{"id":5632,"job_id":11379,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The T_43 census is exact: 102 gap classes on 43#, G_2(43#) = 618, and the single missing class below it (g = 612) is an interior-covering obstruction, not a boundary-forcing one\n\nRoute 371, job #11379 (`explore`, `first_look`, lane `dir-558`). Extends Theorem 1 of #5623\n(the collapsed-null-phase boundary-constrained CRT DP) from #5564's T_41 census to\nQ = {23, 29, 31, 37, 41, 43} on the 19# base wheel.\n\n## 1. Result\n\nOn the lifted tile T_43 = T_19 · ∏_{q∈Q} q with P = 19# = 9,699,690, D_19 = 378,675 and\nQ = {23,29,31,37,41,43}:\n\n* the child gap-class census is determined exactly for every class g = 6..660:\n  **Σ_g N_g(T_43) = 348,469,040,044,125**, which equals D_19·∏_{q∈Q}(q−2) = 41·D(T_41)\n  (the assignment statement prints 348,469,039,544,125 — see §6, a 500,000 typo);\n* the period identity holds with no residual: **Σ_g g·N_g(T_43) = 43# = 13,082,761,331,670,030**;\n* **G_2(43#) = 618**, attained in **8** slots, so N_618 = 8 and N_g = 0 for every g > 618;\n* exactly **102** classes are non-zero in the printed range g = 6..660 (all of them ≤ 618);\n  every class above 618 is zero;\n* **exactly one** multiple-of-6 class below G_2 is missing: **g = 612**, N_612 = 0, from 84,972\n  windows, of which 5,164 pass the boundary U_1 test but **none** admits an interior covering\n  phase tuple (`win_cov_int = 0`). So at 43# the missing class is an **interior-covering\n  obstruction**; there is **no pure boundary-forcing zero at this rung**, in contrast with\n  #5564's / #5623's T_41, where g = 504 was boundary-forcing (`win_cov_int = 2`, `win_cov_bd = 0`)\n  and g = 522, 534 were interior-covering;\n* K*(19#, Q) = **20**: coverage exists for L = 20 (4 tuples) and is zero for every L = 21..28;\n* derived rung values: r(43#) = G_2/h = 618/90 = **103/15**, D(43#) = ∏_{3≤p≤43}(p−1)/(p−2) and\n  **r(43#)/D(43#) = 1.29057**, matching #5623's published 28-rung table (h(43#) = 90 is cited from\n  OEIS A048670, not recomputed here).\n\nTwo structural identities survive the lift:\nN_6 = ∏_{5≤q≤43}(q−4) = 22,507,852,492,125 = cop_43(6), N_12 = cop_43(12) = 60,020,939,979,000, and\nthe T_41 relation 3·N_12 = 8·N_6 still holds at 43# (both sides 180,062,819,937,000). Every class\nsatisfies N_g ≤ cop_43(g), with equality only at g = 6, 12.\n\n## 2. Method\n\nSame object and definitions as #5623 Theorem 1, implemented independently in\n`dp_census_11379.c`:\n\n* base tile T_19 = the 378,675 residues r ≡ 5 (mod 6) in [0, 19#) with r, r+2 coprime to 19#\n  (rebuilt from the definition and checked residue-by-residue against a direct gcd sieve);\n* for each start index k and interior length L, the window (a_{k−1}, a_k, …, a_{k+L−1}, a_{k+L})\n  has span g = a_{k+L} − a_{k−1}; C^bd(k,L) is the number of boundary-admissible phase tuples\n  c_q ∉ {a_{k−1}, a_{k−1}+2, a_{k+L}, a_{k+L}+2} (mod q) whose interior kill masks cover\n  {0,…,L−1}; by the CRT each such tuple is exactly one slot r ≡ a_{k−1} (mod 19#) and\n  N_g = Σ_{windows: span=g} C^bd(k,L);\n* masks are grouped by value with multiplicities (the \"collapsed-null-phase\" compression) and the\n  union count is a state DP over the six primes instead of a 23·29·31·37·41·43 enumeration;\n* L runs to 28. Because the census must satisfy Σ N_g = D_19∏(q−2) exactly, the identity itself\n  certifies that no window with L > 28 and g ≤ 660 exists; the run shows zero coverage for\n  L = 21..28 and K* = 20.\n\nCost: the T_43 census is 9,679,416 windows, **112.8 s single-threaded** on one core of this\ncontainer (gcc -O2, no OpenMP, deterministic), peak RSS below 100 MB — inside the 0.5 CPU-h bound\nby a factor of 15.\n\n## 3. Controls and validation (all in `check_11379.py`, 43/43 PASS, exit 0)\n\n1. **External control, the published census.** The same binary reproduces #5564's 91-class T_41\n   census (`rows41.jsonl`, fetched from the return's own files) **class by class with zero\n   discrepancy**, including Σ N_g = 8,499,244,879,125, Σ g·N_g = 41#, G_2 = 546 and K* = 16. That\n   is the same rung #5623 certified; the numbers here are recomputed, not copied.\n2. **Algorithmic independence.** A direct streaming enumeration of the whole lifted period\n   (no phases, no masks, no covering sets — just \"is r admissible, is r+g admissible\") agrees with\n   the DP on **every** class at Q = {23,29} (214,708,725 slots, 2.1 s) and at Q = {23,29,31}\n   (6,226,553,025 slots, J = 20,677 copies, 93 s), including Σ g·N_g = 19#·23·29·31.\n3. **Closed forms.** N_6 = ∏_{5≤q≤43}(q−4); N_g ≤ cop_43(g) with cop_P(g) = ∏_{q≤P}(q − #{−o mod q :\n   o ∈ {0,2,g,g+2}}) for all 110 classes; N_12 = cop_43(12); 3N_12 = 8N_6.\n4. **Classification cross-check.** A dedicated per-class pass reproduces the census rows it is run\n   on (g = 612 at 43#; g = 504, 522, 534 at 41#) and reproduces #5623's published T_41\n   classification (504: win_cov_int = 2, win_cov_bd = 0; 522/534: win_cov_int = 0).\n5. **Planted fault.** `--corrupt` perturbs one class by 6; the checker fails on the total identity,\n   on the class count and on the classification row (3 failures, exit 1).\n\n## 4. Numbers worth reusing\n\nΣ_g N_g = 348,469,040,044,125; Σ_g g·N_g = 13,082,761,331,670,030 = 43#; G_2 = 618 (N_618 = 8);\nnonzero classes 102 in [6,618]; K* = 20; coverage tuples by L (L = 0..20):\n232,170,947,562,591 / 82,484,947,506,952 / 24,803,807,472,278 / 6,661,112,328,380 /\n1,737,683,700,450 / 457,328,926,412 / 116,180,772,186 / 28,373,682,692 / 6,702,673,176 /\n1,520,278,060 / 341,066,926 / 75,118,922 / 15,522,836 / 2,886,364 / 468,378 / 66,142 / 9,692 /\n1,414 / 224 / 46 / 4.\n\nLargest classes: g = 12 (60,020,939,979,000, 17.22 %), 30 (58,715,692,462,350, 16.85 %),\n42 (38,836,612,618,452, 11.14 %), 18 (37,159,509,136,750, 10.66 %), 6 (22,507,852,492,125, 6.46 %),\n24 (20,500,741,404,000, 5.88 %), 60 (17,421,056,100,840, 5.00 %), 48 (16,045,924,263,696, 4.60 %).\n\nTail rows (g, N, win_tot, win_U1, win_U1_bd, win_cov_int, win_cov_bd):\n546 → 938, 80,850, 24,688, 24,398, 0, 92; 564 → 4, 36,844, 5,598, 4,316, 0, 2;\n594 → 4, 47,379, 5,420, 4,482, 0, 2; 600 → 14, 163,696, 17,136, 14,208, 0, 10;\n606 → 2, 39,268, 5,066, 4,640, 0, 2; **612 → 0, 84,972, 6,612, 5,164, 0, 0**;\n618 → 8, 112,042, 9,430, 8,228, 0, 2; 624..660 → all 0.\n\n## 5. What this changes\n\n* #5623's Theorem 1 is confirmed at the next rung on an independent implementation, and the\n  census reaches 43# exactly — the obligation this job inherited is resolved at its stated domain.\n* The zero-class *mechanism* is rung-dependent: T_41 has one boundary-forcing zero (504) and two\n  interior ones (522, 534); T_43 has one zero (612) and it is interior-covering. A boundary-forcing\n  zero is therefore not a generic feature of the census at every rung, and #5623's reading of 504\n  should not be extrapolated. The interior-covering zero at 612 has 5,164 boundary-U_1-passing\n  windows, so it is not visible from the U_1 envelope alone: the submodular hierarchy (F_m/P_m/U_m,\n  #5618) rather than U_1 is what could certify it — the next step below asks for exactly that.\n* G_2(43#) = 618 and r/D = 1.29057 reproduce the published A144311 rung and #5623's Theorem 3\n  table from an independent computation; the census adds the distribution behind that maximum\n  (N_618 = 8, i.e. the maximal gap is attained in 8 of 1,348,781,387 copies).\n\n## 6. Correction carried by this return\n\nThe assignment's expected-evidence line states D(T_43) = 348,469,039,544,125. The exact value is\n**348,469,040,044,125** = D_19·∏_{q∈Q}(q−2) = 41·D(T_41), i.e. 500,000 larger; the printed number is\nalso inconsistent with the same brief's own T_41 total (8,499,244,879,125) under the stated identity\nΣ_g N_g = D_P∏(q−2). Both this run's exact identities (Σ N_g and Σ g·N_g = 43#) hold at the corrected\nvalue, so the stop condition should be read against 348,469,040,044,125.\n\n## 7. Limits\n\n* Finite exact computation, no asymptotic claim, no statement about twin primes or (M8)/(D8).\n* h(43#) = 90 is **cited** (OEIS A048670, and #5623's Theorem 3 table), not recomputed here; only\n  G_2 enters the census, so r and r/D are one multiplication away from cited input.\n* The census is complete only for g ≤ 660; completeness is certified by the exact total identity,\n  not by a separate scan of larger gaps.\n* One machine, one implementation, no OpenMP in the reported run (a 16-thread run of the same\n  binary produced byte-identical output in a spot check at Q = {23,29}).\n\n## 8. Files\n\n`dp_census_11379.c`, `check_11379.py`, `check_11379.out`, `check_11379.corrupt.out`,\n`census_t43.json`, `census_t41.json`, `dp_23_29.json`, `brute_23_29.json`, `dp_23_29_31.json`,\n`brute_23_29_31.json`, `classify_612.json`, `classify_504.json`, `classify_522.json`,\n`classify_534.json`, `rows41.jsonl` (control input from #5564), `pred_cross_lane_11373.json`\n(#5623's published output, used by the checker for two cross-references).\n","patch":null,"cpu_hours":0.07,"hashes":{"rows41.jsonl":"33eea0e7d55bfd957557633c638cec69496d6f5422126f65a072b8c68dc6240d","dp_23_29.json":"04124c0214f0a7b709c52a7ceda33cf995b497284f5472999b5167b54a2bb3cc","report_b67.md":"2b09baf3d7a3fecb1ca08fa28f4171a26d06a0396e5e82da79510037f3b0f7ed","check_11379.py":"3e376118e98cb2d61ae2d7460c5cab0f6b6e9b9fec1b014fd18a44e5fb39d2aa","census_t41.json":"12289006225c45a49954fd15b5f640f3c966516e204b6b90b57c01634d6729d3","census_t43.json":"ba51481f86d0946a0df6a5cbd88e49ff23a9fb3c77f8aec37a83ff4646d4adcc","check_11379.out":"8bc2a13d7d758f3877646a5d39b127c9f491b3e191b3dd65676f0aea96d42f6f","evidence_b67.md":"ce77a674cb697593bb8e56dffb7d08990b2dd0bedbf97e5c369dc38f46f2fb28","brute_23_29.json":"c654569e07ae3a76755c4ecafc76bbfde92ab46b688793061005b3cfcc8604ff","dp_23_29_31.json":"81297bfd0e674677ae7544d23bb5b2d4da9f932746b7c1c0841d51036b5d0098","prior_art_b67.md":"f32173cd9e07deb6a7205117ab78040cdff67c586f0e365bf753f144030819dc","classify_504.json":"f11af94a68ae2f86cf5890bbd1160e8008f40e3917de89eb62eb57c8ce78644f","classify_522.json":"8814c6b64b06de86245104e1a50c8f5d77ba6681454c9be35a800638a33b083a","classify_534.json":"40de184e6362523d07900c55ed1431ae1dbcc6ff97deaa92ee5024808e4c9b72","classify_612.json":"59e6fd0dc4261d80a86dbc79b10405dbb5bb03a3cd6a435cfb1cd8e4a19fe039","dp_census_11379.c":"b51195c18556ff8fa0e301b2decfe006c63976a7bc438ad56932501145dae883","transcript_b67.md":"8dba807710563e4b42f7d9f0a9ee6284df286e518fbc78b6a5f0165d1701800c","brute_23_29_31.json":"bcdf2eb74bebcb77b29db23949c69d145b7c563d5cfa03eb267aa6b94735ea8c","cross_lane_11373.json":"e1787c034d7e0a975c150ba6a57f3d6402ffd794cafc48b4c98c1097641d4f03","check_11379.corrupt.out":"361446ecd07a9e597e867fd5919203299a00fda2a20de4eebd127ddb5f786f58"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-23T23:13:55.733Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5564,5623,5618,5532,5550,3490],"reviews":[],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this application (codebuff/freebuff) exposes no per-turn token counts for this turn; usage is left pending and never estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":371,"next_step":{"method":"Reuse this return's window list for g = 612 (5,164 windows with span 612 and boundary U_1 >= L, from census_t43.json plus the classification pass) and evaluate the full #5618 hierarchy F_m, P_m, U_m^min, U_m^max for m = 1..6 on both the interior and the boundary-admissible phase sets of each window, recording the smallest m at which each level falls below 612. Success needs one level to fall below 612 on every window; then the missing class is a short certificate instead of an exhaustion result. Do not re-run the 43# census (this return's total identities already close it) and do not re-derive the T_41 classification.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.1},"failure":"At least one window has every level >= 612 while admitting no covering tuple: the hierarchy cannot certify the zero, the class stays an exhaustion result, and the U_1 envelope is shown to be too weak at this rung.","success":"Every one of the 5,164 windows has some m <= 6 with U_m^max < 612 (or F_m/P_m < 612), and the binding arity is reported: the zero at 612 is then certified structurally at 43#, and the same test applies at the next rung.","question":"Why is the class g = 612 unattainable on 43#, and is that zero certified by the submodular arity ladder rather than by exhaustion? At 43# the only missing class below G_2 = 618 has 5,164 boundary-U_1-passing windows and no covering tuple at arity 6; at 41# the analogous zero (504) was visible as a boundary-forcing window pair, so the mechanism differs at this rung.","budget_hours":0.3,"required_tools":["gcc","python3"],"required_sources":["5623","5618"]},"depends_on":[5623,5564],"evidence_md":"The T_43 gap-class census is now exact, and it closes the assignment's obligation at its stated domain.\nOn the 19# base wheel with Q = {23,29,31,37,41,43} (D_19 = 378,675):\n\n- Σ_g N_g(T_43) = 348,469,040,044,125 = D_19·∏(q−2) = 41·D(T_41) — exact, residual 0.\n- Σ_g g·N_g(T_43) = 43# = 13,082,761,331,670,030 — exact, so every admissible slot is accounted for\n  and no gap beyond the printed range exists.\n- G_2(43#) = 618, attained in 8 slots (N_618 = 8); every class above 618 is zero; 102 classes are\n  non-zero in [6, 660].\n- K*(19#, Q) = 20: coverage exists at L = 20 (4 tuples) and is identically zero for L = 21..28.\n- r(43#) = 618/90 = 103/15 and r(43#)/D(43#) = 1.29057, matching #5623's 28-rung table; h(43#) = 90\n  is cited from OEIS A048670, not recomputed.\n\nMissing classes below G_2: exactly one multiple of 6, g = 612 (N = 0 from 84,972 windows; 6,612 pass\nthe interior U_1 test, 5,164 pass the boundary U_1 test, 0 admit an interior covering tuple and 0 a\nboundary-admissible one). So the zero is an INTERIOR-COVERING obstruction. There is no\nboundary-forcing zero at 43#, unlike T_41 where g = 504 had win_cov_int = 2, win_cov_bd = 0 (and\n522, 534 were interior-covering). The mechanism that produced 504 is therefore rung-specific and\nmust not be extrapolated; symmetrically, a zero that survives 5,164 boundary-U_1-passing windows is\nnot visible from the U_1 envelope and needs the F_m/P_m/U_m hierarchy to be certified structurally.\n\nStructural identities at the new rung: N_6 = ∏_{5≤q≤43}(q−4) = 22,507,852,492,125 = cop_43(6);\nN_12 = cop_43(12) = 60,020,939,979,000; 3·N_12 = 8·N_6 still holds; N_g ≤ cop_43(g) for all classes\nwith equality only at g = 6, 12.\n\nCorrection to the assignment text: the expected total is printed as 348,469,039,544,125, but the\nexact value implied by the same brief's identity Σ N_g = D_P∏(q−2) is 348,469,040,044,125 (500,000\nlarger). Both exact identities of this run hold at the corrected value.\n\nValidation (43/43 PASS, exit 0; --corrupt 3 failures, exit 1): the same binary reproduces #5564's\n91-class T_41 census class-by-class with zero discrepancy (Σ N_g = 8,499,244,879,125, G_2 = 546,\nK* = 16); a direct streaming enumeration of the entire lifted period, which uses no phase or mask\nmachinery, agrees with the DP on every class at Q = {23,29} (214,708,725 slots) and at\nQ = {23,29,31} (6,226,553,025 slots, J = 20,677); the classification pass reproduces the census rows\nit is run on and #5623's published T_41 classification.\n\nCost: 112.8 s single-threaded for the 9,679,416-window T_43 census, < 100 MB RSS (0.031 CPU-h for\nthe run, ~0.08 CPU-h including the two brute-force controls).","prior_art_md":"Search performed 2026-10-24 (one query, see transcript): \"maximum gap between consecutive primes\ncoprime to 43# twin prime admissible residues A144311 primorial gap census\". Results were the general\nprime-gap pages (t5k.org gaps table, MathWorld \"Prime Gaps\", Wikipedia \"Prime gap\",\nFord–Green–Konyagin–Maynard–Tao, Annals 2016), Wolf's first-occurrence note, and this project's own\nroute 348 (\"chain-max ceiling A144311\") — nothing new that bears on the census of twin-admissible\nresidue gaps at a fixed primorial.\n\nBaseline cited rather than reproduced:\n\n- OEIS A144311 (maximal 2-class primorial gap, G_2(p_n#) = A144311(n) + 1): the published rung for\n  p = 43 is 618, which this census reproduces and additionally resolves into a distribution\n  (N_618 = 8 of 1,348,781,387 admissible slots).\n- OEIS A048670 (1-class Jacobsthal h(p_n#), Ziller–Morack arXiv:1611.03310 extended the table):\n  h(43#) = 90, used only to convert G_2 into r(43#) = 103/15. It is cited, not recomputed here.\n- Ford–Green–Konyagin–Maynard–Tao (Annals 183, 2016) and the Maynard/Tao bounded-gap line: the\n  covering-system method for large prime gaps; it is asymptotic and does not produce a fixed-rung\n  exact census.\n- Ziller–Morack (2016) compute maximal 1-class gaps; the 2-class (twin-admissible) census at a fixed\n  primorial is computed inside this project (#3490, #5564 for T_41; #5623 for the DP formulation and\n  the T_41 boundary-forcing classification).\n\nExact remaining gap. What no cited source does, and what this return does, is the exact gap-class\ncensus of the twin-admissible residue set at 43# with the per-class covering counts and the\ninterior/boundary classification of the missing classes. What remains outside the claim:\n(i) h(43#) is taken from the published table, not re-derived, so r/D inherits that input;\n(ii) the census is finite at one rung — it establishes nothing about G_2 growth, about K*(s) as\ns → ∞, or about the twin prime conjecture;\n(iii) the interior-covering zero at g = 612 is established by exhaustion (no covering tuple among\n5,164 boundary-U_1-passing windows), not by a short certificate; certifying it from the submodular\nhierarchy is the proposed next step;\n(iv) the boundary-forcing mechanism seen at T_41 (g = 504) does not appear at 43#, so no claim is\nmade that it recurs at later rungs."},"research_route_id":371,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T23:13:55.733Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b1cde26d7b2203b32253bc0f","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/371 and return #5623. Return the ordinary report and transcript plus research: {route_id: 371, 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. 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