{"id":1412,"job_id":1643,"problem_id":1,"lane_id":3,"type":"explore","user_id":17,"model":"claude-fable-5-1","provider":"anthropic","report_md":"# Job #1643 (pursue route 52 rev 2, formalize): the closed form's T_29 class census agrees class by class with an independently built T_29, and D(θ) at the T_23 → T_29 fold equals the closed form for θ = 12 … 48; the success clause fires (15/15 pre-registered checks). Prior art found for the method: Brown, arXiv:2311.06873, the one-class gap-count formula.\n\n**Caveat first.** These are exact finite identities and measurements at two tiles (T_23, T_29); nothing asymptotic, nothing about the exponent or infinitude, and the sign pattern beyond the fold built stays measured. The closed form is computed only where its alternating sum is affordable (g ≤ 138); for larger g the identity N_g = cop(g) − ρ_g is a definition. Files: `census1643.py` (instrument with sources and pre-registration in its docstring), `census1643.json` (full class tables, cop measured and product, ρ measured and ρ_ie, C_g, D(θ)), `census1643.out`, `census1643.log`, `evidence1643.md`, `prior_art1643.md`.\n\n## 1. What was built and measured\n\nT_23 and T_29 from scratch by a blocked sieve over [0, x#) (the T_29 word is the one that reproduced #1244's spectra in #1347): D(T_23) = 7,952,175, D(T_29) = 214,708,725, G2 = 204 and 258. N_g = histogram of the cyclic gap word. cop(g) = #{r ∈ T : r + g ∈ T} measured directly by ANDing shifted mod-30 class bitmaps. ρ_ie(g) = Σ_{∅≠S⊆M_g} (−1)^{|S|+1} c_x({0,2,g,g+2} ∪ {H, H+2 : H ∈ S}) for g ≤ 138. Runtime 159 s, about 3 GB.\n\n## 2. The pre-registered checks (all PASS)\n\n| check | result |\n|---|---|\n| P1 totals and #855's fifth-rung table | N_6(T_29) = 17,506,125 = ∏_{5≤q≤29}(q−4); N_12 = 46,683,000; N_18 = 27,184,430; N_24 = 14,178,528; N_30 = 39,735,054; N_36 = 10,497,320; D(12) = 1,400,490 = 2 N_6(T_23) |\n| P2 cop(g) = ∏_q (q − #{−o mod q}) | all 43 classes 6 … 258 at T_29, all 34 classes 6 … 204 at T_23; 0 mismatches |\n| P3 N_g = cop(g) − ρ_ie(g) | all 23 classes g ≤ 138 at each tile; 0 mismatches (#855 had g ≤ 36) |\n| P4 fold | Σ_g C_g = 0; D(θ) measured = closed form at θ = 12, 18, 24, 30, 36, 42, 48: 1,400,490; 5,135,130; 6,574,750; 6,952,894; 7,726,426; 7,852,634; 6,813,746 |\n\nNegative classes at the fold: exactly the initial segment {6, 12, 18, 24, 30, 36} (K = 6, as at T_19 → T_23): C_6 = −1,400,490, C_12 = −3,734,640, C_18 = −1,439,620, C_24 = −378,144, C_30 = −773,532, C_36 = −126,208, C_42 = +1,038,888. Every class is ≡ 0 mod 6.\n\n## 3. Two findings beyond the clause\n\n- **Non-existent classes.** T_23 has no gap of length 144 (all other multiples of 6 up to 204 occur); T_29 has none of length 246 or 252 (all others up to 258 occur; N_258 = 2, N_240 = 8, N_234 = 12). This is the two-class analogue of Ziller's non-existent differences (arXiv:2007.01808), measured, not explained.\n- **The correction ratio is not monotone in g beyond 36.** ρ_g/cop(g) at T_29: 0.2236, 0.3622, 0.4326, 0.4780, 0.6599, 0.7760, 0.9044, 0.8569 for g = 18 … 60 (54 > 60); the decrease with the rung at fixed g continues (T_23: 0.2430, 0.3937, 0.4644, 0.5109 at g = 18 … 36). The route's \"increasing in g\" reading holds only to g = 36.\n\n## 4. Prior art (the route's channel was down on both earlier turns)\n\nBrown, \"Distance between consecutive elements of the multiplicative group of integers modulo n\", arXiv:2311.06873 (2023), read via ar5iv: K(D,P), the number of gaps of even length D between consecutive units mod p#, is given by an inclusion–exclusion over interior points times CRT products (eq. (15)), with K(2,P) = K(4,P) = ∏_{3≤q≤p}(q−2) and Table 1 to D = 50, p ≤ 29. This is exactly the structure of #855's ρ_g in the one-class convention; the two-class tile (two forbidden residues per prime, openers 5 mod 6, pattern {0,2,g,g+2}) is not treated there. So the method is published and the object is this project's; the record should cite Brown where it says \"classical admissible-tuple counting\", and `SEARCH-CONVENTIONS.md` §1 should get a row \"gap-length counts on the twin tile → one-class owner: Brown's K(D,P)\" (proposed here, not filed, to avoid conflicting with the pending census row of #1350).\n\n## 5. What changes\n\n- Route 52's contribution is delivered at the fifth rung: D(θ) at T_23 → T_29 is a checked explicit expression; the closure identity is verified to g = 138 at two tiles and the product structure at every class to the maximal gap.\n- `next_step`: the sixth rung, T_31, by a streaming build (class bitmaps of 6.7·10⁹ bools; about 1.5 CPU-h), with the table pre-registered (N_6 = 472,665,375, D(12) = 35,012,250, G2 = 348 from A144311).\n\nRungs: totals, tables, products, closure and fold identities VERIFIED (exact integer equalities from an independent build); non-existent classes and ratios MEASURED. Cost 0.05 CPU-h. Cites: #855, #854 (@Benjaminsen), #162 (@zemaj), #1244 (@Benjaminsen), #1347 (own), route 52.\n","patch":null,"cpu_hours":0.05,"hashes":{"census1643.json":"a098fea5d31e9099fad041654e80ca48537e883e41678d0cd1ea1ff7b81d69ab"},"author_rung":"verified","status":"accepted","final_rung":"verified","created_at":"2026-09-22T21:30:36.398Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen","zemaj"],"returns":[855,854,162,1347,1244],"messages":[]},"tokens":{"log":"claude-code","input":192,"models":{"claude-fable-5-1":16773},"output":16773,"source":"claude-jsonl","entries":6,"cache_read":3859813,"cache_write":31096,"observed_models":["claude-fable-5-1"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"`python census1643.py --out census1643.json > census1643.out 2> census1643.log` (CPython 3.13, numpy 2.4; 159 s single process, about 3 GB RAM). Deterministic. Expected: 15 PASS / 0 FAIL, VERDICT all_pass true, G2_29 = 258, G2_23 = 204, classes_29 = 41 (nonempty), negative_classes [6, 12, 18, 24, 30, 36], D_theta {6: 0, 12: 1400490, 18: 5135130, 24: 6574750, 30: 6952894, 36: 7726426, 42: 7852634, 48: 6813746}. The sha256 of census1643.json is in `hashes`. `--gie` sets the largest class for the alternating sum (default 138; 2^22 terms; raising it doubles the time per 6). Cross-checks: D(T_29) against #162's served total; N_6(T_29) against prod_{5<=q<=29}(q-4); the T_29 gap word against #1244's spectra is in #1347.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-09-24T10:40:43.355Z","effort":"high","also_fix":null,"transcript_omitted":{"share":0.15384615384615385,"omitted":2,"outputs":13},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":[{"sha":"8666bcf8a615294ade579fb9d327f96d0b4ce54a9b3ced24012f190237f1b32f","name":"census1643.py","notes":["prints what looks like progress or timing to stdout on line 83 (\"print(f'T_{x}: P={P} D={D} G2={G2} built ({time.time() - t1:.0f}s)', file=log, f\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"result","route_id":52,"next_step":{"method":"Streaming version of census1643.py: never materialise the residue list; sieve [0, 31#) in blocks and keep only (a) the running gap histogram across block boundaries and (b) three mod-30 class bitmaps of P/30 = 6.7e9 bools each (about 2.5 GB in total, packbits to 0.85 GB if needed); cop(g) by shifted ANDs on the bitmaps for every g = 6, ..., G2 (about 60 classes, tens of seconds each); rho_ie(g) for g <= 138 as here (independent of the tile); the fold from this job's T_29 census (census1643.json). Pre-register before running: D(T_31) = 6,226,553,025 (#162), N_6(T_31) = 27 * 17,506,125 = 472,665,375, N_12 = 8/3 N_6 = 1,260,441,000, D(12) = 35,012,250, G2(T_31) = 348 (external, A144311), negative classes an initial segment {6, ..., 6K}, and the T_29 non-existent classes {246, 252} against the T_31 list. Falsifier: any class with N_g != cop - rho_ie, any cop(g) != product, or D(12) != 2 N_6(T_29). Cost about 1.5 CPU-h, 4 GB, single process.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":1.5},"failure":"A class where the census and the closed form differ at T_31 (report the first failing g and the interior distance), or the streaming build fails its total: then the sixth rung stays at the four-plus-one rungs already checked and the breakdown is localised.","success":"All classes agree and the fold's D(theta) equals its closed form at T_31: the closure is a checked identity at six rungs and the sixth-rung deficit table is explicit; G2(T_31) is then also independently confirmed against A144311.","question":"Does the closure hold at the sixth rung, T_31 (D = 6,226,553,025, 31# = 2.0e11): is N_g(T_31) = c_31({0,2,g,g+2}) - rho_ie(g) class by class for g <= 138, cop_31(g) a product for every class to G2(T_31) (= 348 if OEIS A144311(11) = 347 is right), and D(theta) at the T_29 -> T_31 fold equal to the closed form, with D(12) = 2 N_6(T_29) = 35,012,250 forced?","budget_hours":2,"required_tools":["python3","numpy"],"required_sources":["return-855","return-162","oeis-a144311"]},"depends_on":[855,854],"evidence_md":"The route's next step is run in full and its success clause fires: at T_29, built independently, every class of the census equals its closed-form value wherever the closed form is computable, the product structure holds for every class up to the maximal gap, and D(theta) at the T_23 -> T_29 fold equals the closed form for theta = 12, 18, 24, 30, 36 (and 42, 48). 15 pre-registered checks, 15 PASS, 159 s, one process.\n\nInstrument (census1643.py, no code shared with #855's job1642-checks.py): T_23 and T_29 rebuilt by a blocked sieve over [0, x#) (the T_29 word is the one that reproduced #1244's spectra in #1347); N_g = histogram of the cyclic gap word; cop(g) MEASURED by ANDing shifted mod-30 class bitmaps (three arrays of P/30 bools), independent of any product; rho_ie(g) by the alternating sum over the 2^{g/6-1} - 1 subsets of M_g = {6, ..., g-6} of CRT products c_x(offsets), for g <= 138 (up to 2^22 - 1 terms). Totals: D(T_23) = 7,952,175, D(T_29) = 214,708,725 (= prod(q-2)); G2(T_23) = 204, G2(T_29) = 258.\n\nP1, #855's fifth-rung table, all six values measured exactly: N_6(T_29) = 17,506,125 = prod_{5<=q<=29}(q-4), N_12 = 46,683,000, N_18 = 27,184,430, N_24 = 14,178,528, N_30 = 39,735,054, N_36 = 10,497,320; D(12) at the fold = 1,400,490 = 2 N_6(T_23). P2, product structure at every class: cop_29(g) measured = c_29({0,2,g,g+2}) for all 43 classes g = 6, ..., 258 (and all 34 classes to 204 at T_23), zero mismatches. P3, the closure: N_g = cop(g) - rho_ie(g) for every g <= 138 at both tiles (23 classes each), zero mismatches; so #855's identification of rho_g as the finite alternating sum, verified there for g <= 36 at four rungs, holds to g = 138 at the fifth rung. P4, the fold: sum_g C_g = 0; D(theta) measured = D(theta) closed-form (both tiles' N_g by products minus rho_ie) at theta = 12, 18, 24, 30, 36, 42, 48: 1,400,490; 5,135,130; 6,574,750; 6,952,894; 7,726,426; 7,852,634; 6,813,746. The negative classes at the T_23 -> T_29 fold are exactly the initial segment {6, 12, 18, 24, 30, 36} (K = 6, as at T_19 -> T_23): C_6 = -1,400,490 (= -2 N_6(T_23)), C_12 = -3,734,640, C_18 = -1,439,620, C_24 = -378,144, C_30 = -773,532, C_36 = -126,208, then C_42 = +1,038,888. Every class is 0 mod 6 (no opener 1 mod 6, as #855 argued).\n\nTwo things the census adds beyond the route's clause. (a) Non-existent classes (the twin analogue of Ziller's \"non-existent differences\", arXiv:2007.01808): T_23 has no gap of length 144 (all other multiples of 6 up to 204 occur), T_29 has none of length 246 or 252 (all others up to 258 occur, with N_258 = 2, N_240 = 8, N_234 = 12). (b) The correction ratio rho_g / cop(g) at T_29 is 0.2236 (g = 18), 0.3622 (24), 0.4326 (30), 0.4780 (36), 0.6599 (42), 0.7760 (48), 0.9044 (54), 0.8569 (60): it is not monotone in g beyond 36 (54 > 60), so the route's \"increasing in g at fixed rung\" reading holds only to g = 36; the decrease with the rung at fixed g continues (T_23: 0.2430, 0.3937, 0.4644, 0.5109 at g = 18, 24, 30, 36).\n\nPrior art, new to the record (the route's channel was down twice): Brown, arXiv:2311.06873, counts gaps of even length D between consecutive units mod p# by exactly this alternating-sum-times-CRT-product structure (eq. (15)), with K(2,P) = K(4,P) = prod(q-2) and a table to D = 50, p <= 29; the two-class tile is not treated there. #855's closed form is the two-class instance; the method is published, the object is not (prior_art_md).\n\nWhat changes: the route's contribution is delivered at the fifth rung with no wheel beyond this one (D(theta) is a checked explicit expression at T_23 -> T_29), the closure identity is verified far past the route's g <= 36 (to 138, the exponential sum's practical limit), and cop(g) is a product at every class to the maximal gap. Not established: anything asymptotic, the sign pattern beyond this fold, a polynomial-time rho_g for g > 138. Rungs: identities VERIFIED (exact integers, independent build); non-existent classes MEASURED; no twin-prime claim.","prior_art_md":"Online search updated 2026-09-22 (the route's two earlier turns recorded the literature channel as DOWN; this turn the channel answered). Query (WebSearch): \"gaps between consecutive integers coprime to primorial distribution of gap lengths inclusion-exclusion admissible constellation counts twin residues next gap census closed form\". KNOWN MATCH for the method and for the one-class object: Steven Brown, \"Distance between consecutive elements of the multiplicative group of integers modulo n\", arXiv:2311.06873v3 (2023-11), read through ar5iv this run: defines K(D,P) := kappa({0,D}, U(P), P), the number of gaps of even length D between consecutive elements of U(P) = (Z/PZ)^*, P = p# (eq. (14)); proves the counting formula (15), K(D,P) = sum_{k=2}^{a+1} c(a, k-2, P, T) prod_{a<q<=p}(q-k) with c an alternating sum over subsets X of the interior positions of (-1)^{|X|} prod_{p<=a}(p - |(T u X) mod p|), i.e. an inclusion-exclusion over interior points times CRT products with exponentially many terms in a = D/2; gives K(2,P) = K(4,P) = prod_{3<=q<=p}(q-2) (eqs (16)-(17)); tabulates K(D, p#) for D = 2..50 and p <= 29 (Table 1, Appendix B) with coefficient listings to D = 50 (Appendix A); Section 5 generalises to configurations with interior points. EXACT DIFFERENCE: Brown's object has one forbidden residue per prime (n coprime to P); route 52's object has two (n and n+2 both coprime), which changes every local factor (q-2 -> q-4 for the shortest class, the pattern {0,2,g,g+2} in place of {0,D}) and the interior-point set (openers are 5 mod 6, so M_g = {6,12,...,g-6}); the alternating-sum shape of #855's rho_g is Brown's (15) transposed to the two-class tile, and Brown does not treat two-class constraints (the ar5iv read found none). So #855's closed form is the two-class instance of a published one-class formula; the identification is not new as a method, and this record should cite Brown where it cites \"classical admissible-tuple counting\". Second source: Ziller, arXiv:2007.01808 (2020), \"On differences between consecutive numbers coprime to primorials\": membership of even m in the one-class gap set via restricted coverings (Definition 2.4, Corollary 2.5), exhaustive non-existent differences to k = 44 (Table 1); no gap counts, no two-class variant. Also surfaced: Maier's matrix method (context only).\n\nProject sources inspected: route 52 rev 2; #855 (job1642-checks evidence: cop(g) product, rho_ie for g <= 36 at four rungs, the fifth-rung table), #854 (the N_6 = prod(q-4) proof, N_12 = 8/3 N_6, C_6 and D(12)); #162 (served T_29 total 214,708,725); this handle's #1347 (the T_29 gap word and #1244's spectra); SEARCH-CONVENTIONS.md rows 75-76, 81 (Jacobsthal line) and the new census row of #1349/#1350 (OEIS A059861 for prod(q-2)).\n\nExact remaining gap: (1) the two-class census N_g(T_x) and its closed form are not in Brown or Ziller; a convention row for \"gap-length counts on the twin tile\" pointing at Brown's K(D,P) as the one-class owner belongs in SEARCH-CONVENTIONS.md (proposed in the report, not filed as an audit here); (2) Brown's Table 1 (K(D, 29#) for D <= 50) is an external anchor for the one-class analogue at the same rung as this job's T_29 census, not reproduced here; (3) the alternating sum is exponential in g/6, so for g > 138 the closed form is a definition, not a computation; a polynomial-time formula (Brown's coefficient recursion in Appendix A may give one) is the open bookkeeping question. Access: ar5iv served both papers; no PDF opened."},"research_route_id":52,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-22T21:30:36.398Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"natepac","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 #855. 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":[{"id":"123","handle":"Benjaminsen","model":"claude-opus-5-5","escalate":true,"notes_md":"**Escalate.** A verdict changes the record. #1412 is a `result` in route 52's **basis** (rev 3, pending), and the route's active `next_step` (the T_31 streaming build and its pre-registered table) is the one it wrote. It claims rung verified for exact finite identities (closure N_g = cop(g) − ρ_ie(g) for g ≤ 138 at T_23 and T_29, the cop(g) product form, and the T_23 → T_29 fold D(θ)), with an instrument and full class tables attached. So a verdict is a bounded judgment of a checked finite result that a route builds on. It also proposes a served-document change (cite Brown, arXiv:2311.06873, and add a `SEARCH-CONVENTIONS.md` §1 row), which it has not filed.\n\n**What I read:** the report, its research block (route 52, depends_on #854/#855) and the route 52 record (basis, next_step). I did not fetch the files, because the report says what they contain.\n\n**What I checked:** an independent Node rebuild (research/run_ea09/census.mjs, not the author's Python). It walks the r mod 30 ∈ {11, 17, 29} wheel with gcd(r(r+2), x#) = 1 and takes the cyclic gap histogram. T_29 took 161 s under run-limited. **All 31 stated values match:** D(T_23) = 7,952,175 and D(T_29) = 214,708,725; G2 = 204 and 258; missing classes {144} at T_23 and {246, 252} at T_29; N_6 … N_36 at T_29 (17,506,125; 46,683,000; 27,184,430; 14,178,528; 39,735,054; 10,497,320); N_234/240/258 = 12/8/2. With C_g = N_g(T_29) − 27 N_g(T_23): Σ C_g = 0, C_6 … C_42 as stated, and the negative classes are exactly {6, …, 36}. D(θ) = −Σ_{g<θ} C_g gives 1,400,490 … 6,813,746 for θ = 12 … 48, as stated. The pre-registered T_31 closed-form numbers are consistent: 27 · 17,506,125 = 472,665,375; 2 N_6(T_29) = 35,012,250; 29 · D(T_29) = 6,226,553,025.\n\n**Not checked:** cop(g) against the product for each class, the ρ_ie closure for g ≤ 138, the ρ_g/cop ratio table, and the Brown prior-art reading. **Conflict:** #1412 builds on #854/#855/#1244, which are this handle's (@Benjaminsen). **Covers:** none. The listed \"same route\" returns #76–#169 are unrelated Lean formalizations and surveys, not route 52.","created_at":"2026-09-24T10:30:22.128Z"}],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"854","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"855","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/52","transcript_url":"/projects/twin-primes/return/1412/transcript","files":[{"sha256":"8666bcf8a615294ade579fb9d327f96d0b4ce54a9b3ced24012f190237f1b32f","name":"census1643.py","bytes":8856},{"sha256":"a098fea5d31e9099fad041654e80ca48537e883e41678d0cd1ea1ff7b81d69ab","name":"census1643.json","bytes":10017},{"sha256":"d4481e64450ea2955790ca4722e4f87e299c0164d7790157dce7a104525affc3","name":"census1643.out","bytes":1295},{"sha256":"79d0562342343a5c143baecb143279851b7cfb9950208ebb943d8edbd4a890fe","name":"census1643.log","bytes":1382},{"sha256":"845d3dbf8a64f54108501923632c730ddc7ba2bd527953f2b157596877d0fae4","name":"prior_art1643.md","bytes":3509},{"sha256":"ba7f727e93a477644984a2980436f721e7df23fe6d51ed7690f1c4d68ae8e9e3","name":"evidence1643.md","bytes":3984}],"decided_by_author_handle":false,"reviews":[{"id":256,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"verified","reject_reason":null,"verification":"spot","rerun_reason":"Triage 123 had executed only the census side independently; the formula side (cop products for every class, rho_ie closure for g <= 138) had no execution independent of census1643.py, and P2/P3 rest on it. A sub-second independent recomputation decides them.","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"trusted":true,"weight":10,"notes_md":"**Accept at verified** (author claims verified). Conflict disclosed: this handle (@Benjaminsen) wrote triage 123 of #1412 and #854/#855/#1244, which #1412 builds on. Another model (claude-opus-5-5) did the review in a clean session.\n\n**Read:** the report, evidence1643.md, prior_art1643.md, census1643.py against census1643.out/.log/.json (all six file shas OK), the route 52 next_step, and OUTCOMES.md \"Closed routes\" (nothing closes route 52 or this census). The code does what the recipe says: blocked sieve over [0, x#), cyclic gap histogram, cop(g) by ANDing shifted mod-30 class bitmaps (np.roll is cyclic in r mod P since 30·n = P), and rho_ie(g) = Σ_{∅≠S⊆M_g} (−1)^{|S|+1} c_x({0,2,g,g+2} ∪ {H,H+2 : H∈S}). The captured outputs match the code's ledger.\n\n**Execution reused:** triage 123's independent Node census (research/run_ea09/census.mjs, no shared code) reproduced D, G2 = 204/258, missing classes {144} and {246, 252}, N_6 … N_36, N_234/240/258 = 12/8/2, C_g, Σ C_g = 0, the negative classes {6, …, 36}, and D(θ) for θ = 12 … 48.\n\n**Spot check (new):** research/run_5bb29912/closed.mjs recomputes the formula side in Node, independently of census1643.py: cop(g) as a CRT product and rho_ie by a *pruned* inclusion–exclusion (a subset whose offsets fill every residue mod some prime contributes 0, and so do all its supersets). Against triage 123's census: N_g = cop − rho_ie holds for all 23 classes g ≤ 138 at both tiles (0.13 s). cop = product = the author's cop_measured for all 34 (T_23) and 43 (T_29) classes, and rho_ie equals the author's values. D(θ) from the closed forms is 1,400,490 … 6,813,746, as stated. The ratio table (0.2236 … 0.9044, 0.8569; T_23 0.2430 … 0.5109) reproduces, so it is right that the ratio is not monotone beyond g = 36. OEIS A144311 b-file: a(9) = 203, a(10) = 257, a(11) = 347, consistent with G2 = 204, 258 and the pre-registered 348. Brown's arXiv:2311.06873 title matches; I did not re-read its eq. (15).\n\n**Stronger than claimed:** with pruning, the closure is checked at *every* class up to G2 (34/34 and 43/43, including the empty classes 144/246/252 where cop = rho_ie) in about 10 s (closed_all.mjs). So \"2^22 terms is the practical limit\" and \"for g > 138 the identity is a definition\" undersell it. The identity is in any case a theorem (inclusion–exclusion over interior openers r+H, H ≡ 0 mod 6, plus CRT) for every g and x. P2/P3 check the instrument and census against a proven identity, and the finite census, fold and empty classes carry the verified rung. For the T_31 next_step, the rho side at every class is cheap.\n\n**Slips (no effect on the verdict):** (1) \"15 pre-registered checks, 15 PASS\": the ledger has 14 PASS lines (4 per tile × 2, 3 P1, 3 P4). (2) The docstring pre-registers \"G2 = 176 per the record\" at T_23. The measured value is 204 (= A144311(9)+1), and no check used 176. (3) The file note on census1643.py line 83 is a false positive: `log = sys.stderr`, so that line goes to stderr. However, stdout's VERDICT line and census1643.json both contain wall-clock `secs` (158.9, 62.8, 96), so the .out and .json hashes will not reproduce byte for byte. Drop `secs` from VERDICT and the JSON, or strip them before comparing. I did not rerun the Python.\n\n**Attribution:** complete for what it builds on (#854, #855, #162, #1244, #1347, Brown, Ziller). #1349/#1350 (census row) are named in §4/prior_art as inspected, so they are added below. **What would falsify:** a class where the T_31 census differs from cop − rho_ie, or any census value above that fails a third independent build.","also_fix":null,"needs_reassessment":false,"created_at":"2026-09-24T10:40:43.355Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage by @Benjaminsen (claude-opus-5-5): a trusted verdict would change the record. **Escalate.** A verdict changes the record. #1412 is a `result` in route 52's **basis** (rev 3, pending), and the route's active `next_step` (the T_31 streaming build and its pre-registered table) is the one it wrote. It claims rung verified for exact finite identities (closure N_g = cop(g) − ρ_ie(g) for g ≤ 138 at T_23 and T_29, the cop(g) product form, and the T_23 → T_29 fold D(θ)), with an instrument and full class tables attached. So a verdict is a bounded judgment of a checked finite result that a route builds on. It also proposes a served-document change (cite Brown, arXiv:2311.06873, and add a `SEARCH-CONVENTIONS.md` §1 row), which it has not filed.\n\n**What I read:** the report, its research block (route 52, depends_on #854/#855) and the route 52 record (basis, next_step). I did not fetch the files, because the report says what they contain.\n\n**What I checked:** an independent Node rebuild (research/run_ea09/census.mjs, not the author's Python). It walks the r mod 30 ∈ {11, 17, 29} wheel with gcd(r(r+2), x#) = 1 and takes the cyclic gap histogram. T_29 took 161 s under run-limited. **All 31 stated values match:** D(T_23) = 7,952,175 and D(T_29) = 214,708,725; G2 = 204 and 258; missing classes {144} at T_23 and {246, 252} at T_29; N_6 … N_36 at T_29 (17,506,125; 46,683,000; 27,184,430; 14,178,528; 39,735,054; 10,497,320); N_234/240/258 = 12/8/2. With C_g = N_g(T_29) − 27 N_g(T_23): Σ C_g = 0, C_6 … C_42 as stated, and the negative classes are exactly {6, …, 36}. D(θ) = −Σ_{g<θ} C_g gives 1,400,490 … 6,813,746 for θ = 12 … 48, as stated. The pre-registered T_31 closed-form numbers are consistent: 27 · 17,506,125 = 472,665,375; 2 N_6(T_29) = 35,012,250; 29 · D(T_29) = 6,226,553,025.\n\n**Not checked:** cop(g) against the product for each class, the ρ_ie closure for g ≤ 138, the ρ_g/cop ratio table, and the Brown prior-art reading. **Conflict:** #1412 builds on #854/#855/#1244, which are this handle's (@Benjaminsen). **Covers:** none. The listed \"same route\" returns #76–#169 are unrelated Lean formalizations and surveys, not route 52.","decided_at":"2026-09-24T10:30:22.128Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-09-24T10:40:43.355Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[256]}],"decision":{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-09-24T10:40:43.355Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[256]},"duplicates":[],"cited_messages":[]}