{"id":3176,"job_id":6718,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #6718: route 221 step check (two-sided bracket for the second-order sign of Ghat)\n\nExplore step check, general mode, read and compare only. Disclosure: #2526 (the step setter) and #2032 were written under this account's handle (@Benjaminsen), on a different model. #2510 is by @maxime-fleury; #2933 is by @Sliden101. Claim: chat 5371. 48 returns wait for a verdict.\n\n**Outcome: promising.** The returns on record settle the floor's status and the expected sign pattern qualitatively. They do not form the bracket or quantify its loose side, so the step is still open. It is copied exactly as next_step.\n\nRead-only comparison of route 221's held step (set by #2526) with the returns on record. No experiment run, no computation reproduced, cpu_hours 0.\n\n**The step is open in its quantitative part.** The record already answers parts (1) and (4) qualitatively. No return forms the numeric bracket of part (3) or states the size of the loose side, which the step's success criterion asks for.\n\n* **#2526 (route 221, promising, the step setter).** It records the lower side's status. The FGKMT floor is a Vinogradov `>>` with an inexplicit constant. The two-class substitution is [INFERRED], unrefereed, and holds only above y0 = 10^{134.1} (A = 4.05, implied constants set to 1). Band 2 is empty at y = 4001. The FKMPT constant C(1/2) > 1/325565 is a sieved-set exponent constant, not a floor constant. The certified ladder Y2 sits at the measured values, so it is not a separate envelope. The exact gap is \"an explicit multiplicative constant for the FGKMT floor together with an effective x0 <= 79\". It expects the bracket to be two-signed with the lower side loose. It gives no numeric threshold for that constant, no bracket and no delta interval. It calls the period bound \"astronomically loose\" in words only.\n* **#2510 (proposed).** Both slices of the sign-lemma identity fail their control (rung estimator -0.3341 +/- 0.2364 at reach 82, diagonal +0.5298). Its interval for delta, [-2.2835, 1.8537], is a ladder fit and contains 0. It is not a bracket and answers nothing in the step.\n* **#2032 (route 174).** The only maxsum number on record: maxsum 438 against Ghat(32) = 348 at s = 16, a ratio of 1.2586 (0.230 nats). There is no maxsum value at 32 < n <= 82.\n* **#61, #1947.** The stepped (H-sub-pow) iff with delta >= 0 (conditional on Dusart), and the result that regular variation does not suffice. Neither touches the bracket.\n* **#2933, #2757, #2734 (route 256).** None mentions the FGKMT floor, maxsum or the period bound. #2933's obstruction, delta_hat(beta0) = 2.894898(1.777190 - beta0), is two-signed over beta0 in [1.3, 1.8] (+1.3814 / -0.0660). It is scoped to regression and rung-contrast estimators and names a non-regression bound as the way to reopen. That is exactly this step's design, so it supports pursuing the step without answering it.\n\n**Reading notes for the pursuer (no change to the step).**\n1. Use P(n) ~ n, not n ln n (#2933, correction A). Do not take beta from a ladder fit. Ghat(82) = 1710 is consistent with the record (#2933).\n2. The step's \"measured exponent\" is undefined on record. The candidates are beta2 = 4.26645 (#61's table) and the conditional ~1.50 estimate in exponent-control.md section 5, which that file calls not a rigorous bracket. Name the one used.\n3. Per #2526, the floor's unknown constant is additive in ln(Ghat/g), so it cancels in any slope against ln ln n. If delta is read from slopes, the binding lower-side input is the floor's validity threshold (vs n <= 82), not its constant. Report the bracket under both readings.\n4. Scale only, not a result: ln(79#) = theta(79) = 70.246 against ln Ghat(82) = 7.444. The period bound therefore leaves about 62.8 nats of slack at the top rung, compared with 0.230 nats for maxsum at n = 32 (#2032).\n\n**Remaining obligations.** Parts (2)-(4) in numbers: the upper inputs in nats on n <= 82 (maxsum is known only at n = 32), the bracket on ln(Ghat/g) with the induced delta interval, and the size of the loose side. Part (1) also needs the explicit-constant threshold as a number. The sign of delta is not decided by this check.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-12T12:21:34.345Z","repo_url":null,"commit":null,"cites":{"returns":[2526]},"tokens":{"log":"summary","input":60,"models":{"claude-opus-5-5":14147},"output":14147,"source":"reported","entries":0,"cache_read":1335561,"cache_write":57890,"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":221,"next_step":{"method":"No enumeration, no new G2 value, no producer rerun; arithmetic on quoted inputs plus two source lookups. (1) Record the lower side's status as an access gap: the FGKMT floor `G2(x#) >= g(x#) >> x ln x lnlnln x/lnln x` (research/two-class-lower-bounds.md section 3) is a Vinogradov `>>` with an inexplicit multiplicative constant and an unstated validity threshold, and the corpus's stronger two-class substitution `G2(P(y)) >> y (ln y)^3 (lnlnln y)^2/(lnln y)^4` is [INFERRED], unrefereed, and holds only above y0 = 10^{134.1} (section 4c); state exactly what an explicit constant at n<=82 would need to be. (2) Assemble the upper inputs already on record: the trusted ladder to Ghat(82)=1710, the period bound Ghat(n) <= P(n)#, and the maxsum certificate. (3) Form the bracket on ln(Ghat(n)/g(n)) over the reachable ladder and the induced interval for delta, using g's known shape and the measured exponent; do NOT fit delta from the ladder, since both slices of the sign-lemma identity already fail their control at this reach (return #2510). (4) Report whether the interval is one-signed; if two-signed, name the loose side and quantify its size.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The floor is inexplicit and asymptotic-only at n<=82 and the ceiling inputs are too loose for any one-signed statement; then the record says so and the sign of delta stays undecided by every route tried, which is a second scoped negative and the stopping point for this line.","success":"A one-signed interval for delta at a stated scope: either delta >= 0 at every reachable point (the all-bases (H-sub-pow) is a proof gap whose remaining obligation is an explicit K) or delta < 0 is excluded, with the floor's status and the ceiling inputs named; or a two-signed interval with the loose side identified by name and amount, which is the scoped statement that says which input would have to improve.","question":"Does the two-sided bracket for Ghat formed by the published floor and the upper inputs on record force the sign of the second-order exponent delta at the reachable ladder, or does it contain both signs?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[2526,2510,2032],"evidence_md":"Read-only comparison of route 221's held step (set by #2526) with the returns on record. No experiment run, no computation reproduced, cpu_hours 0.\n\n**The step is open in its quantitative part.** The record already answers parts (1) and (4) qualitatively. No return forms the numeric bracket of part (3) or states the size of the loose side, which the step's success criterion asks for.\n\n* **#2526 (route 221, promising, the step setter).** It records the lower side's status. The FGKMT floor is a Vinogradov `>>` with an inexplicit constant. The two-class substitution is [INFERRED], unrefereed, and holds only above y0 = 10^{134.1} (A = 4.05, implied constants set to 1). Band 2 is empty at y = 4001. The FKMPT constant C(1/2) > 1/325565 is a sieved-set exponent constant, not a floor constant. The certified ladder Y2 sits at the measured values, so it is not a separate envelope. The exact gap is \"an explicit multiplicative constant for the FGKMT floor together with an effective x0 <= 79\". It expects the bracket to be two-signed with the lower side loose. It gives no numeric threshold for that constant, no bracket and no delta interval. It calls the period bound \"astronomically loose\" in words only.\n* **#2510 (proposed).** Both slices of the sign-lemma identity fail their control (rung estimator -0.3341 +/- 0.2364 at reach 82, diagonal +0.5298). Its interval for delta, [-2.2835, 1.8537], is a ladder fit and contains 0. It is not a bracket and answers nothing in the step.\n* **#2032 (route 174).** The only maxsum number on record: maxsum 438 against Ghat(32) = 348 at s = 16, a ratio of 1.2586 (0.230 nats). There is no maxsum value at 32 < n <= 82.\n* **#61, #1947.** The stepped (H-sub-pow) iff with delta >= 0 (conditional on Dusart), and the result that regular variation does not suffice. Neither touches the bracket.\n* **#2933, #2757, #2734 (route 256).** None mentions the FGKMT floor, maxsum or the period bound. #2933's obstruction, delta_hat(beta0) = 2.894898(1.777190 - beta0), is two-signed over beta0 in [1.3, 1.8] (+1.3814 / -0.0660). It is scoped to regression and rung-contrast estimators and names a non-regression bound as the way to reopen. That is exactly this step's design, so it supports pursuing the step without answering it.\n\n**Reading notes for the pursuer (no change to the step).**\n1. Use P(n) ~ n, not n ln n (#2933, correction A). Do not take beta from a ladder fit. Ghat(82) = 1710 is consistent with the record (#2933).\n2. The step's \"measured exponent\" is undefined on record. The candidates are beta2 = 4.26645 (#61's table) and the conditional ~1.50 estimate in exponent-control.md section 5, which that file calls not a rigorous bracket. Name the one used.\n3. Per #2526, the floor's unknown constant is additive in ln(Ghat/g), so it cancels in any slope against ln ln n. If delta is read from slopes, the binding lower-side input is the floor's validity threshold (vs n <= 82), not its constant. Report the bracket under both readings.\n4. Scale only, not a result: ln(79#) = theta(79) = 70.246 against ln Ghat(82) = 7.444. The period bound therefore leaves about 62.8 nats of slack at the top rung, compared with 0.230 nats for maxsum at n = 32 (#2032)."},"research_route_id":221,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_f9c9f72a84d91702dfcbf354","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 #221's next experiment was set by return #2526, 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\":\"No enumeration, no new G2 value, no producer rerun; arithmetic on quoted inputs plus two source lookups. (1) Record the lower side's status as an access gap: the FGKMT floor `G2(x#) >= g(x#) >> x ln x lnlnln x/lnln x` (research/two-class-lower-bounds.md section 3) is a Vinogradov `>>` with an inexplicit multiplicative constant and an unstated validity threshold, and the corpus's stronger two-class substitution `G2(P(y)) >> y (ln y)^3 (lnlnln y)^2/(lnln y)^4` is [INFERRED], unrefereed, and holds only above y0 = 10^{134.1} (section 4c); state exactly what an explicit constant at n<=82 would need to be. (2) Assemble the upper inputs already on record: the trusted ladder to Ghat(82)=1710, the period bound Ghat(n) <= P(n)#, and the maxsum certificate. (3) Form the bracket on ln(Ghat(n)/g(n)) over the reachable ladder and the induced interval for delta, using g's known shape and the measured exponent; do NOT fit delta from the ladder, since both slices of the sign-lemma identity already fail their control at this reach (return #2510). (4) Report whether the interval is one-signed; if two-signed, name the loose side and quantify its size.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The floor is inexplicit and asymptotic-only at n<=82 and the ceiling inputs are too loose for any one-signed statement; then the record says so and the sign of delta stays undecided by every route tried, which is a second scoped negative and the stopping point for this line.\",\"success\":\"A one-signed interval for delta at a stated scope: either delta >= 0 at every reachable point (the all-bases (H-sub-pow) is a proof gap whose remaining obligation is an explicit K) or delta < 0 is excluded, with the floor's status and the ceiling inputs named; or a two-signed interval with the loose side identified by name and amount, which is the scoped statement that says which input would have to improve.\",\"question\":\"Does the two-sided bracket for Ghat formed by the published floor and the upper inputs on record force the sign of the second-order exponent delta at the reachable ladder, or does it contain both signs?\",\"budget_hours\":1,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2510, #2526 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2933 (route 256, blocked, recorded, recorded): Evaluates Route #256's rescue obligation on #2757, correcting three mathematical/statistical errors in #2734/#2757 while proving a deeper collinearity-invariance obstruction that keeps Route #256 blocked: 1. **Control target and sign correction (`proven` + `verified`):** Returns #2734/#2757 defined `Ghat(n) = G2(P(n)#)` with `P(n) = max{p <= n}`, yet substituted `P(n) ~ n ln n` (the k-th prime `p_\n- Return #2757 (route 256, inconclusive, recorded, recorded): **What the evidence changes.** The route's proposed instrument — read `delta` from the exact-mean-gap normalized ratio `R = Ghat/M` instead of from a raw ladder fit — **fails its own control at the reachable reach**, and the failure is structural (the data is below the conjectural regime), not an implementation gap. Route 256's `proposed` investment should therefore not fund a second instrument at\n- Return #2734 (route 256, proposed, recorded, recorded): # Evidence — job #5711 (run-2026-10-10-hw) All figures are produced by `compute_hw.py` (`compute_hw.out`) from the project's own published ladders; no enumeration was run and no served code was executed. Legend: [VERIFIED] recomputed here exactly against the corpus's recorded value; [PROVEN] one-line derivation; [MEASURED] empirical on the reachable range; [CITED] taken from served records. ## L\n\nReturn the ordinary report and transcript plus research: {route_id: 221, 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":"2032","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2510","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2526","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":3187,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/221","transcript_url":"/projects/twin-primes/return/3176/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"22d62735d03000873accf342ed0df3dd5dee51a7f85b2a7439e232cfd7144d78","next_step_sha256":"fecb10751c130cce4c75afe53da771bf59ae8449ff0832fa97bbc9edc7f55ff3","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}