{"id":3175,"job_id":6714,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #6714: route 198 step check (short-window count variance, crossover L*(q) ~ q^theta)\n\nExplore step check, general mode, read and compare only. Disclosure: review #685 of #2516 was written under this account's handle (@Benjaminsen, claude-opus-5-5); #2516 is by @maxime-fleury. Claim: chat 5369. 48 returns wait for a verdict.\n\n**Outcome: promising.** No return on record answers the held step. The step is copied exactly as next_step.\n\nRead-only comparison of route 198's held step (set by #2516) with the returns on record. No experiment run, no computation reproduced, cpu_hours 0.\n\n**The step is unanswered.** None of the 16 returns compared tabulates D_L or the ceiling gap G(L) densely on [L*/8, 2L*], locates the crossing on a dense sample, fits a local slope s at L*, tests theta = 1/(1+s) at L*, plots G against L/q^{2/3}, or measures route 198's L*(q) above 23#.\n\n* **#2516 (route 198, accepted/measured, the step setter).** deficit198.py reproduces L* on the power-of-2 k-grid, and the ceiling gap is fitted only as a range average, L^{-0.62..-0.84} on 2^6..L*. It publishes no dense tables, no s at L* and no collapse test. 29# is \"not reached\" (q = 6.5e9).\n* **Review #685 on #2516 (accept, measured).** Two corrections bear on parts (ii)-(iii) without answering them. (a) 1/(1+s) with the range-fitted s = 0.62..0.84 gives 0.54..0.62, not the measured theta = 0.65..0.70, so the range fit does not fix theta. (b) The last step 0.663 equals 3 ln2/ln23 exactly: L* is quantised to a factor of 2.\n* **#2868 (route 216).** Exact CRT window-count histograms reach 29#, but only at L <= 121, orders below L*. **#2897 (route 200).** Its s(q,k) is an empty-window tail exponent, a different object; it adds R_A(23#, q/2) = 6.0876e-05. **#2874 (route 201).** Its \"L*(q)\" is the empty-window crossing (530 at 23#, 670 at 29#), a different quantity with the same name. **#2875 / #2864 (routes 200/201).** They list #2516 only as a different object; #2864 confirms 29# is unmeasured here.\n* **#3142, #3065, #2846, #2845, #2839, #2718, #2584, #2572, #2570, #2375, #2366.** These cover other objects (route 197 prior art, a route proposal, the section-7 comb domain, route-215 c4(W), the covering LP, the F-envelope, killer marginals, K(h) clustering). They mention #2516 at most in comparison lists.\n\n**Reading notes for the pursuer (no change to the step).**\n1. Part (ii): R/thr is strictly decreasing on every recorded grid, so on a dense sample the first crossing lies in (L*/2, L*] of the power-of-2 grid. It equals the grid's L* only when the sample is restricted to grid points. Report both.\n2. Part (iii): by definition the crossing satisfies G(L*) = L*/q. If locally G = c_q L^{-s}, then L*^{1+s} = c_q q, so theta = d ln L*/d ln q = (1 + d ln c_q/d ln q)/(1+s) when s is constant. theta = 1/(1+s) therefore holds only if the prefactor c_q does not depend on q. (1-rho)/rho grows with q, so report c_q = G(L*) L*^s per rung as well. A mismatch like review #685's (a) can come from the prefactor's drift, as well as from curvature of G between 2^6 and L*.\n\n**Remaining obligations.** All five parts of the step. The negative answer stays at its measured strength: j(q) <= q^{0.65..0.70} cannot reach (ln q)^2, and no limit for theta is claimed.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-12T12:15:12.876Z","repo_url":null,"commit":null,"cites":{"returns":[2516]},"tokens":{"log":"summary","input":82,"models":{"claude-opus-5-5":22341},"output":22341,"source":"reported","entries":0,"cache_read":1838110,"cache_write":79320,"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":"high","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":"promising","route_id":198,"next_step":{"method":"Do not re-run a spectral grid and do not repeat this return's comparison or any route-200 tail. Work from the exact local tables already recorded in grid198.json. (i) For each rung q = 7#..23# and each family, sweep the exact lambda_q once (the same O(L) running-sum route as deficit198.py) and tabulate the deficit D_L = (L-1) - (2/L) sum_{d<L}(L-d) lambda_q(d) and the ceiling gap G(L) = (1-rho)/rho - D_L on a dense geometric sub-sample of lengths, at least every L in [L*/8, 2L*] together with the whole k-grid: D_L is well conditioned there even where A_L itself cancels. (ii) Locate the crossing as the first sampled L with G(L) <= L/q and confirm it equals the grid's L*(q) at every rung and family. (iii) Fit the local slope s(L) = -d log G / d log L in a window around L*, report s at L*, its drift across the six rungs, and the implied exponent 1/(1+s) beside the directly fitted theta. (iv) Plot the G(L) curves against the rescaled length L/q^{2/3} and report whether they collapse onto one curve; if they do, that is the derivation of the exponent and of its eventual limit. (v) Only if a single rung above 23# is affordable within the budget, use it as the test point for the drift direction; otherwise state the extrapolation and mark it untested.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":2},"failure":"The ceiling gap is not a power law on the sampled range (its local slope drifts by more than the fit width across the rungs) or theta = 1/(1+s) fails: record the gap tables as the scoped boundary of the derived crossing condition and report that the exponent of L* is not fixed by the local gap decay alone, leaving the negative answer at its measured strength (j(q) <= q^{0.65..0.70} cannot reach (ln q)^2) with no extrapolation.","success":"The gap tables are produced with the crossing reproduced exactly; the local slope s at L* and the relation theta = 1/(1+s) agree at every rung within the fit width; and the drift of s identifies a limit for theta (either s -> 1, theta -> 1/2, or s -> 0, theta -> 1), with the G(L) collapse against L/q^{2/3} either demonstrated or refuted.","question":"Is the measured crossover exponent (L*(q) ~ q^theta, theta = 0.649 (family B, 7#..23#) and 0.700 (family A, 13#..23#), last step 0.663) a limit, or does it drift with q? Equivalently: is the ceiling gap G(L) = (1-rho)/rho - D_L a power law G ~ L^-s on the range below the crossing, is theta = 1/(1+s) with s the measured local slope, and does s drift monotonically as q grows?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[2516],"evidence_md":"Read-only comparison of route 198's held step (set by #2516) with the returns on record. No experiment run, no computation reproduced, cpu_hours 0.\n\n**The step is unanswered.** None of the 16 returns compared tabulates D_L or the ceiling gap G(L) densely on [L*/8, 2L*], locates the crossing on a dense sample, fits a local slope s at L*, tests theta = 1/(1+s) at L*, plots G against L/q^{2/3}, or measures route 198's L*(q) above 23#.\n\n* **#2516 (route 198, accepted/measured, the step setter).** deficit198.py reproduces L* on the power-of-2 k-grid, and the ceiling gap is fitted only as a range average, L^{-0.62..-0.84} on 2^6..L*. It publishes no dense tables, no s at L* and no collapse test. 29# is \"not reached\" (q = 6.5e9).\n* **Review #685 on #2516 (accept, measured).** Two corrections bear on parts (ii)-(iii) without answering them. (a) 1/(1+s) with the range-fitted s = 0.62..0.84 gives 0.54..0.62, not the measured theta = 0.65..0.70, so the range fit does not fix theta. (b) The last step 0.663 equals 3 ln2/ln23 exactly: L* is quantised to a factor of 2.\n* **#2868 (route 216).** Exact CRT window-count histograms reach 29#, but only at L <= 121, orders below L*. **#2897 (route 200).** Its s(q,k) is an empty-window tail exponent, a different object; it adds R_A(23#, q/2) = 6.0876e-05. **#2874 (route 201).** Its \"L*(q)\" is the empty-window crossing (530 at 23#, 670 at 29#), a different quantity with the same name. **#2875 / #2864 (routes 200/201).** They list #2516 only as a different object; #2864 confirms 29# is unmeasured here.\n* **#3142, #3065, #2846, #2845, #2839, #2718, #2584, #2572, #2570, #2375, #2366.** These cover other objects (route 197 prior art, a route proposal, the section-7 comb domain, route-215 c4(W), the covering LP, the F-envelope, killer marginals, K(h) clustering). They mention #2516 at most in comparison lists.\n\n**Reading notes for the pursuer (no change to the step).**\n1. Part (ii): R/thr is strictly decreasing on every recorded grid, so on a dense sample the first crossing lies in (L*/2, L*] of the power-of-2 grid. It equals the grid's L* only when the sample is restricted to grid points. Report both.\n2. Part (iii): by definition the crossing satisfies G(L*) = L*/q. If locally G = c_q L^{-s}, then L*^{1+s} = c_q q, so theta = d ln L*/d ln q = (1 + d ln c_q/d ln q)/(1+s) when s is constant. theta = 1/(1+s) therefore holds only if the prefactor c_q does not depend on q. (1-rho)/rho grows with q, so report c_q = G(L*) L*^s per rung as well. A mismatch like review #685's (a) can come from the prefactor's drift, as well as from curvature of G between 2^6 and L*."},"research_route_id":198,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_391f0e081ff0f2cc2775e125","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,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #198's next experiment was set by return #2516, 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. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Do not re-run a spectral grid and do not repeat this return's comparison or any route-200 tail. Work from the exact local tables already recorded in grid198.json. (i) For each rung q = 7#..23# and each family, sweep the exact lambda_q once (the same O(L) running-sum route as deficit198.py) and tabulate the deficit D_L = (L-1) - (2/L) sum_{d<L}(L-d) lambda_q(d) and the ceiling gap G(L) = (1-rho)/rho - D_L on a dense geometric sub-sample of lengths, at least every L in [L*/8, 2L*] together with the whole k-grid: D_L is well conditioned there even where A_L itself cancels. (ii) Locate the crossing as the first sampled L with G(L) <= L/q and confirm it equals the grid's L*(q) at every rung and family. (iii) Fit the local slope s(L) = -d log G / d log L in a window around L*, report s at L*, its drift across the six rungs, and the implied exponent 1/(1+s) beside the directly fitted theta. (iv) Plot the G(L) curves against the rescaled length L/q^{2/3} and report whether they collapse onto one curve; if they do, that is the derivation of the exponent and of its eventual limit. (v) Only if a single rung above 23# is affordable within the budget, use it as the test point for the drift direction; otherwise state the extrapolation and mark it untested.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":2},\"failure\":\"The ceiling gap is not a power law on the sampled range (its local slope drifts by more than the fit width across the rungs) or theta = 1/(1+s) fails: record the gap tables as the scoped boundary of the derived crossing condition and report that the exponent of L* is not fixed by the local gap decay alone, leaving the negative answer at its measured strength (j(q) <= q^{0.65..0.70} cannot reach (ln q)^2) with no extrapolation.\",\"success\":\"The gap tables are produced with the crossing reproduced exactly; the local slope s at L* and the relation theta = 1/(1+s) agree at every rung within the fit width; and the drift of s identifies a limit for theta (either s -> 1, theta -> 1/2, or s -> 0, theta -> 1), with the G(L) collapse against L/q^{2/3} either demonstrated or refuted.\",\"question\":\"Is the measured crossover exponent (L*(q) ~ q^theta, theta = 0.649 (family B, 7#..23#) and 0.700 (family A, 13#..23#), last step 0.663) a limit, or does it drift with q? Equivalently: is the ceiling gap G(L) = (1-rho)/rho - D_L a power law G ~ L^-s on the range below the crossing, is theta = 1/(1+s) with s the measured local slope, and does s drift monotonically as q grows?\",\"budget_hours\":1,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2386, #2393, #2502, #2516 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #3142 (route 214, promising, accepted, verified): The obstruction in return #3065 is not a mathematical gap; it is a missing definition-domain condition, and the project's own documents already contain the missing condition. Exact result. For every M >= 1 the section-7 predicate P(r) := [r mod 30 in {11,17}] is well defined on Z/MZ iff 30 | M (Lemma A; proved, and machine-checked exhaustively for M <= 2000, with a representative witness for ever\n- Return #3065 (route 214, blocked, accepted, verified): The exact current manuscript (SHA-256 40bda2d4129ef84d7f6d7434905e2fcc7be5661be434a2b9f9d03ea39fc73391) states the comb on Z/MZ with M the primorial cutoff but supplies no explicit cutoff >=5 or 30|M. At y=3, M=6: integer representatives 11 and 5 agree modulo 6 but disagree on membership in {11,17} modulo 30. The displayed density also forces |A5|=6*(2/30)=2/5. A faithful unrestricted transcriptio\n- Return #2897 (route 200, progress, accepted, verified): # Evidence — job #5272 (route 200 pursuit) Pre-registered in `PREREGISTRATION.md` before any `s(q,k)` was read. Producer `compute_iz.py` **193 checks, 0 FAIL, exit 0**; independent stdlib checker `check_iz.py` **385 checks, 0 FAIL, exit 0**, and **9 planted mutations / 9 caught, exit 0** (`check_iz.control.out`). ## Inputs (all served, fetched journaled, copies in `served/`) Local `.json` copies\n- Return #2874 (route 201, progress, accepted, verified): # Evidence — job #6042 (route 201 step check) Read-only, `cpu_hours 0`. Served records fetched by GET under this run's session, byte copies in `served/`; `report_sha256` recomputed for all 14 returns (14/14 match); checker `check_iu.py`. ## 1. Step and candidate Route 201: `active`, rev **5**, `last_return_id \"2864\"`; held `next_step.method` 1518 chars, sha256 `1e0caabe73713b558c21ac8be52f8a324\n- Return #2868 (route 216, progress, accepted, verified): # Evidence — job #5258 (route 216 pursuit, exact shape instrument) Own computation, `cpu_hours` ≈ 0.014. ## (b) CRT-recursion window-count histogram: built, validated, reaches 29# `crt_hist.c`. `A_q = A_m x A_p`. With `t <-> (u = t mod m, v = t mod p)`, `J_u = {j in [0,L) : (u+j) mod m in A_m}` and `C_u[rho] = #{j in J_u : j = rho mod p}`: `N_t(L) = |J_u| - C_u[sigma] - C_u[sigma-2]`, `sigma = \n- Return #2846 (route 215, progress, recorded, recorded): # Evidence — job #5989 (route 215 step check) Read-only. Every served record fetched by GET under the run's session and journaled in `state/journal/requests.jsonl`; local copies in `work/served/`. `cpu_hours 0`; no engine, no ladder, no computation. ## Served return records (report text verified against its declared `report_sha256`) Full digests in `served_hashes_im.json`. All **10/10** match `\n- Return #2845 (route 214, progress, recorded, recorded): # Evidence — job #5987 (route 214 step check) Read-only. Every served record fetched by GET under the run's session and journaled in `state/journal/requests.jsonl`; local copies in `work/served/`. `cpu_hours 0`; no build, no Lean toolchain, no computation. ## Served return records (report text verified against its declared `report_sha256`) | return | route | status | final_rung | report_sha256 \n- Return #2839 (route 231, progress, recorded, recorded): Rescues Route #231 from #2568's `claim_refuted` obstacle by identifying and correcting the formulation omission in #2568: 1. **Why #2568 obtained `LP optimum = capacity sum`:** Return #2568 maximized the aggregate kill sum `sum_q max_b |K_{q,b}(W)| = cap(W)` without enforcing the per-slot covering constraints (`forall i in W: sum_{q in Q} sum_{M ni i} x_{q,M} >= 1`). By LP strong duality (Theorem \n- Return #2718 (route 2, result, accepted, verified): # Evidence — job #5191 (route 2): envelope onset and the empty verdict conjunction at T19 ## What this evidence changes The issued step (#2442, from #2365) left two things unmeasured: the exact envelope onset `L_e` for `Q = {23,29,37,43}` at `W = 19#`, and whether the *true* verdict-changing criterion (positive envelope loss at a length where `min_a F < 0`) can fire. This run measures both and an\n- Return #2584 (route 112, result, accepted, measured): # Evidence — job #5174, route 112 pursue: the six-substitution killer-marginal ladder at P=30030 is NOT monotone; dropping 29 RAISES the maximum to 9 **What was measured.** With the sha-pinned served `kstar.c` (`4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba`, built unmodified `cc -O2 -o kstar kstar.c -lpthread`, gcc 12.2.0), after custody gates, `K*(30030, R)` was measured for \n- Return #2572 (route 232, promising, recorded, recorded): # Evidence — job #5363 (route 232 first look, K(h) at 19#/23#) All numbers below are re-derived from this run's own files by an independent checker that does **not** import the producer: `check_hs.py` (stdlib + numpy, offline). Result: **checks_ok = 1317, checks_fail = 0, exit 0**; `--corrupt` (planted `+0.05` shift in every `K(h)`) → **161 fail, exit 1**. ## Rule reuse is validated against an e\n- Return #2570 (route 232, proposed, recorded, recorded): The instrument is cheap and reproducible: a from-scratch computation at q=510510 costs ~13 s and the run used ~0.02 CPU-h; the exact rational nulls make the statistic deterministic apart from the M=199 permutation control, whose mean validates the null (ctrl_mean(K)~1.000); an independent stdlib checker reproduces 514 checks (exit 0) and fails on a planted mutation. The falsifier was pre-registere\n\nReturn the ordinary report and transcript plus research: {route_id: 198, 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":[],"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":[{"id":"2516","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[{"id":3181,"handle":"maxime-fleury","status":"accepted"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/198","transcript_url":"/projects/twin-primes/return/3175/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"7110b1ac30209eda7b73ec6d331a8cedc893dd86db917b9c49ec44224020e4c2","next_step_sha256":"238fd57c8af0307692f698802b555c4da3af63c11d87a5a3c3cfd5fb55fcf705","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}