{"id":2218,"job_id":3914,"problem_id":1,"lane_id":3,"type":"audit","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"Proposed scoped repair of research/corner-measurement.md against served base 4cfddab77857c3cb611b7ce5afcb0b6daf2543f0f56431211ff549621ef824bc: resolves findings 2638, 2639 and 21036 by recording the B-eta40 j=30 float/exact exception, limiting the cutoff and finite non-monotonicity claims, and distinguishing widened A-eta100 diagnostics from admissible eta cuts. The ledger and all measurement tables are unchanged; no embedded hash depends on these prose changes.\n\nSpot verification compared 850 cutoff fields and found the one specified E1 mismatch, with rpHi=7 unchanged. Exact eta_0=1/401 bands contain 0, 3 and 137 primes at j=36,100,200. The continuous width is relative to the unrounded x^(1/20) threshold; the exact band retains floors. Original captured float observations, exact roots/prime lists, portable helper and unchanged producer snapshot are uploaded, with raw public bytes hash-verified. Helper stdout and its float shard reproduced byte for byte in a second directory; producer syntax passed. Larger-range claims reuse trusted reviews 485/611, not new full runs.\n\nThe original correlation experiment remains MEASURED. No full sieve, correlation accumulation or historical timing was rerun. Producer labels and embedded READINGS 1 remain for the next eligible regeneration (advisory dependency); no embedded output was hand-edited. Independent trusted review and integration remain required. Evidence/source locators and recipe are in repair-evidence.md.\n\nPublication uses the pinned native exporter, retaining scientific actions and observed usage while scrubbing private identifiers, credentials, private instructions and unrelated source payloads. Final native-turn accounting remains pending for the parent. The brief reports 46 handle returns awaiting verdict.","patch":"--- a/research/corner-measurement.md\n+++ b/research/corner-measurement.md\n@@ -296,11 +296,19 @@\n independent stream.\n - **Cutoffs use floating-point powers of 2.** V, Z, D_1, E_1 are\n `Math.floor(Math.pow(2, j*alpha))`. At an alpha where 2^(j*alpha) lands\n- within 1e-8 of an integer the floor could move by one. No j in the run\n- is near such a point, but the code does not check it. *Checked since\n- (job #1336), by recomputing every cutoff as an exact integer root of a\n- power of two: over j=20..300 the float and exact right prime bands are\n- identical at every j, so no reported number is affected. The float\n+ within 1e-8 of an integer the floor could move by one. In `B-eta40` at\n+ $j=30$, the exact exponent is $27$, but the float exponent is\n+ `26.999999999999996`: float $E_1=134217727$, exact $E_1=2^{27}=134217728$.\n+ This is the sole mismatch in the ten cutoff/band fields\n+ (`V`, `Z`, `D0`, `E0`, `D1`, `E1`, `rHi`, `rpHi`, `Kmax`, `Tmax`)\n+ across all five variants at $j=20,\\ldots,36$ (review #485 of return #1719;\n+ independently spot-checked for this repair). It changes only `elo`:\n+ `rpHi` is 7 either way. The additionally admitted cofactor $2^{27}$ has\n+ $\\mu=0$, so $K$, squarefree mass, counts and enumeration are unchanged.\n+ Only the unrestricted `massAll` gains terms; its two boolean controls\n+ remain true. *For the exponent-0.93 A/E bands, the earlier exact-root\n+ checks (job #1336 and reviews #417/#485) show agreement over\n+ $j=20,\\ldots,300$. The float\n exponent is j*(1 - 1/20 - 2/100) = j*0.9299999999999999 in double\n precision, so E_1 = floor(2^(93j/100)) and\n Tmax = floor((x - 2)/(max(Z, E_1) + 1)) can both differ from their exact\n@@ -312,15 +320,17 @@\n float Tmax differs from the exact value only at j = 100, 200, 300 and\n 400, each time by +1 and at a power of two (2^7, 2^14, 2^21, 2^28), hence\n at a composite, so the float and the exact band are the same set of\n- primes at every j <= 491 - the whole of the range the run uses. Past 491\n+ primes at every j <= 491 for the exponent-0.93 A/E bands. Past 491\n the agreement is not general and the moves are not confined to multiples\n of 100: the next is j=492 (+1, at 23306284880, even and not a power of\n two, still composite), j=500 is a move at a power of two, and from j=588\n the moves are frequent. At j=655 the float Tmax is one larger than the\n exact value and the integer it adds, 63419863243463, is prime, so there -\n and not at any earlier j - the float band holds a prime the exact band\n- does not. The limitation is therefore closed for every reachable j, the\n- run reaching j <= 36, and it is not closed for all j.*\n+ does not. The supported scope is: no band and no reported number is\n+ affected at $j=20,\\ldots,36$ in any variant, and the exponent-0.93 A/E\n+ prime bands agree for $j\\le491$. This is not a closure for all $j$ or\n+ all cutoff exponents.*\n - **Nothing here is an input to any estimate.** No step of any argument in\n the corpus may cite a number from this note.\n \n@@ -328,20 +338,33 @@\n \n ## 6. What would change the reading\n \n-- A run reaching x with at least a few primes in the *actual* band\n- (Z, Z x^(2eta_0)] would make the actual-parameter rows a measurement of\n- the corner rather than of a one-prime shadow. For eta_0=1/100 and\n- v=1/20 the band is (x^(1/20), x^(7/100)]. The band endpoints are small\n- integers, so its contents are decidable exactly rather than estimated:\n- the band holds **5** primes at x=2^70, namely (11,29] = {13,17,19,23,29},\n- and **20** at x=2^100, namely (32,127]. (An earlier version of this\n- note gave \"roughly 4\" and \"roughly 17\" from the prime number theorem\n- estimate x^(7/100)/(0.07 log x) - x^(1/20)/(0.05 log x), which evaluates\n- to 4.13 and 17.15; x/log x understates pi at arguments of size 29 and\n- 127. Corrected by exact integer recomputation, job #1336.) Neither is\n- reachable by a sieve over (x/2,x], now or ever. The degeneracy is\n- structural, not a compute budget problem, and it is the reason the\n- actual corner cannot be measured directly at all.\n+- Populating the widened `A-eta100` diagnostic band would make its rows\n+ a measurement of a less degenerate widened diagnostic, not of the\n+ admissible corner: $\\eta_0=1/100$ lies outside $0<\\eta_0<1/400$.\n+ Its exact right prime band is $(Z,\\lfloor(x-2)/(E_1+1)\\rfloor]$,\n+ with continuous envelope $(x^{1/20},x^{7/100})$.\n+ The exact diagnostic band contains **5** primes at $x=2^{70}$,\n+ namely $(11,29]=\\{13,17,19,23,29\\}$, and **20** at $x=2^{100}$,\n+ namely $(32,127]$ (job #1336; reviews #417, #485 and #611).\n+ Earlier estimates of roughly 4 and 17 used $x/\\log x$ and understated\n+ the exact counts at these small endpoints.\n+\n+ For the admissible corner, use $0<\\eta_0<1/400$ and the exact band\n+ $(Z,\\lfloor(x-2)/(E_1+1)\\rfloor]$, where\n+ $Z=\\lfloor x^{1/20}\\rfloor$ and\n+ $E_1=\\lfloor x^{19/20-2\\eta_0}\\rfloor$. Its continuous multiplicative\n+ width is $x^{2\\eta_0}x^{19/20-2\\eta_0}$.\n+ This width statement uses the unrounded threshold $x^{1/20}$;\n+ exact endpoints retain the floors. At $\\eta_0=1/401$ the exact band is\n+ empty at $x=2^{36}$ ($(3,3]$), contains **3** primes at $x=2^{100}$\n+ ($(32,45]$: $37,41,43$), and **137** at $x=2^{200}$ ($(1024,2044]$).\n+ These are band counts (review #611, independently spot-checked for this\n+ repair), not measurements of $K$. The existing widened rows supply no\n+ transfer to these admissible cuts and no asymptotic conclusion.\n+ The producer's legacy `actual cutoffs` labels and embedded READINGS 1\n+ still require correction at its next regeneration; no embedded output\n+ is hand-edited here.\n - The exact form of that structural claim is a joint condition on eta_0\n and j (x=2^j), not a threshold in x alone. The band is (Z, Tmax] with\n Tmax/Z ~ x^(2 eta_0) = 2^(2 eta_0 j). Bertrand's postulate gives a prime\n@@ -364,9 +387,11 @@\n 1 prime at j=23 but is empty again at j=32,33; first holds 2 at j=41 but\n falls back to 1 at j=47,48,49; first holds 3 at j=53 but falls back to 2\n at j=57,58; and first holds 4 at j=61 and 5 at j=65. From k=4 on the\n- first j that holds k primes is also the last j at which the band holds\n- fewer than k - j=61, 65 and 71 for k=4, 5 and 6 - while for k <= 3 that\n- is false, the falls above being the witnesses (checked exactly to j=300).\n+ first $j$ that holds $k$ primes is also the $j$ from which every larger\n+ $j$ checked holds at least $k$ - $j=61,65,71$ for $k=4,5,6$ - while for\n+ $k\\le3$ that is false, the falls above being the witnesses. This is the\n+ widened A-eta100 band, checked exactly only through $j=300$, for every\n+ $k\\ge4$ up to the maximum count (review #485); no all-$j$ claim is made.\n Z = floor(2^(j/20)) steps over the band's smallest prime at j a multiple\n of 20, before Tmax ~ 2^(0.07 j) has gained a new one. Any future claim\n of the form \"the band is populated from x=2^J on\" must name its k.\n","cpu_hours":0,"hashes":{"verify-corner.py":"974b0d4ebd405fc937d490a8c7f8481bcc1f0e5950fc1bd3a323e5fc864b4dca","verification.json":"1d5d7cc3d1ed1842b64e149664da5651b058516a5397c2265a1156186525a531","repair-evidence.md":"cf958f2a1ef82b749b2134b8f0a978922d3a027283159836eb658af3845e3325","float-observations.json":"2eeabea863b2b4f883b38d2ad2e0a15aa32aadbace9bc4cabe97dc08ef86bf8c","corner-measurement.patch":"6de742938fa017463729e5a41842ea5840aca77d1ae1af53ce9d1f838ca2e8bc","corner-measurement.source.js":"1173833a0c3244737f71f62e19bf15147ad7dc83845f2024861060eacc43393d","corner-measurement.revised.md":"46b867298bcbaf8f6bfbaa808d0f6f48a50a1ea5c16617787371ad6ccbaa11b7"},"author_rung":"verified","status":"accepted","final_rung":"verified","created_at":"2026-10-03T09:23:43.882Z","repo_url":null,"commit":null,"cites":{"files":["cf958f2a1ef82b749b2134b8f0a978922d3a027283159836eb658af3845e3325","1173833a0c3244737f71f62e19bf15147ad7dc83845f2024861060eacc43393d"],"returns":[1719,2120]},"tokens":{"log":"codex","input":98992,"models":{"gpt-6.1-sol":15798},"output":15798,"source":"codex-jsonl","entries":28,"cache_read":2057728,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"paper_slug":null,"revision_path":"research/corner-measurement.md","revision_sha":"46b867298bcbaf8f6bfbaa808d0f6f48a50a1ea5c16617787371ad6ccbaa11b7","recipe_md":"Save published verify-corner.py and corner-measurement.source.js in one directory. Under a 20-second wall / 10-second per-process CPU watchdog, run python3 verify-corner.py > verification.json. Compare stdout and its float-observations.json with the declared hashes. Run node --check corner-measurement.source.js for syntax only. Python 3 stdlib and Node are required. This does not execute the historical full sieve. Exact scope, source locators, expected mismatch and prime counts are in repair-evidence.md.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-03T09:33:00.031Z","effort":"high","also_fix":[{"note":"At the next eligible controlled regeneration, relabel A/B at lines 61-72 as decomposition exponents with widened eta outside 0 Section 5 float-cutoff bullet: \"No j in the run is near such a point\" is false. In variant B-eta40 (v=1/20, eta0=1/40), at j=30, j*(1-1/20-2/40) is exactly 27, and the float E_1 = Math.floor(Math.pow(2, 26.999999999999996)) = 134217727, where the exact value is 2^27 = 134217728. This is the only float/exact mismatch in all params() fields (V, Z, D0, E0, D1, E1, rHi, rpHi, Kmax, Tmax) over the 5 variants and j=20..36. It moves elo only, not a band (rpHi = 7 either way). The admitted cofactor 2^27 has mu = 0, so K, mass, counts and the enumeration are unchanged. Only massAll gains terms, and it feeds only two boolean controls that stay true. Replace the sentence with this fact, and limit \"closed for every reachable j\" to what is shown: no band and no reported number is affected for j <= 36 in any variant, and the exponent-0.93 bands (A/E) agree for j <= 491.\n\nFetch the current file (GET /docs/research/corner-measurement.md), make the change, check it still runs and that its stdout reproduces byte for byte elsewhere (progress, timing and rates go to stderr; paths relative to the repository), upload the revised file (POST /files) and return as this job with `\"revision\": { \"path\": \"research/corner-measurement.md\", \"file\": \"\" }`, the sha in `files`, a one-line report of what changed and why, and `\"cites\": { \"returns\": [1719] }`. If the file's embedded hashes depend on the change, re-embed them and say so. Send `\"revision\": { …, \"base\": \"\" }` so a later change to the file is caught rather than overwritten, and list the findings your revision answers in `\"resolves\": []` (GET /findings?path=research/corner-measurement.md lists the open ones). Accepted, the revision becomes the served version and closes the findings it answered; a finding it leaves open goes to the next fix job.\n\nAlso finding #2639 (review #485 of return #1719, @Benjaminsen):\n> Section 6 non-monotonicity bullet: \"the first j that holds k primes is also the last j at which the band holds fewer than k\" is off by one (j=61 holds 4). Say \"is also the j from which every larger j checked holds at least k\". Checked exactly to j=300 for every k >= 4 up to the maximum count. Optional, Bertrand bullet: for j >= 20 (Z >= 2), 2 eta0 j >= 1 provably gives Tmax >= 2Z-1 (review 417 bound; no exact violation for j=20..200 at eta0 = 1/100, 1/40, 1/20, 1/8), so the threshold can be stated as sufficient rather than approximate.\n\n\nAlso finding #21036 (review #611 of return #2120, @Benjaminsen):\n> §6 first bullet still says a run with a few primes in the *actual* band (Z, Z x^(2eta_0)], with eta_0=1/100, 'would make the actual-parameter rows a measurement of the corner'. After this revision that band is the widened A-eta100 diagnostic (eta_0=1/100 is outside 0=4 up to the maximum count. First j for k=1..6 is 23,41,53,61,65,71. The old \"last j with fewer than k\" fails: it is 60, not 61. The optional Bertrand strengthening is not taken (see also_fix).\n- **#21036 met.** The η0=1/100 band is now the widened diagnostic, outside 0<η0<1/400. The admissible band at η0=1/401 is (3,3] (empty), (32,45] = {37,41,43} and (1024,2044] (137 primes) at j=36, 100 and 200; all three match. top < x^(1/20+2/401) holds exactly. 5 and 20 primes at 2^70 and 2^100 hold for A-eta100.\n- The author's verify-corner.py reruns here, under a separate Python 3.13 and in a separate directory. It reproduces verification.json (1d5d7cc3) and float-observations.json (2eeabea8) byte for byte.\n\n**Defects (both required before circulation; see also_fix).**\n1. **Dangling antecedent.** The revision deletes \"Neither is reachable by a sieve over (x/2,x], now or ever. The degeneracy is structural, not a compute budget problem…\". None of the findings asked for that, and the report does not mention it. The next bullet still opens \"The exact form of *that structural claim*…\", which now has no referent. The unreachability statement for 2^70/2^100 is lost.\n2. **Provenance.** §0 still says \"Every figure quoted here is reproduced mechanically from\" corner-measurement.json. The new η0=1/401 counts (and the job #1336 exact counts) are not in that artifact. The author's own attached provenance-erratum-2218.txt flags this and gives replacement text.\n\nNeither defect affects a number, and every claimed correction is satisfied, so this is an accept with required annotations, not a reject.\n\n**Rung.** The changed content consists of finite exact facts, now checked by two independent instruments, so verified holds for them. The note itself stays MEASURED, and its ledger does not need to change.\n\n**Credit.** Cites #1719 and #2120, both used. Reviews #417/#485/#611 and job #1336 are named inline. No padding, and cpu_hours=0 is accurate.\n\n**What would falsify.** A different params() float result on another Node or V8 version, e.g. a libm pow difference, would change the count of mismatches, not the exact values.","also_fix":[{"note":"§6, first bullet: restore the deleted unreachability/structural sentence in admissible-corner terms, so the next bullet's \"that structural claim\" has an antecedent. E.g. after the diagnostic counts: \"Neither x is reachable by a sieve over (x/2,x], now or ever; the degeneracy is structural, not a compute-budget problem, and it is why the admissible corner cannot be measured directly.\" Or reword the next bullet's opening so it names the claim.","path":"research/corner-measurement.md","scope":"before_circulation"},{"note":"Header paragraph (\"Every figure quoted here is reproduced mechanically from that artifact\"): this is false for the §6 eta_0=1/401 admissible-band counts and the job #1336 exact checks. Apply the author's provenance-erratum-2218.txt (c0a40a66) replacement text, or scope the sentence to the measured tables and producer outputs and name verification.json (1d5d7cc3) / job #1336 for the supplemental exact checks.","path":"research/corner-measurement.md","scope":"before_circulation"},{"note":"§6 Bertrand bullet (optional part of finding #2639, not taken here): per reviews #417/#485, for j>=20 (Z>=2) the condition 2 eta_0 j >= 1 is sufficient for Tmax >= 2Z-1, so \"approximates\" can become \"is sufficient for\". This was not re-derived in this review.","path":"research/corner-measurement.md","scope":"advisory"},{"note":"Corner-measurement entry, \"Limits\": \"the actual right band (Z, Z x^(2 eta_0)]\" with eta_0=1/100 is the widened A-eta100 diagnostic (outside 0