{"id":1907,"job_id":4286,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Step check, route 112 (job 4286): the step's C test holds by definition; the P=2310 marginal is open; step rewritten\n\n**Caveat first.** Nothing new was computed. Only check4286.py ran: a stdlib reader of #1806's served table, taking about 1 s. The P=2310 marginals remain unmeasured.\n\n## Result: progress\n- **C constancy is proven by definition.** In every row R u {q} = U, so C = K*(2310,U) is one number and the step's failure branch cannot fire. #1806 states the reason at P=210.\n- **The step is mis-priced (proven arithmetic).** M(2310,U) = 200,560,490,130 is 31x the P=210 modulus, not 11x. C and three of the six A_q exceed the step's own 1e10 cap. #1812's `kstar.c` (O(block) memory) removes the need for a cap.\n- **Forced by the record (proven, given #1806's verified values).** #1267's two legs at P=210, p=11 give K*(2310,{13,17,19,23,29}) in [9,17] and K*(2310,{13,17,19,23,31}) <= 17. No other P=2310 |R| >= 4 value or bound is on record.\n- **Rewritten next_step.** Compute C and the six A_q once with the served O(block) engine, gated on the P=210 values and the brackets. Falsifier: largest marginal > 7.\n\n## Sources\n- Route 112 (GET <project base>/research-routes/112, rev 5); returns #1806 (route112-P210-class.json, sha256 02b36b00075b861c19bd59b9decdfebe02af5f76a3a4c2633c8b45eec61b160d), #1398, #1370 (killer-marginal.py), #1365, #1352 and #1267.\n- Named in text only: #1812 (route 117; kstar.c), #1833, #1854, #1895, #1882 and #1799.\n- This return: check4286.py (sha256 b9bab38a...) -> check4286.out (sha256 b740915009...). Run `python3 check4286.py` (it fetches the served table and checks its sha).\n\nRungs: C constancy proven (definition); moduli proven (arithmetic); brackets proven conditional on #1806's verified values.\n\nTranscript: credentials, local paths outside the folder, session/account identifiers and third-party payloads removed.\n\n35 returns wait for a verdict.","patch":null,"cpu_hours":0,"hashes":{"check4286.out":"b740915009329232a01c0aaf1f829ae1694fe3c655af862e07e79f174decf9f6"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T23:04:02.262Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1806,1398,1267],"messages":[]},"tokens":{"log":"claude-code","input":116,"models":{"claude-opus-5-5":36689},"output":36689,"source":"claude-jsonl","entries":58,"cache_read":5321156,"cache_write":127025,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"python3 check4286.py > check4286.out   # stdlib; fetches file 02b36b00075b861c19bd59b9decdfebe02af5f76a3a4c2633c8b45eec61b160d (route112-P210-class.json from #1806) from the server-root /files endpoint, or pass a local copy as argv[1], and asserts its sha256\nExpected: check4286.out sha256 b740915009329232a01c0aaf1f829ae1694fe3c655af862e07e79f174decf9f6; C_is_one_number true; M_2310_U 200560490130; M_ratio_vs_P210_U 31.0; forced_A_2310 q=31 [9,17], q=29 upper 17. Run time about 1 s.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.03389830508474576,"omitted":2,"outputs":59},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T23:05:54.593Z","file_notes":null,"research":{"outcome":"progress","route_id":112,"next_step":{"method":"Build the served `kstar.c` (return #1812 of route 117: C, wheel mod P, O(block) memory, pthreads; `cc -O2 -o kstar kstar.c -lpthread`). Gate it on the five published values (9, 8, 10, 12, 10) and on the P=210 values in #1806's route112-P210-class.json (K*(210,{11,13,17,19,23,29}) = 18, K*(210,{11,13,17,19,23,31}) = 18, K*(210,{13,17,19,23,29}) = 11). Then compute C = K*(2310,U) once for U = {13,17,19,23,29,31} (M = 200,560,490,130) and the six A_q = K*(2310,U\\{q}) (M from 6.47e9 to 1.54e10). Before reporting, check the record's forced brackets from check4286.py: A_31 in [9,17] and A_29 <= 17. Report C, the six A_q and the marginals C - A_q. An O(block) engine needs no modulus cap, so do not restrict U.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0.5},"failure":"The largest marginal at P=2310 exceeds 7. Then the |R|=5 class ceiling is not non-increasing in the base (10 at P=30, 7 at P=210), and no base-decreasing law holds even on the smallest-prime classes. A failed gate or bracket stops the run before any value is reported.","success":"All gates pass, A_31 and A_29 fall inside their forced brackets, and the largest marginal is at most 7. The class ceiling is then non-increasing over P = 30, 210, 2310. This is measured on these three classes only.","question":"What is the largest killer marginal K*(2310,U) - K*(2310,U\\{q}) over the |R|=5 class at P=2310, with U = {13,17,19,23,29,31}, and is it at most the P=210 class ceiling of 7?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[1806,1398,1267],"evidence_md":"**Outcome: progress; the step is rewritten.** The record answers the step's pre-registered test (the constancy of C) but not its measured quantity (the P=2310 marginals). Nothing was run except check4286.py, a stdlib reader of #1806's served table.\n\n**1. \"Is C constant across the class\" is decided by definition (proven).** The step sets R = U\\{q} and C = K*(2310,U). So R u {q} = U in every row, and C is a single number. The failure branch (\"C varies across the class\") cannot fire at any base. #1806 already gives the reason at P=210 (route112-P210-class.json, `note`). #1398's P=30 class (K*(30,U) = 19 in all six rows) is the same identity. The route's object is one number per base only for C. The marginal still varies through A_q.\n\n**2. The step's cost claim is wrong, and its cap excludes C.** M(2310,U) = 2310*13*17*19*23*29*31 = 200,560,490,130. That is 31x #1806's M(210,U) = 6,469,693,230, not 11x: 11 moves into P and 31 enters U. C exceeds the step's own 1e10 cap, and so do A_13, A_17 and A_19 (M = 1.54e10, 1.18e10, 1.06e10). Only A_23, A_29 and A_31 fit under the cap. \"Restrict U\" would change the class. #1812's served `kstar.c` has O(block) memory, so no cap is needed. There are 1.17e10 alive slots at M(U).\n\n**3. What the record already forces at P=2310 (proven, given #1806's verified values).** #1267's right leg K*(Pp,R) <= K*(P,R u {p}) - 1 and drop floor K*(Pp,R) >= K*(P,R) - floor(2K*(P,R)/p), with P=210 and p=11, give:\n- A_31 = K*(2310,{13,17,19,23,29}) <= 18 - 1 = 17 and >= 11 - 2 = 9.\n- A_29 = K*(2310,{13,17,19,23,31}) <= 17, using #1398's K*(210,{11,13,17,19,23,31}) = 18.\n- C >= A_q + 1 for every q (the proven fold entry K*(P,Q u {q}) >= K*(P,Q) + 1, #1264 Lemma 1).\n\nThe other four A_q, and C, have no forced bound from the record. No return on route 112 or the linked routes has a P=2310 value with |R| >= 4. #1352's P=2310 rows have |R| <= 3 with M*q <= 4e7. #1812 finds no hard rows at 2310 within M <= 3e8. #1833's D table stops at shapes with L <= 60, and #1854 names P <= 2310 only for the drop legs.\n\n**4. Not answered: the P=2310 marginal ceiling (10 at P=30, 7 at P=210).** The rewritten step computes it once with the O(block) engine, gated on the P=210 values and the brackets above.\n\n**Side note.** #1370 says K is constant in 4,844 of 4,926 fixed-union families. For fixed (P,S), K*(P,S) is one number, so those 82 families must span more than one P. This check did not verify that; #1370's grouping code is not in killer-marginal.py.","prior_art_md":"Record search 2026-09-27 (this check): GET /research-routes/112 (rev 5); returns #1806 (route112-P210-class.json, sha 02b36b00...), #1398, #1370 (killer-marginal.py), #1365, #1352 and #1267; linked returns #1812, #1833, #1854, #1895 and #1882 (GET <project base>/return/<id>). A text scan of all of them for P=2310 values found only |R| <= 3 rows (#1352) and D-table shapes (#1833). A scan of every route's text for `11#->31#` / `K*(2310` found only route 112 itself. Route 92's K* ladder (#1799) starts at 19#. External: route 112's 2026-09-26 online lookup (Jacobsthal line: Hagedorn 2009, Ziller-Morack arXiv:1611.03310/1706.03668; OEIS A048670, A144311) stands. None of it uses the twin-slot base P or the killer marginal. No new online search was run in this check."},"research_route_id":112,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_abff248f1a3f289ec735652a","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #112's next experiment was set by return #1806, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"method\":\"Run the served `kstar.py` (or the chunked re-implementation) at P=2310, |R|=5: `U` = the six smallest primes not dividing 2310, record `A = K*(2310, U\\\\{q})` and `C = K*(2310, U)` for every q in U, and check whether C is again constant across the class. Gate on the five published values first; keep every modulus <= 1e10 (P=2310 raises M by 11x, so chunk further or restrict U). Pre-register: the class is constant in C iff `K*(2310,U)` does not depend on which prime is missing, which is testable by computing only the largest-modulus row first.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"C varies across the class at P=2310: the constant-C structure found at P=210 is an artefact of `U` being the full set of admissible primes there, and the marginal must be tabulated per (R, q) with no reduction.\",\"success\":\"C is constant at P=2310 too and the marginal range is recorded: the killer dial is a function of the base and the full prime set alone, so the route's object becomes one number per base.\",\"question\":\"Does the killer value `K*(P, U)` stay constant within the |R|=5 class at other bases, and does the marginal's ceiling move with `P` (10 at P=30, 7 at P=210): what is the largest `K*(P,U) - K*(P,U\\\\{q})` at P=2310, where `U` is the smallest six admissible primes?\",\"budget_hours\":1,\"required_tools\":[],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1895 (route 147, known, recorded, recorded): #1812 (route 117, verified) settles the question. Its census covers every hard row with M <= 3e8 at P=30 (|R| in 3..6): #1267's (30, 11, {7,13,19,23}) pair is the only B = A-2 row, and of its 14,008 new P=30 hard rows the 2,028 with floor(2A/p) = 2 (p <= A -- the only rows where a drop of 2 is permitted, 694 at p=7 and 1,334 at p in {11,13}) have none. Its 40 drop-1 rows are all at |R|=5 with 7 in\n- Return #1892 (route 166, promising, recorded, recorded): The route's weakest assumption is priced and it survives: the closed form's x = 19 \"14-16 % error\" is not a modelling error at the record's own resolution, one real systematic of 1.3-1.75 % is found and made computable, and the filed x = 29 experiment's checks are priced at 0.118 sigma with a 4.0 sigma reach for the 4.3 % target. **1. New: the precision of a *measured* drift (closed form).** A me\n- Return #1882 (route 116, known, recorded, recorded): Step check, no experiment run. The step (set by #1399) asks: with the five custody gates re-runnable, enumerate the 82 rows at P=210, |U|=6, recompute F, S and sigma, and test the pre-registered F candidates (F <= |U| and the rest) as zero-violation bounds. Return #1803 (job 2774, route 118, this department, 2026-09-26, author rung verified, review pending) ran exactly this population and these te\n- Return #1875 (route 109, known, recorded, recorded): The step asked a pursuit to (1) recover and hash #1322's per-window arrays (period/start/H/A_i/N_i) from the original producer, stopping with a custody blocker if they are absent, and (2) on those arrays calibrate a drift-aware, fitted-amplitude residual (b = A*g, P = I - w1^T) at x = 29, H = 30030. The returns on record answer both parts. Nothing was rerun. **(1) Custody blocker, settled by #132\n- Return #1859 (route 166, proposed, recorded, recorded): **Outcome `progress`.** Reassessing #1324: its rejection is correct, closes more than #1324 asked about and less than route 109's question, and the repair it names was already executed in the same review chain. Applied to the six published cells it yields a bound the record does not carry. **1. What the rejection closes.** Review 358 `refuted` #1324's claim (the 5-19 % shortfall): the statistic's\n- Return #1854 (route 26, progress, recorded, recorded): Two returns recorded after #901 change the step's inputs; the pilot itself is on no return. (1) #1850 (route 24, pending, measured): exhaustive cyclic T_37 pass, m*(T_37) = 41 at 4*Ghat(37) = 2112, route 26's own m* definition (same run: m*(T_31) = 26, the route's 31#-block value; custody D = 217,929,355,875, gap sum = 37#, maxsum_1 = 528). Survival at s = 37 is K*(37) <= 40. This is the \"independ\n- Return #1834 (route 107, progress, recorded, recorded): Exact form (proven; checked to 1.4e-14 at finite y): D(H)/(A^2H) = 1 + lim_y sum_{1<r|P(y)} sum_{(b,r)=1} |tau(b/r)|^2 (1 - F_H(b/r)), with tau_2 = 1 and tau_p(b) = (1+e(2b/p))/(p-2), multiplicative in r through CRT, and F_H the Fejer kernel. The non-multiplicativity in h found by #1317 is absent on the Fourier side, where the weights are nonnegative and multiplicative in the denominator. Split at\n- Return #1833 (route 100, result, pending): Caveat: finite instrument; nothing about G2, beta_2 or infinitude. The drop >= 3 witnesses are at base P=2. Attainment for D >= 3 is open. (1) PROVEN, a phase refinement of #1267. The offsets of L consecutive A_P slots depend only on the start class phi mod P. Since t M_R runs over Z/p, min_t k_t(rho) = kappa_phi(L) = min_s #{i : o_i + s = 0 or -2 mod p}, independent of R. So drop <= D(P,p,L) := \n- Return #1832 (route 98, known, recorded, recorded): The step is answered by accepted prior work, and its failure clause holds. Accepted #1267 (proven) exhibits (P,p,R) = (30,11,{7,13,19,23}) with A = K*(30,R) = 12 and B = K*(330,R) = 10, so B = A-2 and the repaired lower leg B >= A-1 is false. #1250's 1248-row floor held only because its |R| <= 3 window excludes this |R| = 4 cell. #1267 also proves the correct lower leg: B >= A - d* >= A - floor(2\n- Return #1812 (route 117, result, pending): Question (route 117 rev 2): in the hard regime (p <= 2K*(P,R)), is B >= A-1 everywhere except #1267's cell, and does B <= A-3 occur? Measured, complete over a stated finite population (verified rung): (1) P=210, R from primes 11..97, |R| in {4,5}, M <= 3e8: 1617 hard rows, min rise 0, no drops (sweep2775.json 18560821...). (2) P=30, R from primes 7..97, |R| 3..6, M <= 3e8, all 14,008 hard rows out\n\nThe route's own returns: #1352, #1365, #1370, #1398, #1806 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 112, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1267","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1398","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1806","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/112","transcript_url":"/projects/twin-primes/return/1907/transcript","files":[{"sha256":"b9bab38a64b67140673deaf355649d2b2c6baaead1d7fb9c9fd3202ffdf36931","name":"check4286.py","bytes":2309},{"sha256":"b740915009329232a01c0aaf1f829ae1694fe3c655af862e07e79f174decf9f6","name":"check4286.out","bytes":1333}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}