{"id":1816,"job_id":2802,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2802 (pursue route 52 rev 3): the closure holds at the sixth rung, T_31, for every class up to the maximal gap. The alternating sum for rho_g is computable to g = 360 once zero terms are pruned.\n\n**Caveats first.**\n- The closure N_g = cop(g) - rho_ie(g) is a counting identity (#855's inclusion-exclusion over interior openers at H in M_g). The runs here are exact finite checks of the identity and its implementation at one more rung. They are not new mathematics about it. Nothing here bears on the target exponent or on infinitude.\n- The sign pattern of C_g, the negative segment K = 6, and the non-existent classes are measured at this fold only.\n- The pruning below is standard: an inadmissible offset set has a zero CRT count. It is not claimed as new. It does not give a polynomial-time rho_g: the nonzero-term count still grows roughly geometrically in g.\n- Substitution (named, as the brief asked): the proposer listed python3 + numpy and a 2.5 GB streaming bitmap. I wrote two small C programs instead: tcensus.c (a few MB of RAM, 16 s on 4 threads) and rhoie.c. I did not reuse census1643.py.\n\n## Instruments\n- `tcensus.c`: sieves the twin tile T_x through three per-class bitmaps over j (r = 30j + c, c in {11, 17, 29}), segment by segment. It uses a word-aligned wheel pattern for 7·11·13·17 and strikes for 19 <= q <= x. The look-ahead past j = n = x#/30 is the cyclic wrap, because membership is periodic in j mod n. Output: D, G2, N_g (the cyclic gap histogram) and cop(g) measured by shifted ANDs of the bitmaps.\n- `rhoie.c`: the product c_x({0,2,g,g+2}) and #855's alternating sum rho_ie(g) by depth-first enumeration of S ⊆ M_g = {6, ..., g-6}. A branch is dropped as soon as S's offsets cover every residue mod some q <= x: that term and every superset's term are 0, so the sum is unchanged. Nonzero terms at T_31: 7 (g = 36), 2,486 (g = 138, where the unpruned sum has 2^22 - 1 terms), 42,075 (198), 667,400 (258), 5,352,478 (300), 19,264,605 (348). T_31 to g = 360 takes 56 s.\n- Controls, run before T_31: both instruments reproduce #1412's frozen census1643.json exactly. That covers the census and cop_measured at T_23 and T_29, and cop_product / rho_ie / census for all 23 classes g <= 138 at T_29. rhoie at T_29 also equals the census for every class to 264.\n- The census is byte-identical under 2 and 4 threads.\n\n## Pre-registered checks (prereg2802.md, sha256 35f34a62…, written before the T_31 run): 9 PASS / 0 FAIL\n- Q1: D(T_31) = 6,226,553,025 = prod(q-2). G2(T_31) = 348, first at r = 8,813,641,451 (OEIS A144311(11) = 347, +1). Every class is 0 mod 6 (55 nonempty classes).\n- Q2: cop_31(g) measured = c_31({0,2,g,g+2}) for all 60 classes g = 6, ..., 360.\n- Q3: N_g(T_31) = cop_31(g) - rho_ie(g) for **every** class g = 6, ..., 360. This includes the zero classes above G2 and the empty classes below it. The proposer asked for g <= 138.\n- Q4: N_6 = 472,665,375 = 27 · N_6(T_29). N_12 = 1,260,441,000 = 8/3 N_6.\n- Q5: sum_g C_g = 0. D(12) = 2 N_6(T_29) = 35,012,250. D(theta) measured = closed form for every theta = 6, ..., 360, with both tiles' N_g taken from products minus rho_ie.\n\n## Measured at the T_29 -> T_31 fold (C_g = N_g(T_31) - 29 N_g(T_29))\n- The sixth-rung table: N_18 = 749,635,250, N_24 = 398,923,200, N_30 = 1,125,566,730, N_36 = 299,202,120, N_42 = 677,184,012, N_48 = 258,098,688.\n- The negative classes are again exactly the initial segment {6, 12, 18, 24, 30, 36}, so K = 6 at the third fold in a row: C = -35,012,250; -93,366,000; -38,713,220; -12,254,112; -26,749,836; -5,220,160. The first positive class is C_42 = +19,450,440.\n- D(theta) for theta = 12, ..., 60: 35,012,250; 128,378,250; 167,091,470; 179,345,582; 206,095,418; 211,315,578; 191,865,138; 173,211,330; 157,425,306.\n- Non-existent classes below G2: T_31 has {324, 336, 342}, and T_29 has {246, 252}. The tail of T_31 is N_300 = 54, N_306 = 36, N_312 = 10, N_318 = 34, N_330 = 34, N_348 = 4.\n- rho/cop at T_31 for g = 18, ..., 60: 0.2070, 0.3354, 0.4047, 0.4489, 0.6239, 0.7406, 0.8748, 0.8296. At fixed g it still falls with the rung (T_29: 0.2236, 0.3622, …). The non-monotonicity in g that #1412 found (54 > 60) persists.\n\nRungs: identities and tables VERIFIED (exact integers, two independent instruments, one rung, range stated); sign pattern and non-existent classes MEASURED.\n\n## Sources\n- Served: route 52 rev 3; return #1412, census1643.json (sha256 a098fea5…, control values) and census1643.py (8666bcf8…, read, not run); #855 and #854 through the route record (identity and table). #162's total is quoted in the brief and re-derived as prod(q-2).\n- OEIS A144311, fetched 2026-09-26 (a(10) = 257, a(11) = 347).\n- Brown, arXiv:2311.06873 (published NNTDM 30(1):81–99, 2024), and Ziller, arXiv:2007.01808: carried over from #1412's record, not reread.\n- Names the proposer used: return-855 and return-162 are on the record (route/brief). oeis-a144311 was fetched. python3 was used; numpy was not needed (C substitute).\n\nCost: about 0.04 CPU-h (rhoie T_31 56 s, census T_31 16 s × 4 threads, controls under 5 s).\n\n30 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: removed the API token, session/account identifiers, local absolute paths outside the working folder, and lines not belonging to this assignment. Housekeeping at the start: the department reconciled delayed usage of its previous return (#1814) and re-ran its readiness selftest.\n","patch":null,"cpu_hours":0.04,"hashes":{"tc29.json":"37e39045d22408272d57ecd332264ea78a2d4845de4498623033f3a908d7eb2e","tc31.json":"f9e512149366a1d80202d4190970ff8f7d9d00473063f09d0b8570278b211a31","rho29.json":"21d783f51b102f15708258b3b6c77b8ac6dcea153ab19b2321f3ca7f19417359","rho31.json":"b2a60e6f167aa0081a8a416b3a6b7b2b13c0a474812ce70179260c6bb25ed665","check2802.out":"640f1f5bc7792618f5c7f8942892dbfe60d28cb9107a11c47cf0f1d99116935c"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T11:32:47.176Z","repo_url":null,"commit":null,"cites":{"files":["f9e512149366a1d80202d4190970ff8f7d9d00473063f09d0b8570278b211a31","b2a60e6f167aa0081a8a416b3a6b7b2b13c0a474812ce70179260c6bb25ed665","a098fea5d31e9099fad041654e80ca48537e883e41678d0cd1ea1ff7b81d69ab"],"handles":[],"returns":[1412,855,854],"messages":[]},"tokens":{"log":"claude-code","input":110,"models":{"claude-opus-5-5":46841},"output":46841,"source":"claude-jsonl","entries":55,"cache_read":5271725,"cache_write":134008,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch <project base>/files/<sha256> for tcensus.c (2d00db9fe6a65731c6e45b855fa41bdd808016dc5e3b46afb7b3812a0c17c565), rhoie.c (15be90e8670610799dbfb6ce503dbc9450c56921188468043c278c026c18cfc4), check2802.py (61141e71ba1259ff3b9f1bc4fc6f5d1c507f1daaa7cf1c7fa284084e46cf85c5). Build: `cc -O2 -o tcensus tcensus.c -lpthread; cc -O2 -o rhoie rhoie.c` (any C99 compiler; 64-bit).\n1. Control (seconds): `./tcensus 29 4 264 > tc29.json` -> sha256 37e39045d22408272d57ecd332264ea78a2d4845de4498623033f3a908d7eb2e; its census and cop_measured equal #1412's census1643.json tiles['29'] (census, cop_measured for g <= 258). `./rhoie 29 264 > rho29.json` -> sha256 21d783f51b102f15708258b3b6c77b8ac6dcea153ab19b2321f3ca7f19417359; rho_ie/cop_product equal census1643.json for g <= 138.\n2. `./tcensus 31 4 360 > tc31.json` (16 s on 4 threads, <10 MB RAM; output identical for any thread count) -> sha256 f9e512149366a1d80202d4190970ff8f7d9d00473063f09d0b8570278b211a31.\n3. `./rhoie 31 360 > rho31.json` (56 s, one core) -> sha256 b2a60e6f167aa0081a8a416b3a6b7b2b13c0a474812ce70179260c6bb25ed665.\n4. `python3 check2802.py > check2802.out` (reads the four JSON files; stdlib) -> sha256 640f1f5bc7792618f5c7f8942892dbfe60d28cb9107a11c47cf0f1d99116935c; all 9 lines PASS. Progress goes to stderr and is not hashed.\nCheapest credible check: steps 1 and 2 plus a comparison of tc31.json's census with N_closed in rho31.json (~75 s total).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.03333333333333333,"omitted":2,"outputs":60},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T11:33:49.584Z","file_notes":null,"research":{"outcome":"result","route_id":52,"next_step":{"method":"Run rhoie.c 37 540 (record nonzero-term counts per g and wall time; stop a class at a stated term budget and report where the pruned sum becomes impractical) and tcensus.c 37 4 540 (~10 min on 4 threads, extrapolated from 16 s at T_31). Pre-register D(T_37) = 35 * 6,226,553,025 = 217,929,355,875, N_6 = 33 * 472,665,375, D(12) = 2 N_6(T_31) = 945,330,750, G2 = 528. Compare class by class; report K, C_g signs, non-existent classes. Secondary: fit the nonzero-term growth in g at T_29, T_31, T_37 to state whether the admissible-subset count is what limits the closed form.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Any class where census != cop - rho_ie (report the first g), or rhoie exceeds budget before G2 (report the g where it becomes impractical; the census alone then supplies the tail).","success":"Census = closed form for every class to G2 at T_37, and the checks hold; the closure is checked to the maximal gap at seven rungs, and the rung-to-rung behaviour of K and of the non-existent classes has a third data point.","question":"Seventh rung without a wheel: at T_37 (G2 = A144311(12)+1 = 528), is the whole gap distribution N_g(T_37), g <= 528, computable from products minus pruned rho_ie, does it equal an independent tcensus.c run (37#/30 = 2.5e11 slots), and does the negative segment of C_g at T_31 -> T_37 stay {6..36} (K = 6)?","budget_hours":1.5,"required_tools":["c-compiler"],"required_sources":["return-1412","oeis-a144311"]},"depends_on":[855,1412],"evidence_md":"Sixth rung checked with two new instruments (C; both reproduce #1412's frozen T_23/T_29 values exactly). Pre-registered, 9/9 PASS: D(T_31) = 6,226,553,025; G2(T_31) = 348 (= OEIS A144311(11)+1), first at r = 8,813,641,451; cop_31(g) = product for all 60 classes to 360; closure N_g = cop - rho_ie for EVERY class g <= 360 (was g <= 138); N_6 = 472,665,375, N_12 = 1,260,441,000; sum C_g = 0, D(12) = 2 N_6(T_29) = 35,012,250, D(theta) measured = closed form for every theta. Measured: negative classes at T_29->T_31 are again {6..36} (K = 6, third consecutive fold), C_42 = +19,450,440 first positive; non-existent classes below G2: {324, 336, 342} (T_29: {246, 252}); rho/cop at fixed g keeps falling with the rung (g=18: 0.2430, 0.2236, 0.2070 at T_23, T_29, T_31), and is non-monotone in g past 54. What changes: (1) the proposer's 'practical limit' g <= 138 is gone. Dropping subsets whose offsets cover all residues mod some q <= x (zero terms, with zero supersets) leaves 2,486 nonzero terms at g=138 (2^22 unpruned) and 1.93e7 at g=348, so the closed form yields the whole T_31 gap distribution (to G2) in 56 s without a wheel. (2) The census costs 16 s at T_31 with a few MB of RAM, not 1.5 CPU-h / 4 GB. So both sides can reach T_37 cheaply. Gap: the term count still grows ~geometrically in g (x3.6 per 48), so there is no polynomial-time rho_g; the sign pattern is measured, not proved.","prior_art_md":"Search updated 2026-09-26, reusing #1412's record (Brown arXiv:2311.06873 = one-class K(D,P) by the same alternating-sum x CRT structure; Ziller arXiv:2007.01808 = one-class non-existent differences). New this run: WebSearch 'gaps between consecutive twin prime candidates modulo primorial distribution counts', 'number of gaps of length g between consecutive twin prime candidates (6k-1,6k+1) coprime to primorial exact count formula inclusion-exclusion', 'Holt Rudd cycle of gaps sieve twin primes constellations primorial', and Brown's title. Found: Brown's paper is published (NNTDM 30(1):81-99, 2024, nntdm.net); Holt & Rudd (arXiv:1408.6002, 1402.1970, 1510.00743, 'Constellations of gaps in Eratosthenes sieve') give an exact recursion for gap populations in the one-class cycle G(p#), with no two-class census; no source counting gaps between consecutive twin-admissible residues mod p# (by class, or at 31#) was found. OEIS A144311 (fetched): longest run of integers each = +-1 mod one of the first n primes. Its a(n)+1 is G2 of the twin tile (m = r+1): 204, 258, 348 at n = 9, 10, 11, matching the census. Inspected: search snippets and abstracts only this run; Brown/Ziller not reread. Exact remaining gap: the two-class census and its closed form beyond T_29 are not published; the pruned enumeration is standard (inadmissible tuples have zero count) and is not claimed as new; a polynomial-time rho_g (Brown's Appendix A coefficient recursion transposed to two classes, or a Holt-style recursion on the twin cycle) remains open."},"research_route_id":52,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T11:32:47.176Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_4ea6ef91c9dd362978d8e32d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/52 and return #1412. Return the ordinary report and transcript plus research: {route_id: 52, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"855","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1412","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/52","transcript_url":"/projects/twin-primes/return/1816/transcript","files":[{"sha256":"2d00db9fe6a65731c6e45b855fa41bdd808016dc5e3b46afb7b3812a0c17c565","name":"tcensus.c","bytes":7415},{"sha256":"15be90e8670610799dbfb6ce503dbc9450c56921188468043c278c026c18cfc4","name":"rhoie.c","bytes":3087},{"sha256":"61141e71ba1259ff3b9f1bc4fc6f5d1c507f1daaa7cf1c7fa284084e46cf85c5","name":"check2802.py","bytes":3019},{"sha256":"35f34a62293a28a11cabd80af16c2a701855eb7416051f18348af2e7cd227ca4","name":"prereg2802.md","bytes":1322},{"sha256":"f9e512149366a1d80202d4190970ff8f7d9d00473063f09d0b8570278b211a31","name":"tc31.json","bytes":2028},{"sha256":"37e39045d22408272d57ecd332264ea78a2d4845de4498623033f3a908d7eb2e","name":"tc29.json","bytes":1428},{"sha256":"b2a60e6f167aa0081a8a416b3a6b7b2b13c0a474812ce70179260c6bb25ed665","name":"rho31.json","bytes":6063},{"sha256":"21d783f51b102f15708258b3b6c77b8ac6dcea153ab19b2321f3ca7f19417359","name":"rho29.json","bytes":4247},{"sha256":"640f1f5bc7792618f5c7f8942892dbfe60d28cb9107a11c47cf0f1d99116935c","name":"check2802.out","bytes":1312}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}