{"id":5451,"job_id":10973,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-fable-5-1","provider":"anthropic","report_md":"# #4816's collision-dropping control recomputed from the definition by independent exact code: all six values at x = 19, 23, 29 agree to 10 places for h = 7 and h = 12, as do the x = 13, 17, 31 anchors and the exact kappa4 at x = 31; extension to x = 37, 41, 43, 47: h = 7 stays in [0.02, 0.06] through x = 41 and leaves it from below at 43, h = 12 enters the band at 43; every consecutive difference negative; 0 overlap-class sign changes; the control numerator is tuple-independent, -6 c_x^4, because p = 2 zeroes every block of size two or more\n\n## Result\n\nProven by exact finite computation (rung: finite exact table, independent code). All six values\n#4816 section 5 prints at x = 19, 23, 29 agree to all ten printed decimals: h = 7 gives 0.0477490640,\n0.0385992282, 0.0325913451 and h = 12 gives 0.1261865693, 0.1053112950, 0.0911025968, each equal to the\nexact ratio rounded to 10 places. The x = 13, 17 anchors (#4796, via #4816 gate G3), both x = 31 values,\n#4816's exact control `kappa4` at x = 31 for both h, and its exact main-path `kappa4` at x = 31 also agree\nexactly. Extension: h = 7 gives 0.0243017459, 0.0215210876, 0.0191546784, 0.0172109410 at x = 37, 41,\n43, 47, so it stays in the band [0.02, 0.06] through x = 41 and leaves it from below at x = 43; h = 12\ngives 0.0707099117, 0.0636213264, 0.0574736948, 0.0523351763, so it enters the band at x = 43 and is in\nit at x = 47 (0.0076648 below the top edge). Every consecutive difference from x = 13 to x = 47 is\nnegative for both h. Route 333's sign-change count is 0: no overlap-class tuple changes the sign of its\nmain-path `K_x(d)` between any two consecutive rungs from x = 13 to x = 47 (none is ever 0), for either\nh. One new fact about the definition: the control numerator is tuple-independent,\n`K^cd_x(d) = -6 c_x^4` for every `d` (lemma and proof in `DEFINITION.md`), because p = 2 zeroes every\nblock of two or more elements. The control ratio is therefore `6 h^4 c_x^4 / B_abs(x)` and measures\nno collision at all; its decay and band crossings are facts about the main-path `B_abs`.\nNeither #4816 nor the pages of routes 209 and 333 state this.\n\n## How it was computed\n\n`python3 -I research/control-4816/control.py` (Python 3.9.6, stdlib only) enumerates all `h^4` tuples\nand, per tuple, the integer block numerators `prod_{p<=x} (p - w_B(p))` and `prod_{p<=x} (p - n_B(p))`\nfor the 15 blocks at the ten rungs x = 13 to 47. It forms `q^4 K_x(d)` as an exact integer from the 15\nset partitions, so every sum is exact; ratios are `Fraction`s. Decimals are printed two ways:\nround-half-even to the printed precision of #4816 (the agreement test), and a directed interval\n[floor, ceiling] at 12 places. The script was written from `DEFINITION.md`; #4816's attachments and\n#2475's source were not opened (they are not in this repo's state snapshot).\n\nChecks inside the run, all passing (`control.out`, last line `ALL CHECKS PASS`):\n\n- the lemma `K^cd_x(d) = -6 c_x^4` holds tuple by tuple at every rung (0 mismatches of 2401 x 10 and\n  20736 x 10);\n- class counts 72 / 2329 and 3552 / 17184, and the 2-separated class has `|kappa4| = B_abs` at every\n  rung (#4816 section 3);\n- main-path ratios `|kappa4| / B_abs` agree with every value #4816 prints (h = 7 at x = 19, 23, 29 to\n  7 places; x = 31 to 47 for both h to 10 places), and the main-path `kappa4` sign turns positive at\n  x = 29 for h = 7 and stays negative for h = 12, as #4816 section 3 says;\n- the main-path block numerators were recounted by direct enumeration of `n mod 13#` (no CRT product)\n  for 9 fixed h = 12 tuples and all 15 blocks: 0 mismatches.\n\nA separate exploratory script (not archived) recomputed the h = 7 overlap-class signs at x = 13 and 47\nin plain `Fraction`s with set unions instead of bitmasks: 1101 positive of 2329, 0 sign changes, which\nagrees.\n\nRuntime: 6.9 CPU seconds on the Studio under `nice -n 10`, one thread, maximum resident set 17 MB\n(`/usr/bin/time -l`). The brief's bound was 2 CPU hours and 2 GB.\n\n## Extended table\n\n`r_h(x) = |kappa4_ctl(x)| / B_abs(x)`, band [0.02, 0.06] closed. The rows x = 13, 17 are #4796's\nanchors; x = 19 to 31 are #4816's rows; x = 37 to 47 are new.\n\n### h = 7\n\n| x | r (10 dp) | directed 12 dp | in band | r(x) - r(previous rung), directed 12 dp | #4816 printed | agree |\n|---|---|---|---|---|---|---|\n| 13 | 0.0818523903 | [0.081852390309, 0.081852390310] | no |  | 0.0818523903 | yes |\n| 17 | 0.0616824780 | [0.061682478020, 0.061682478021] | no | [-0.020169912289, -0.020169912288] | 0.0616824780 | yes |\n| 19 | 0.0477490640 | [0.047749063972, 0.047749063973] | yes | [-0.013933414049, -0.013933414048] | 0.0477490640 | yes |\n| 23 | 0.0385992282 | [0.038599228239, 0.038599228240] | yes | [-0.009149835733, -0.009149835732] | 0.0385992282 | yes |\n| 29 | 0.0325913451 | [0.032591345134, 0.032591345135] | yes | [-0.006007883106, -0.006007883105] | 0.0325913451 | yes |\n| 31 | 0.0277875728 | [0.027787572795, 0.027787572796] | yes | [-0.004803772339, -0.004803772338] | 0.0277875728 | yes |\n| 37 | 0.0243017459 | [0.024301745900, 0.024301745901] | yes | [-0.003485826896, -0.003485826895] |  |  |\n| 41 | 0.0215210876 | [0.021521087576, 0.021521087577] | yes | [-0.002780658325, -0.002780658324] |  |  |\n| 43 | 0.0191546784 | [0.019154678380, 0.019154678381] | no | [-0.002366409196, -0.002366409195] |  |  |\n| 47 | 0.0172109410 | [0.017210940980, 0.017210940981] | no | [-0.001943737401, -0.001943737400] |  |  |\n\nExact ratios:\n\n- x = 13: `5294745225/64686507077`\n- x = 17: `804139431046875/13036756253111551`\n- x = 19: `619723454860125/12978756090824441`\n- x = 23: `250987999218350625/6502409780273911117`\n- x = 29: `26677062658519894900125/818532114849134106256873`\n- x = 31: `830941099845008578185375/29903335061332524616301251`\n- x = 37: `45800954562164040317334375/1884677535106971523799488411`\n- x = 41: `55648159793029308985561265625/2585750352799706906785274295989`\n- x = 43: `31181518870694089468242829171875/1627880053680048881011508507994103`\n- x = 47: `127863715819139950625713251372919921875/7429211219001636169291243428868509504199`\n\nExact consecutive differences r(x') - r(x):\n\n- 13 to 17: `-17009331923342842843359600/843302225628024347031946427`\n- 17 to 19: `-2357545913224100893165739868000/169200879624665160352667130217991`\n- 19 to 23: `-772183830368606264736703459884000/84393190540766438999541256085210597`\n- 23 to 29: `-31976544662209646117727049875815787906000/5322431229063297882934948796945556772357141`\n- 29 to 31: `-117581167358682251538308051392584809431595724000/24476840088794772722542509687870239390329757248123`\n- 31 to 37: `-196454733473708889129533528160498909030807424386000/56358143814870061627374870951522348094380344859302161`\n- 37 to 41: `-13550997785713656141651928050580876609212118408554150000/4873305601316533617206594937038375808701180186346389283479`\n- 41 to 43: `-9960905930344529007790231795202133563935859980255850212500000/4209291423118792211798632408267358179186315950197271011888552867`\n- 43 to 47: `-23507297248279699934306814642071437565449418667381002571390506284500000/12093864757988804870632754076221767661936029807856164961396112624545738497`\n\n### h = 12\n\n| x | r (10 dp) | directed 12 dp | in band | r(x) - r(previous rung), directed 12 dp | #4816 printed | agree |\n|---|---|---|---|---|---|---|\n| 13 | 0.1981384593 | [0.198138459279, 0.198138459280] | no |  | 0.1981384593 | yes |\n| 17 | 0.1565435089 | [0.156543508854, 0.156543508855] | no | [-0.041594950426, -0.041594950425] | 0.1565435089 | yes |\n| 19 | 0.1261865693 | [0.126186569333, 0.126186569334] | no | [-0.030356939521, -0.030356939520] | 0.1261865693 | yes |\n| 23 | 0.1053112950 | [0.105311295037, 0.105311295038] | no | [-0.020875274296, -0.020875274295] | 0.1053112950 | yes |\n| 29 | 0.0911025968 | [0.091102596777, 0.091102596778] | no | [-0.014208698261, -0.014208698260] | 0.0911025968 | yes |\n| 31 | 0.0794111246 | [0.079411124613, 0.079411124614] | no | [-0.011691472164, -0.011691472163] | 0.0794111246 | yes |\n| 37 | 0.0707099117 | [0.070709911686, 0.070709911687] | no | [-0.008701212928, -0.008701212927] |  |  |\n| 41 | 0.0636213264 | [0.063621326441, 0.063621326442] | no | [-0.007088585245, -0.007088585244] |  |  |\n| 43 | 0.0574736948 | [0.057473694766, 0.057473694767] | yes | [-0.006147631675, -0.006147631674] |  |  |\n| 47 | 0.0523351763 | [0.052335176292, 0.052335176293] | yes | [-0.005138518475, -0.005138518474] |  |  |\n\nExact ratios:\n\n- x = 13: `714492900/3606028343`\n- x = 17: `72342406125000/462123320567281`\n- x = 19: `4645990082250/36818419795307`\n- x = 23: `22579511799735000/214407313020689047`\n- x = 29: `1199967833036296813500/13171609542227486531879`\n- x = 31: `74753551672853749641000/941348608732745306752171`\n- x = 37: `1545138408802701901584375/21851793786056003539225154`\n- x = 41: `5006248444520754161133375000/78688212341976488461753823309`\n- x = 43: `467527979735521541381400187500/8134642841962927464647671655713`\n- x = 47: `11502941531417838123762624863203125000/219793690329485774576533341781834582939`\n\nExact consecutive differences r(x') - r(x):\n\n- 13 to 17: `-69315064582179445929900/1666429791926890124445383`\n- 17 to 19: `-516512713580813393206512750/17014650413847381259141550267`\n- 19 to 23: `-164792305640261491546360470750/7894138457979521897701836902429`\n- 23 to 29: `-40126634287656431753039449760800330500/2824089410106663470928658184676911629313`\n- 29 to 31: `-144963455623209279416410037939014810201669500/12399076317346796740562346535831347382143959309`\n- 31 to 37: `-178985304504530635727232170817903856435298785875/20570155678818668082541462546823780708729647309334`\n- 37 to 41: `-12188670558179531476982774112020601147579851990446875/1719478589490257155913104022215644412409202410884314586`\n- 41 to 43: `-3935102129064792888505240581031969811061793420979250937500/640100503274517926463841886639661684593181197594348082414317`\n- 43 to 47: `-9187379008293130396064650763546228027479748459316214928533197937500/1787943169747367768344863959991432246788247936823538662654784951680507`\n\n## Band membership and crossings\n\n- h = 7: outside (above) at x = 13, 17; inside from x = 19 through x = 41; outside (below) at x = 43\n  and 47. It leaves from below between x = 41 (0.02152) and x = 43 (0.01915). At x = 47 it is 0.0027891\n  below the bottom edge.\n- h = 12: above the band from x = 13 through x = 41 (0.06362 at x = 41); inside at x = 43 (0.05747) and\n  x = 47 (0.05234). It crosses 0.06 between x = 41 and x = 43.\n- Every one of the 18 consecutive differences (9 per h) is negative, so on the finite table x = 13 to 47\n  both series are strictly decreasing. Route 333's failure clause (a non-monotone step between two new\n  rungs) does not occur. This is a statement about these ten rungs only.\n\nRoute 333's success clause asked for the h = 12 series to name a crossing rung of 0.06 (x = 43) and\nfor the h = 7 series to stay in the band on all four new rungs or name the rung where it leaves from\nbelow (x = 43).\n\n## Sign-change counts (route 333)\n\nOverlap class, main-path `K_x(d)`, consecutive rungs. \"Strict\" counts tuples with\n`K_x(d) K_x'(d) < 0`; \"through zero\" counts tuples where exactly one value is 0.\n\n| rungs | h = 7 strict | h = 7 through zero | h = 12 strict | h = 12 through zero |\n|---|---|---|---|---|\n| 13 to 17 | 0 | 0 | 0 | 0 |\n| 17 to 19 | 0 | 0 | 0 | 0 |\n| 19 to 23 | 0 | 0 | 0 | 0 |\n| 23 to 29 | 0 | 0 | 0 | 0 |\n| 29 to 31 | 0 | 0 | 0 | 0 |\n| 31 to 37 | 0 | 0 | 0 | 0 |\n| 37 to 41 | 0 | 0 | 0 | 0 |\n| 41 to 43 | 0 | 0 | 0 | 0 |\n| 43 to 47 | 0 | 0 | 0 | 0 |\n\nThe overlap class has 1101 positive and 1228 negative tuples at every rung for h = 7, and 5856 positive\nand 11328 negative for h = 12; no `K_x(d)` is 0. The control `K^cd_x(d)` is the same negative number for\nevery tuple (the lemma), so it has no sign changes either. The control's decay is therefore not a\nrepartitioning artefact in either the numerator or the sign pattern of `B_abs`. The h = 7 main-path\n`kappa4` sign change at x = 29 comes from magnitudes moving within a fixed sign pattern.\n\n## Reading, with its limits\n\n- #4816 section 5 says h = 7 \"leaves the band at x=19\"; its section 0 and the numbers say it enters the\n  band at x = 19. The numbers are right; the section 5 word is a slip.\n- By the lemma, `r_7(x) / r_12(x) = (7/12)^4 B_abs^(12)(x) / B_abs^(7)(x)`, and each `r_h` is the ratio of\n  one fixed product-of-marginals term to the main-path absolute mass. Any argument that used the\n  control as a collision-sensitive discriminator should be re-read with that in mind. This lane does not\n  judge what route 209 should conclude from it.\n- Nothing here is a rate, an asymptotic statement, or a bound on `G2`, `kappa4` or a twin-prime count.\n  The band crossings are facts about the ten rungs computed; no rung beyond 47 was computed.\n\n## What this rests on\n\n- The definitions as restated in `DEFINITION.md` from #4816 section 1 and the route 209 and 333 pages.\n  The p = 2 convention and the main-path denominator were pinned by #4796's x = 13 anchor and #4816's\n  exact x = 31 fractions; #2475's `R_nodedup` source itself was not read by this lane.\n- The six verified values and the x = 31 fractions as printed in the repo copy\n  `research/sah-state/2026-10-22/returns/04816.md`.\n- Exact integer and `Fraction` arithmetic in CPython 3.9.6; no floating point enters any stated value\n  or comparison. The decimals are derived from the exact fractions.\n- Not checked: #4816's gate G1 against #2370's served C/A sequences (those artifacts are not in this\n  repo). `B_abs` here is computed directly from the definition, and it reproduces #4816's main-path\n  ratios and its exact x = 31 `kappa4`, which #4816 gated against those sequences.\n- Same-definition caveat: this is a separate implementation of the same written definition, so it\n  checks the arithmetic, not whether the definition is the right control for route 209.\n\n## Filing block\n\n```text\nreplicates #4816 (sec 5); addresses routes 209 and 333; bears_on #4796, #4805. Scope key: control-4816-x13-47.\nIndependent exact recomputation of the #2475 collision-dropping control |kappa4_ctl|/B_abs (p=2 included, main-path B_abs), h=7 and 12, all h^4 tuples. All six #4816 values at x=19,23,29 agree to 10 dp; x=13,17,31 and the exact x=31 kappa4 (control and main) agree too.\nExtension x=37,41,43,47: h=7 0.0243017459, 0.0215210876, 0.0191546784, 0.0172109410 (in band through 41, leaves from below at 43); h=12 0.0707099117, 0.0636213264, 0.0574736948, 0.0523351763 (enters [0.02,0.06] at 43). All differences x=13..47 negative. Overlap-class sign changes of K_x(d) between consecutive rungs: 0 for both h.\nNew: p=2 zeroes every block of size >=2, so K^cd_x(d) = -6 c_x^4 for every d; the control is 6 h^4 c_x^4 / B_abs and carries no collision data.\nCheck: python3 -I research/control-4816/control.py (7 CPU s), last line ALL CHECKS PASS.\n```\n\n946 characters.\n\n## Files\n\n`RESULT.md` (this report as kept in the department repo), `DEFINITION.md` (the control quantity restated from #4816 with every convention pinned, and the lemma that p = 2 zeroes every block of two or more elements, so the control numerator is K^cd_x(d) = -6 c_x^4 for every tuple d), `control.py` (stdlib Python written from DEFINITION.md; #4816's attachments and #2475's source were not opened), `control.out` (the archived run, last line ALL CHECKS PASS). Cheapest decisive check: `python3 -I control.py` (7 CPU s, 17 MB), compare the x = 19, 23, 29 rows with #4816 section 5.\n\nOne wording the archived output needs: in the zero-count lines the format prints the count followed by a literal 0, so `00` means zero zeros; no K_x(d) is ever 0 at any rung. The numerator of the control carries no collision data (the lemma); its ratio still divides by the main-path B_abs, which is collision-preserving, so the decay and band crossings are facts about B_abs.\n\nBuilt and tested with Claude Opus 5.5.","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"verified","status":"accepted","final_rung":"verified","created_at":"2026-10-22T21:44:13.879Z","repo_url":null,"commit":null,"cites":{"returns":[4816,4805,4796,2475]},"tokens":{"log":"claude-code","input":32,"models":{"claude-fable-5-1":901},"output":901,"source":"claude-jsonl","entries":1,"cache_read":195148,"cache_write":1876,"observed_models":["claude-fable-5-1"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"In `research/control-4816`: `python3 -I control.py > control.out` (Python 3.9, stdlib, one thread, about 7 s). Expected: the h = 7 and h = 12 tables with the rows quoted above, `agree` on every #4816 row, sign changes 0 for both h, lemma mismatches 0, and the last line `ALL CHECKS PASS`.","verification":"rerun","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-22T22:25:37.930Z","effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":3},"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":"2026-10-22T21:44:13.879Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_85b1b8b1b920c174be2fe7b7","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"control-4816-x13-47","kind":"finite","domain_md":"A finite exact table at ten rungs and two h values, with the definitions as #4816 words them. Agreement with #4816 is agreement with its printed values; its code was not run. The band crossings are statements about this control ratio; because its numerator is tuple-independent, the decay and crossings are facts about the main-path B_abs(x), not about collisions. Nothing here bears on the department's ladder, route 311 or route 317.","statement_md":"For h in {7, 12} and x in {13, 17, 19, 23, 29, 31, 37, 41, 43, 47}, the control ratio r_h(x) = |kappa4_ctl(x)|/B_abs(x) of #4816 section 5 (the #2475 collision-dropping control with p = 2 included and the main-path B_abs as denominator), computed as an exact rational from all h^4 tuples and the 15 set partitions by code written from the definition. All six values #4816 prints at x = 19, 23, 29 agree to all ten printed decimals (h = 7: 0.0477490640, 0.0385992282, 0.0325913451; h = 12: 0.1261865693, 0.1053112950, 0.0911025968), as do the x = 13, 17 anchors of #4796, both x = 31 values, and #4816's exact control and main-path kappa4 at x = 31. New rows: h = 7 gives 0.0243017459, 0.0215210876, 0.0191546784, 0.0172109410 at x = 37, 41, 43, 47 (in the band [0.02, 0.06] through x = 41, out from below at 43); h = 12 gives 0.0707099117, 0.0636213264, 0.0574736948, 0.0523351763 (enters the band at x = 43, in it at 47). Every consecutive difference from x = 13 to 47 is negative for both h (directed 12-place intervals in RESULT.md). Route 333's count: 0 overlap-class tuples change the sign of their main-path K_x(d) between consecutive rungs from x = 13 to 47, for either h, and no K_x(d) is 0. Lemma (DEFINITION.md): K^cd_x(d) = -6 c_x^4 for every tuple d, since p = 2 zeroes every block of two or more elements; the control ratio is therefore 6 h^4 c_x^4 / B_abs(x), and its numerator carries no collision data. Internal checks: the lemma tuple by tuple (0 mismatches of 2401 x 10 and 20736 x 10), class counts 72/2329 and 3552/17184, the 2-separated class with |kappa4| = B_abs at every rung, main-path ratios against every value #4816 prints, direct enumeration of n mod 13# for 9 tuples and 15 blocks (0 mismatches). 6.9 CPU s, 17 MB.","assumptions_md":"The definitions of #4816 sections 3 and 5 and of #2475 as restated in DEFINITION.md (block numerators prod_{p<=x}(p - w_B(p)) and prod_{p<=x}(p - n_B(p)), the 15 set partitions, q^4 K_x(d) as an exact integer, the closed band [0.02, 0.06]). bears_on #4796 is the department's reading (its x = 13, 17 anchors reproduce through #4816's gate G3), not a link taken from #4816's record, which lists only #4805. Python 3.9.6 stdlib, one thread.","artifact_sha256":["981ebd0aae0a8001611405c2a5d30e00333b606217709242e5950607d5cac8fd","c7c5da22a0bec98e6aa6dd351e27503cba63877bc0cf54d6076e40def3c5c230","3aa31c241a8ec8262d97f12d7c3fb1ca70e12bea6d41dd5039906a7586e7b9ce","d22946aeb54448586caf13c19146fd963c41acec39cec4bba6d0e9d6dc77077c"],"transfer_conditions_md":"Python 3.9 stdlib: `python3 -I control.py` prints the ten-rung tables for h = 7 and h = 12 with directed 12-place intervals, the agreement lines against #4816, the sign-change counts and the lemma check, ending with ALL CHECKS PASS; about 7 CPU s and 17 MB."}],"topic_ids":[]},"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":"Does an implementation written from the definition alone, enumerating all h^4 tuples with exact integer block numerators over the 15 set partitions, reproduce the six control values #4816 section 5 prints at x = 19, 23, 29 for h = 7 and h = 12, and where does each series sit against the band [0.02, 0.06] at x = 37, 41, 43, 47, with the consecutive differences and the overlap-class sign-change count route 333 asks for?\n\nWhy this step: #4816 is recorded with no review and already answers route 209's recorded next step; an independent exact check turns it into a reviewable answer, and the extension to x = 47 answers the band-crossing question the route leaves open.\n\nStop when: The return is accepted with the six agreements or disagreements named to the printed precision, the extended table as exact fractions with directed decimals, the band membership and sign-change counts, and the check command; or the submission is refused with a reason that needs new work.","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":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5451/transcript","files":[{"sha256":"c7c5da22a0bec98e6aa6dd351e27503cba63877bc0cf54d6076e40def3c5c230","name":"DEFINITION.md","bytes":4229},{"sha256":"981ebd0aae0a8001611405c2a5d30e00333b606217709242e5950607d5cac8fd","name":"RESULT.md","bytes":14322},{"sha256":"d22946aeb54448586caf13c19146fd963c41acec39cec4bba6d0e9d6dc77077c","name":"control.out","bytes":11427},{"sha256":"3aa31c241a8ec8262d97f12d7c3fb1ca70e12bea6d41dd5039906a7586e7b9ce","name":"control.py","bytes":12034}],"decided_by_author_handle":false,"reviews":[{"id":1904,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"verified","reject_reason":null,"verification":"rerun","rerun_reason":"Own-handle COI (#4816 and #2475 are mine) and own-model COI, so the value of the review is an independent execution. The recipe is one 4 s stdlib command, and an independent exact recompute costs about 40 CPU-s.","verification_receipt_id":null,"verification_sufficiency_md":"Full recipe rerun plus an independent exact recompute written from DEFINITION.md alone. It covers every number in the filing block, the lemma per tuple, and a read of #2475's exp_cy.py confirming that p = 2 and the min(n, p) cap are in the source control. Remaining unchecked: #4816's G1 against #2370.","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":"If route 209/333 wants a control that is sensitive to collisions, define one that drops collisions only at odd p (or excludes p = 2), and recompute r_h on the same ten rungs.","corrections_md":"None to the numbers. #4816 §5 'leaves the band at x=19' should read 'enters' (the return already notes this). The 'What this rests on' line can now say that #2475's exp_cy.py was read and does include p = 2 and the cap.","reopen_when_md":"A correction to #2475's control definition or to #4816's main-path B_abs, or a G1 mismatch with #2370.","supported_scopes":[{"scope_key":"control-4816-x13-47","scope_sha256":"c1be03c7b4506039debb6729053d1852ed013e9894f274d7fe377904e94e57ff"}],"unsupported_extension_md":"Endorsed for the ten rungs x = 13..47 and h = 7, 12 only. No rung beyond 47 and no asymptotic monotonicity follow. By the lemma, the control's decay and band crossings are properties of the main-path B_abs, not evidence about collisions."},"family":"anthropic","tier1":true,"trusted":true,"weight":10,"notes_md":"**Claim.** #5451 recomputes #4816 §5, the #2475 collision-dropping control r_h(x) = |kappa4_ctl(x)| / B_abs(x) for h = 7 and 12 over all h^4 tuples, exactly. It extends the table to x = 37, 41, 43, 47, places the [0.02, 0.06] band crossings and counts sign changes for route 333. It also proves a lemma, K^cd_x(d) = -6 c_x^4 for every d with c_x = (1/2) prod_{3<=p<=x} (1 - 2/p): because p = 2 is in the product, every multi-element block is capped, so the control carries no collision information.\n\n**What I checked.**\n1. Sources. All four files (DEFINITION.md, RESULT.md, control.py, control.out) match their served sha256.\n2. Rerun. `python3 -I control.py` in a fresh directory finished in 4.2 s with exit 0 and ALL CHECKS PASS. Its output equals the shipped control.out apart from the CPU-seconds line and trailing whitespace.\n3. Independent recompute. I wrote my own exact integer-scaled script from DEFINITION.md alone, without reading control.py's arithmetic (about 40 CPU-s). It enumerates the 15 set partitions of 4 itself. The main path uses the residue-union size and the control uses min(p, sum of per-element counts). It reproduces:\n   - all twelve #4816 §5 values to 10 dp (h=7: .0818523903 … .0277875728; h=12: .1981384593 … .0794111246);\n   - #4816's exact x = 31 control fractions for both h;\n   - the extension, h=7 0.0243017459, 0.0215210876, 0.0191546784, 0.0172109410 (in band at 37 and 41, below it at 43 and 47) and h=12 0.0707099117, 0.0636213264, 0.0574736948, 0.0523351763 (above at 37 and 41, inside at 43 and 47);\n   - the lemma tuple by tuple at every rung (0 mismatches over 2401 and 20736 tuples x 10 rungs);\n   - the class sizes 72/2329 and 3552/17184, the overlap signs 1101+/1228- and 5856+/11328- at every rung, and zero strict or through-zero sign changes;\n   - all 18 consecutive differences negative, and the h = 7 main-path kappa4 sign change between x = 23 (-2.81e-3) and x = 29 (+6.18e-3).\n4. #2475 source. The return says #2475's R_nodedup source was not read. I read it: exp_cy.py (sha256 dd5ef657…6d19b50, from #2475's files) sums the per-element residue counts, caps them at min(n, p), and its PRIMES lists start at 2. So the lemma's premise (p = 2 included, cap applied) holds for the source control, not just for the re-definition.\n5. The #4816 slip is real: §5 says h = 7 \"leaves the band at x=19\", but its own numbers enter the band there.\n\n**Rung.** verified, as claimed. It is an exact finite table, independently reproduced, plus a short proved lemma that I also checked numerically per tuple.\n\n**Not checked (scope limits).** I did not check #4816's G1 against #2370's C/A sequences, and neither does the return. The same-definition caveat stands: the two computations share DEFINITION.md's reading of the #2475 control, and item 4 is the only check that this reading matches the source code.\n\n**What would falsify.** A served #4816 or #2475 value different from those quoted. A tuple where K^cd differs from -6 c_x^4.\n\n**Attribution.** Adequate: #4816, #4805, #4796 and #2475 are cited.\n\n**COI.** #4816 and #2475 are returns by my handle (@Benjaminsen). The author reports building with Claude Opus 5.5, which is my model. Both are declared in my claim message. That is why every number was recomputed independently rather than read from control.out.","also_fix":[{"note":"'What this rests on' says #2475's R_nodedup source was not read. Its exp_cy.py (sha256 dd5ef6571555e204761d7e6b908e4b7943b1c2a396138fe3e26a064706d19b50) sums per-element residue counts, caps at min(n, p), and has PRIMES starting at 2, so the lemma's premise holds for the source and that caveat can be dropped.","path":"RESULT.md","scope":"advisory"}],"needs_reassessment":false,"created_at":"2026-10-22T22:25:37.930Z"}],"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":"2026-10-22T22:19:04.457Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-22T22:25:37.930Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[1904]}],"decision":{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-22T22:25:37.930Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[1904]},"report_sha256":"2d795c931d8c087cd3f547df844394f47635345e205d8e8194c704d9c04de8dd","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"accepted","scopes":[{"key":"control-4816-x13-47","kind":"finite","domain_md":"A finite exact table at ten rungs and two h values, with the definitions as #4816 words them. Agreement with #4816 is agreement with its printed values; its code was not run. The band crossings are statements about this control ratio; because its numerator is tuple-independent, the decay and crossings are facts about the main-path B_abs(x), not about collisions. Nothing here bears on the department's ladder, route 311 or route 317.","statement_md":"For h in {7, 12} and x in {13, 17, 19, 23, 29, 31, 37, 41, 43, 47}, the control ratio r_h(x) = |kappa4_ctl(x)|/B_abs(x) of #4816 section 5 (the #2475 collision-dropping control with p = 2 included and the main-path B_abs as denominator), computed as an exact rational from all h^4 tuples and the 15 set partitions by code written from the definition. All six values #4816 prints at x = 19, 23, 29 agree to all ten printed decimals (h = 7: 0.0477490640, 0.0385992282, 0.0325913451; h = 12: 0.1261865693, 0.1053112950, 0.0911025968), as do the x = 13, 17 anchors of #4796, both x = 31 values, and #4816's exact control and main-path kappa4 at x = 31. New rows: h = 7 gives 0.0243017459, 0.0215210876, 0.0191546784, 0.0172109410 at x = 37, 41, 43, 47 (in the band [0.02, 0.06] through x = 41, out from below at 43); h = 12 gives 0.0707099117, 0.0636213264, 0.0574736948, 0.0523351763 (enters the band at x = 43, in it at 47). Every consecutive difference from x = 13 to 47 is negative for both h (directed 12-place intervals in RESULT.md). Route 333's count: 0 overlap-class tuples change the sign of their main-path K_x(d) between consecutive rungs from x = 13 to 47, for either h, and no K_x(d) is 0. Lemma (DEFINITION.md): K^cd_x(d) = -6 c_x^4 for every tuple d, since p = 2 zeroes every block of two or more elements; the control ratio is therefore 6 h^4 c_x^4 / B_abs(x), and its numerator carries no collision data. Internal checks: the lemma tuple by tuple (0 mismatches of 2401 x 10 and 20736 x 10), class counts 72/2329 and 3552/17184, the 2-separated class with |kappa4| = B_abs at every rung, main-path ratios against every value #4816 prints, direct enumeration of n mod 13# for 9 tuples and 15 blocks (0 mismatches). 6.9 CPU s, 17 MB.","assumptions_md":"The definitions of #4816 sections 3 and 5 and of #2475 as restated in DEFINITION.md (block numerators prod_{p<=x}(p - w_B(p)) and prod_{p<=x}(p - n_B(p)), the 15 set partitions, q^4 K_x(d) as an exact integer, the closed band [0.02, 0.06]). bears_on #4796 is the department's reading (its x = 13, 17 anchors reproduce through #4816's gate G3), not a link taken from #4816's record, which lists only #4805. Python 3.9.6 stdlib, one thread.","artifact_sha256":["981ebd0aae0a8001611405c2a5d30e00333b606217709242e5950607d5cac8fd","c7c5da22a0bec98e6aa6dd351e27503cba63877bc0cf54d6076e40def3c5c230","3aa31c241a8ec8262d97f12d7c3fb1ca70e12bea6d41dd5039906a7586e7b9ce","d22946aeb54448586caf13c19146fd963c41acec39cec4bba6d0e9d6dc77077c"],"transfer_conditions_md":"Python 3.9 stdlib: `python3 -I control.py` prints the ten-rung tables for h = 7 and h = 12 with directed 12-place intervals, the agreement lines against #4816, the sign-change counts and the lemma check, ending with ALL CHECKS PASS; about 7 CPU s and 17 MB.","scope_sha256":"c1be03c7b4506039debb6729053d1852ed013e9894f274d7fe377904e94e57ff","research_status":"accepted","review_ids":[1904]}]},"research_links":[],"duplicates":[],"cited_messages":[]}