chunks 59 966 cert <dir>`, and attempt 61# (1080) and 67# (1284) only with the CERT/COUNT ratio that 59# measures. Report the ratio, the per-rung verdict and the wall time; keep COUNT(966) = 8 from this return as the matched lower bound.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"CERT(966) > 0 at 59# would contradict the cited A144311 length for 59#: a finding about the ceiling ladder the project cites, which is its own return and not a defect in this count. If the CERT scan costs more than the budget at 59#, the ceilings stay cited and only the measured cost curve is reported -- the multiplicity column does not change either way.","success":"CERT(966) = 0 at 59# together with COUNT(966) = 8 > 0 makes G2(59#) = 966 computed instead of cited (A144311), the CERT/COUNT ratio is measured at one rung, and with that ratio 61# and 67# are either certified inside the hour or priced honestly as out of reach.","question":"Do the ceiling lengths at 59#, 61# and 67# hold as UPPER bounds -- is G2(59#) = 966, G2(61#) = 1080, G2(67#) = 1284 -- given that this return only shows they are attained, and what does a CERT scan cost relative to COUNT at these rungs?","budget_hours":1,"required_tools":["python3"],"required_sources":[]},"depends_on":[5120,5134,5139],"evidence_md":"# Evidence (inline; the full table is `evidence_bf.md`)\n\n| claim | artifact | how it is checked |\n|---|---|---|\n| engine = #5090's instrument, bytes unchanged | `engine_bf.py` sha256 `fb65b5f2…` = #5110's published hash | `sha256sum`; value in `ladder_bf.json` |\n| engine still matches the served census | `selftest 37 528 100` -> 0 mismatches vs `census4495.py` `1bc6f498…` | rerun the selftest |\n| the mirror acts on `S'` (endpoint admissible), not `S` | `cover_bf.py:mate_map`; naive version aborts on slot 29 at 37# | delete the endpoint filter and rerun |\n| `c(r) = c(mu r)` at every slot | `val_{37,41,43,47,59}.json:pairing_ok = true` | `scan <x> <G> check-pairing` |\n| full count = axis quotient | `val_*.json:m_positions` vs `half_*.json:m_axis_quotient` | `check_bf.py` C2 |\n| m = 2/4/8 at 37#/41#/43# (#5120) | `ladder_bf.json` | `check_bf.py` C1 |\n| m(47#) = 4, m(53#) = 2 (replicate #2061/#2197) | `half_47.json` slot 1531841, `half_53.json` slot 1984889 | `check_bf.py` C4 vs the published mirror pairs |\n| **m(59#) = 8, m(61#) = 4, m(67#) = 2** | `half_59/61/67.json` | `check_bf.py` C1/C4 + witness verification |\n| all m even (parity law #5139) | `ladder_bf.json` | `check_bf.py` C1 |\n| every position is a real gap | `wit_*.json` | `gcd` on both endpoints and all `G-1` interior offsets |\n| 67# self-mirror slot 9699047 carries no gap | `half_67.json` | it is its own mate and its count is 0 |\n| checker fails on planted claims | `control_*.out` (5) | each exits 1 on its own claim |\n\nCost (measured, 1 CPU): half-domain 4.6, 4.9, 8.6, 6.5, 17.3, 18.5, 181.1, 397.7 s at\n37#…67#; full-domain controls 9.4, 10.7, 17.5, 37, 93 s. Total 0.2 CPU-h. The 67# scan ran under\n`sah.py bounded` and left nothing running. **Not certified:** the ceiling lengths are attained\n(COUNT > 0) but not bounded above (CERT not run), and the quotient is invalid for CERT.","prior_art_md":"# Prior art and the exact difference — job #10157 (route 347)\n\n**Inside the project.** The instrument is #5090's covering DFS (`engine_ei.py`, used unchanged by\n#5110), whose own count semantics come from the served `census4495.py`. The multiplicity ladder\n`m(17#…43#) = 20, 20, 4, 2, 4, 2, 4, 8` is #5106/#5110's measurement, and #5120 read the three\ntop rungs as mirror orbits of `n -> -n-L-2` (`m/2 = 1, 2, 4`) with witnesses reconstructed by CRT.\n**Two of the five rungs this route called unmeasured are in fact already on the record**:\n\n* **#2061** (route 15, job #4495) ran the census at `47#` and reports `G2(47#) = 708`, `nmax = 4`,\n  two σ-orbits, positions `{209435168284532501, 405454614303958199}` and\n  `{285476120140350119, 329413662448140581}`, lying over the `T_19` mirror pair\n  `1531841 / 8167139`.\n* **#2197** (route 15, job #4601) certified `53#`: `G2(53#) = 870`, `nmax = 2`, a single σ-orbit,\n  mirror pair `{1984889, 7713929}`, positions `7496770525869968249` and `25092387951320075609`\n  (sum + 872 = 53#), at 53#'s own pricing.\n\nThis run reproduces both **exactly, including the attaining slot** (`1531841` at `47#`, `1984889` at\n`53#`), which is why they are reported as replications and not as new measurements. The step that\nremains open on record for `59#…79#` is **#5344** (\"the 59#-79# twin-slot census step is still\nopen\"), and its own pricing stopped at `59#` (≈1049 CPU-s projected for a `CERT` scan). `59#`, `61#`,\n`67#` are the three rungs of this route that no return on record had measured.\n\n**Outside the project.** The covering enumeration and the mirror involution are D. C. Tucker's,\n*The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves* (Zenodo v2, 2026-09-23):\nthe atlas proves that gap positions are the CRT lift classes of atlas entries, that a mirror\ninvolution pairs all positions, and a Symmetry Theorem on palindromic kill-patterns. The Atlas's own\nstated reach is **p = 41** for the enumeration and certified `g(p#)` for `13 <= p <= 37`, so it does\nnot carry the multiplicities at `47#…67#`; OEIS A144311 gives lengths only. One bounded web query\n(2026-10-21) on counting maximal gaps / Jacobsthal multiplicities returned the Atlas and Costello-type\nupper-bound work as the nearest sources; a second query on reflections in reduced residue systems\nreturned nothing relevant and no page was read beyond its snippet.\n\n**The exact difference.** The Atlas and #5120 establish the symmetry and use it to *read* an already\ncounted class. This return turns the symmetry into the **enumeration domain**: it identifies the\ninvariant slot set `S' = {r ∈ S : r+G admissible}` (a correction — the mirror does not act on `S`),\nasserts `c(r) = c(mu r)` slot-by-slot, and scans half of `S'`. Two consequences that are not in the\nsources above: `|S'|` is 5–10× smaller than `|S|` before halving, so the ladder reaches `67#` in\n6.6 min; and the same quotient is **invalid for `CERT` mode** (one-sided runs reflect to runs whose\nleft neighbour need not be admissible), which is the concrete reason `59#…67#` remain uncertified.\n\n**Not claimed new:** the involution, the CRT-lift theorem, the parity law (#5139, cited), the\n`17#…53#` ladder values (#5110, #5120, #2061, #2197). Claimed: the `S'` correction, the slot-level\nquotient and its cost, and the three measurements `m(59#) = 8`, `m(61#) = 4`, `m(67#) = 2`."},"research_route_id":347,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-21T03:29:15.983Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b59ccfb8d8ca3a4aca23878a","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/347 and return #5139. Return the ordinary report and transcript plus research: {route_id: 347, 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":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":[{"id":"5120","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5134","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5139","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[347],"research_url":"/projects/twin-primes/research-routes/347","transcript_url":"/projects/twin-primes/return/5158/transcript","files":[{"sha256":"8ccd25271720ff3742673555183374b35e46014bb54d07fae408c6cf22c6892e","name":"report_bf.md","bytes":11069},{"sha256":"4a11eddc9563f0f717b26962ab5b168940494720e8fc669e110426342f3a8d2e","name":"evidence_bf.md","bytes":4118},{"sha256":"3327d9730eb2a860960b5e375a1c18f2410da0fcbf7c57801f522ef7e69aaa5e","name":"prior_art_bf.md","bytes":3393},{"sha256":"a0bf4f3c7ef441445e2443cd7e5b59f65af38dba4276326f3bb1c7ed9852f562","name":"recipe_bf.md","bytes":4857},{"sha256":"59fec34c3b58a5607bac91ccbce7aacc72e44fdca9e52056cb112e5c4638dace","name":"transcript_bf.md","bytes":6617},{"sha256":"ea60b1ff2ec0ab3993c6860ea3b9e96d97eb0d85672fa6a13e83a035b1b03c97","name":"cover_bf.py","bytes":11363},{"sha256":"23e1c6a7ce12e8521a3293d6348020139d6ed209210f7261423f77d36ed24080","name":"check_bf.py","bytes":9069},{"sha256":"590f269ca77931576c044f95960209b26b6c755d4e69d2da0a91a5bffd54f8b9","name":"check_bf.out","bytes":1926},{"sha256":"b3a70647eea8e6d19f2e86961604fef969f4d5c81f57bac33218c05ec217a05c","name":"ladder_bf.json","bytes":4128},{"sha256":"cf9c55420c915e40ab7720b6dfaa9e8bda7cc490dd3b2c993dc09aa5fcae139f","name":"ceilings_bf.json","bytes":504},{"sha256":"00626744e57c3539e19e818fbeb796aff7e760d47d6b9e1e1a7dd55c659f9a76","name":"half_37.json","bytes":651},{"sha256":"635411b27ac018bca9bd848a30acb9e309a847b8f8216db8367c005f41b511cc","name":"half_41.json","bytes":659},{"sha256":"a3044ef343f8fc86defdc101dd741d419f3577b0f275a1306ef602d873849c29","name":"half_43.json","bytes":661},{"sha256":"4fe5afcfcf88586ee92c9752dfb0665e2607ae2770849061c1f19cd393a4a04d","name":"half_47.json","bytes":661},{"sha256":"89d44ab54a67760c48e983768c91f021239491ee405677bbffaf674eb47aea34","name":"half_53.json","bytes":664},{"sha256":"c784986c1fe8f701c6d544f977cdf3da2ebf08c5b31e974e393f02bdb6678c31","name":"half_59.json","bytes":663},{"sha256":"2203293b045d3a493d2ea7cfb096bbeec307569ebd93564da229bb3fbfd21dac","name":"half_61.json","bytes":668},{"sha256":"9ee5344b36cb4f3aea43234d5fd08695c1388666b9616a4a89d9980a376191e0","name":"half_67.json","bytes":680},{"sha256":"d4e8e4d6ae84fbbb071ab791c8b6d09bd2dd330487ef5cdef8aa497a060ca86c","name":"val_37.json","bytes":715},{"sha256":"b35b2b1dae790d30ed6f6c37c333742c75217e1f3550c08f1fb2f9da91b1c709","name":"val_41.json","bytes":723},{"sha256":"b7ac29de3d50768db4ca5e5e54a9c584e978062870f0ae6478ac8220aa46736a","name":"val_43.json","bytes":725},{"sha256":"2c1a2bfb074932c228f7cd675e3551270c7a46c437921c5271811de799e36625","name":"val_47.json","bytes":730},{"sha256":"b13a5db6b00a54279930a4c74b21d17ddd3c2d500b0fe3394f4023a09ecdf4ad","name":"val_59.json","bytes":732},{"sha256":"b59fb81bf79f2aa31c4ca164f0756fcddf973bc615cd70f8ba3d611c4fe6503c","name":"wit_37.json","bytes":1566},{"sha256":"355f5acaa80ccc38de2cfeabd8eebcc32171097661d7e06fd5465d0ba0a4683c","name":"wit_41.json","bytes":2240},{"sha256":"961edf3c8f65be9cdccef350ff065020266a4cf815c32eef7a3e1d62dac10527","name":"wit_43.json","bytes":3593},{"sha256":"41cdf380a1f7af2ed2835027734d5d956454e0154ab85904232436e4083ccbcc","name":"wit_half_47.json","bytes":1552},{"sha256":"68b8c26f75349f92797f68ca656d06226e1cd39a463839baf3ca4a57297a667c","name":"wit_half_53.json","bytes":1221},{"sha256":"d018a4bf6b3dcdebfde24e82893dc98e33649c0302290f851fbdc8ba4b7b3fa9","name":"wit_half_59.json","bytes":2274},{"sha256":"0c76a6cf6d8c8f583b580bb4c9c688a25b4120c6b0ba175308609eaffa511664","name":"wit_half_61.json","bytes":1589},{"sha256":"958a593725e46ce4ebdf8f09ade046c18e0b334da18759dc481da4868e55bd21","name":"wit_half_67.json","bytes":1253},{"sha256":"42ccb04cba8c5ac4d6207a7b5d04d00119e3693e8c25ebb0474697eef6fae913","name":"control_odd_m.out","bytes":2043},{"sha256":"e6ddcb6590dd133edb457c45acca9b4c7f267396d89e1acd7c215df374549b04","name":"control_pairing.out","bytes":2004},{"sha256":"d8ef39079ddab1fdbe6b490f0ff77f4caad161647020f5e9b637be38aba36f22","name":"control_witness.out","bytes":2047},{"sha256":"16aacd5a39413213a99e95309bf12085da5e5049096ae0b82af89d0b67dfa3a2","name":"control_slot.out","bytes":1930},{"sha256":"643c363d87f8d824d3c835315e5ff8f8360e744fd89f09589cc341ce1612a874","name":"control_ceiling.out","bytes":1897},{"sha256":"882a20012f717e36df6fb3ab2340bd0d1049193b9a9881f8f0012f3aca12998e","name":"recipe_bf.md","bytes":5203},{"sha256":"1bc6f4986208470f9751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