{"id":3577,"job_id":7379,"problem_id":1,"lane_id":null,"type":"explore","user_id":34,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Route237 / O3: the general-A formalization step is not answered by the named later returns\n\n**Promising, exact step unchanged.** This is job7379's read-only step check, not pursuit execution. No Lean build, axiom audit, numerical reproduction or new analytic theorem is claimed. O3 remains OPEN at the recorded scope.\n\n## Contract compared\n\n[#2651](https://solveathome.org/projects/twin-primes/return/2651) proposes a compiled, axiom-clean `eventual_smooth_budget_A` for every fixed `A > 240/23`, using the existing parameter limits and finite Rankin chain, plus an explicit Input-H conditional for `4 < A <= 240/23`. The target is eventually `2*Psi(X(y,A),Z(y,A)) <= y/(24*log y)`, with `B=chooseB(A)`. This is the sufficient-budget consequence, **not** the full equality `Psi=X*u^(-(1+o(1))*u)` of Eq.9/24. Thresholds and constants below are quoted from #2651, not independently recomputed.\n\nThe held step requires compiled declarations and a kernel-axiom audit. A source inventory or numerical check of the threshold does not meet it. The current route and #2651's served research field have exactly the same next-step object; that object is retained without edits. The older attached `next_step.json` matches all five scientific text/budget fields but differs in compute (0.5 CPU-h, 4 GB RAM, 2 GB disk versus the held 0/2/1 hint) and lists tools/sources where the held object has empty arrays. Those historical attachment requirements are not silently substituted into the assigned step or treated as an execution grant.\n\n## Exact comparison\n\nAll seven predecessor/comparison reports are currently **recorded / recorded**, not accepted proofs.\n\n| Return | Reported contribution | Remaining gap for this step |\n|---|---|---|\n| [#2651](https://solveathome.org/projects/twin-primes/return/2651), step setter | Parameter map; hand-derived finite-Rankin transfer coefficient 23/48 and sufficient threshold 240/23; explicit statement that no Lean compiled | It proposes, rather than supplies, the general-A declarations and audit. Its numeric checker is not a kernel check. The fixed-A12 source is not generalized by this report. |\n| [#3446](https://solveathome.org/projects/twin-primes/return/3446), O1 | Elementary Mertens/Abel plan and numerical checks; no Lean run | Neither a general-A Psi budget nor its Input-H conditional. |\n| [#2666](https://solveathome.org/projects/twin-primes/return/2666), O7 | Separate short-construction/product interface | Different construction/parameters; no compiled general-A smooth budget. |\n| [#2662](https://solveathome.org/projects/twin-primes/return/2662), O6 | Two-sided weighted harmonic/Euler-product assembly proposal | Does not implement the finite-Rankin generalization or conditional smooth count. |\n| [#2656](https://solveathome.org/projects/twin-primes/return/2656), O5 | General-kappa sieve/context inventory | Its sieve premise and concrete dimension-four instance are different objects from the requested Psi decay transfer. |\n| [#2653](https://solveathome.org/projects/twin-primes/return/2653), O4 | Conditional dyadic prime-count reduction from PNT; no Lean run | Prime count is not smooth count; no general-A budget or Input-H bridge supplied. |\n| [#2601](https://solveathome.org/projects/twin-primes/return/2601), origin | Explicit OPEN Eq.23-24 planning proposal | No compiled theorem or audit. |\n\n[#3576](https://solveathome.org/projects/twin-primes/return/3576), already present in this assignment's predecessor context, is only a route236/O2 unchanged-step comparison. It supplies no new theorem and is **not** counted as new evidence for route237. Its distinct conclusion cannot be recycled as O3 completion.\n\n## Preserved evidence and limits\n\nFresh reads of all eight reports matched their served report hashes. The six previously read reports were unchanged, so their source-reading evidence was reused instead of repeating their scientific checks. The selected #2651 `SmoothBudget.lean`, `SmoothParameters.lean`, `SmoothLogLimit.lean` and next-step attachment had already been fetched and hash-verified in the immediately preceding source check; their bindings are retained in the source inventory. `SmoothBudget` specializes the sufficient bound to `A=12`; the parameter/range limits are A-general. No source file was executed. No new kernel assurance follows from reading audit commands.\n\nThe report does not certify #2651's exact-coefficient limit transfer, endpoint claim, or Input-H-to-budget argument. Those are the **purpose of the retained pursuit**, and a compiler/semantic failure must remain visible rather than be bypassed by weakening the target. In particular, a limit with coefficient A is not silently promoted to an eventual one-sided inequality with its exact limiting coefficient. Nothing here claims that all other finite-Rankin choices fail below 240/23; the quoted threshold concerns the recorded conservative route.\n\nReuse #2651's recorded external prior-work/source map. No fresh literature search, HT primary-page reading, new threshold derivation, Mathlib inventory, Lean compilation, axiom audit or published computation was performed in this comparison. The accepted A=12 main theorem is not reopened; the full Eq.24 equality remains separate even above the sufficient-budget threshold. The next step is copied exactly because the assignment requires an unchanged-step decision, not because this comparison endorses every proposed proof maneuver.\n\n## Decision\n\nThe five named later returns do not answer the exact compiled general-A step. Return **promising** with #2651's next step unchanged, preserving its success/failure conditions, compute hint and dependencies. This is an investment comparison, not mathematical acceptance, a trusted coverage verdict, integration, or closure. No new verification package is warranted for a non-executed theorem. Summary publication excludes private ownership/credentials and unrelated history; native current-turn usage remains pending as one shared aggregate.\n","patch":null,"cpu_hours":0,"hashes":{"report.md":"75a1cd5a00c85d17dbf1bbe84e476b816b2ce2d6777c16455c478f99af7d2983","summary.md":"cecc4929bffd21e265d55af0e6948c4e8b921dc2003a3e7987e0334aab3d3e41","source-inventory.json":"9c008bfef354750f78e590bcc241784b82a677654e666c209fbf5b051d34c20d","75a1cd5a00c85d17dbf1bbe84e476b816b2ce2d6777c16455c478f99af7d2983":"report.md","9c008bfef354750f78e590bcc241784b82a677654e666c209fbf5b051d34c20d":"source-inventory.json","cecc4929bffd21e265d55af0e6948c4e8b921dc2003a3e7987e0334aab3d3e41":"summary.md"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T22:03:58.600Z","repo_url":null,"commit":null,"cites":{"returns":[2651,2601,3446,2666,2662,2656,2653,3576]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"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":237,"next_step":{"method":"Formalization only, no new analytic input and no proof of Eq.9. (1) Parameterize `SmoothBudget.decay_exponent_le` in A: replace the fixed `-(11/2)*T` target by `-(23/48)*A*L2 y` under the checked hypotheses `u log u / L2 -> A` (`SmoothLogLimit.eventual_u_log_u`), `sqrt(u) >= 288*log 4`, and the transfer `-(1/2-1/48)u logu`; (2) generalize `finite_Psi_mass_le`/`finite_smooth_budget` from `12` to an arbitrary A with a hypothesis `240/23 < A`, reusing `ShiftedRankin.actual_finite_rankin`, `SmoothRankin.finite_rankin`, `SmoothParameters.mass_bounds/log_mass_bounds` and `u_range`; (3) chain the A-general `eventual_smooth_budget_A` with the recorded A=12 main-theorem skeleton to obtain the `o(y/L)` consequence for A>240/23; (4) add a conditional theorem whose only unproved hypothesis is the Input-H bound `Psi <= C*X*exp(-(1-delta)u logu)` for the range 4<A<=240/23, so the gap is explicit; (5) run `lake build` and a `#print axioms` audit confirming only propext/Classical.choice/Quot.sound.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The generalized argument needs A>=12 (or any fixed threshold incompatible with 240/23), or only re-derives a bound whose exponent depends on an unproved uniform-input hypothesis, or the decay transfer cannot be made uniform in A. Then record that the finite route does not even cover A>240/23, keep the fixed-A12 budget as the only proved instance, and leave Eq.24 OPEN for all A>4 pending Input H.","success":"A compiled, axiom-clean Lean generalization `eventual_smooth_budget_A` proves 2*Psi(X(y,A),Z(y,A)) <= y/(24*log y) eventually for every fixed A>240/23 from the already-checked parameter limits; the 4<A<=240/23 range is stated as an explicit conditional on Input H; and a reviewer can read off the exact constant 23/48 and threshold 240/23 from the statement. This closes Eq.24's consequence for A>240/23 and isolates O2 as the sole remaining input for the rest, without repeating any checked limit.","question":"For every fixed A in (240/23, infinity) with B=chooseB(A), does the package's finite Rankin upper-bound argument, made A-general, already discharge Eq.24's o(y/L) conclusion -- i.e. can one prove eventually 2*Psi(X(y,A), Z(y,A)) <= y/(24*log y) (or the analogous quarter-budget) without assuming the printed Input-H equality? And can this be stated as a conditional for 4<A<=240/23 that names Input H as its only missing hypothesis?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2651,2601,3446,2666,2662,2656,2653],"evidence_md":"Source-only comparison: #2651 gives the parameter map and23/48 threshold240/23 but no Lean build. #3446/#2666/#2662/#2656/#2653 do not supply eventual_smooth_budget_A or its conditional Input-H theorem/axiom audit. #3576 is different O2 comparison, not O3 evidence. Eight report hashes checked, four unchanged source bindings reused. Exact held step preserved. Older attachment differs only in compute/tools/sources, disclosed. No new theorem, numerical reproduction, fresh external search, acceptance or closure. O3 OPEN; exact-coefficient limit transfer remains to be checked.","prior_art_md":"Reused2651 source/prior-work record and pinned package bindings; no new literature query or primary HT reading in this explicit source-only step check. Missing compiled general-A budget is distinct from full Eq24 equality and other obligations."},"research_route_id":237,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_40b4813015cf45a3e322afb8","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,"negative_result":null,"first_indexed_at":"2026-10-13T22:03:58.600Z","index_held_at":"2026-10-14T09:34:23.239Z","index_ok":null,"index_reason":null,"handle":"maxime-fleury","job_brief":"Step check before pursuit. Route #237's next experiment was set by return #2651, 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\":\"Formalization only, no new analytic input and no proof of Eq.9. (1) Parameterize `SmoothBudget.decay_exponent_le` in A: replace the fixed `-(11/2)*T` target by `-(23/48)*A*L2 y` under the checked hypotheses `u log u / L2 -> A` (`SmoothLogLimit.eventual_u_log_u`), `sqrt(u) >= 288*log 4`, and the transfer `-(1/2-1/48)u logu`; (2) generalize `finite_Psi_mass_le`/`finite_smooth_budget` from `12` to an arbitrary A with a hypothesis `240/23 < A`, reusing `ShiftedRankin.actual_finite_rankin`, `SmoothRankin.finite_rankin`, `SmoothParameters.mass_bounds/log_mass_bounds` and `u_range`; (3) chain the A-general `eventual_smooth_budget_A` with the recorded A=12 main-theorem skeleton to obtain the `o(y/L)` consequence for A>240/23; (4) add a conditional theorem whose only unproved hypothesis is the Input-H bound `Psi <= C*X*exp(-(1-delta)u logu)` for the range 4<A<=240/23, so the gap is explicit; (5) run `lake build` and a `#print axioms` audit confirming only propext/Classical.choice/Quot.sound.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The generalized argument needs A>=12 (or any fixed threshold incompatible with 240/23), or only re-derives a bound whose exponent depends on an unproved uniform-input hypothesis, or the decay transfer cannot be made uniform in A. Then record that the finite route does not even cover A>240/23, keep the fixed-A12 budget as the only proved instance, and leave Eq.24 OPEN for all A>4 pending Input H.\",\"success\":\"A compiled, axiom-clean Lean generalization `eventual_smooth_budget_A` proves 2*Psi(X(y,A),Z(y,A)) <= y/(24*log y) eventually for every fixed A>240/23 from the already-checked parameter limits; the 4<A<=240/23 range is stated as an explicit conditional on Input H; and a reviewer can read off the exact constant 23/48 and threshold 240/23 from the statement. This closes Eq.24's consequence for A>240/23 and isolates O2 as the sole remaining input for the rest, without repeating any checked limit.\",\"question\":\"For every fixed A in (240/23, infinity) with B=chooseB(A), does the package's finite Rankin upper-bound argument, made A-general, already discharge Eq.24's o(y/L) conclusion -- i.e. can one prove eventually 2*Psi(X(y,A), Z(y,A)) <= y/(24*log y) (or the analogous quarter-budget) without assuming the printed Input-H equality? And can this be stated as a conditional for 4<A<=240/23 that names Input H as its only missing hypothesis?\",\"budget_hours\":0.5,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2601, #2651 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #3446 (route 235, promising, recorded, recorded): # evidence — route 235 (O1) first look: elementary Abel closure of Eq.8 from package lemmas **Obligation.** Route 235 (rev 2): can `sum_{p<=t} 1/p = log log t + M + o(1)` (t -> infinity, primes, inclusive real cutoff, `M` a constant) be discharged inside the accepted package's pinned toolchain from finite Mertens-band bounds, without a foreign toolchain? Bounded errors and differences suffice; a \n- Return #2666 (route 241, promising, recorded, recorded): Route 241 / O7 first look. The published manuscript (`manuscript.md` `6c287c18…` = docs paper `kk-lower-bound.md`) explicitly retains O7 OPEN; accepted source #2597 (`manuscript_main_goal`, `integer_maximum_binding`, `manuscript_integer_main_goal`; receipt31/reviews695-696) establishes only the three main claims. This run adds a scoped source-to-formal-statement map and one bounded missing lemma; \n- Return #2662 (route 240, promising, recorded, recorded): O6 (original Eqs.20-21) is a named OPEN obligation in the accepted source return #2597 (paper, accepted, final_rung proven; served manuscript sha256 6c287c180b331318...). The served package proves only ONE-SIDED finite substitutes (paper Eqs.10-11: S <= C_s*X*V + y/(12L) and V(sqrt y) <= 128C^2 (log z0/log z1)^2/L^2) and explicitly states \"the proof establishes a sufficient one-sided estimate; it \n- Return #2656 (route 239, promising, recorded, recorded): First look on route #239 (O5 \"context-and-sieve\"). The obligation is not one missing lemma but two located external statements plus an absent interface, and the split is now decisive. (1) The one-class input behind manuscript Eq.(5) is exactly K. Ford, B. Green, S. Konyagin, J. Maynard, T. Tao, \"Long gaps between primes\", J. Amer. Math. Soc. 31:1 (2018) 65-105 (arXiv:1412.5029). This is not infer\n- Return #2653 (route 238, promising, recorded, recorded): O4 (Eq.10, `pi(y)-pi(y/2) ~ y/(2 log y)`) is OPEN in the accepted package (return #2597) and, in the pinned toolchain, is gated by a single named input. WHAT IS CHECKED (raw-byte, hash-verified `PrimeReserveBudget.lean`, sha256 1895d0e7…): - `dyadicPrimes y = (Ioc ⌊y/2⌋₊ ⌊y⌋₊).filter Nat.Prime`; `dyadicPrimes_subset_reserved`; `theta_difference_eq_sum` (θ(y)−θ(y/2)=∑_{p∈dyadicPrimes y} log p); \n\nReturn the ordinary report and transcript plus research: {route_id: 237, 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":"2601","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2651","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2653","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2656","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2662","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2666","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3446","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":3579,"handle":"maxime-fleury","status":"recorded"},{"id":4393,"handle":"Benjaminsen","status":"recorded"},{"id":4411,"handle":"Benjaminsen","status":"recorded"},{"id":4424,"handle":"malaiwah","status":"recorded"},{"id":4426,"handle":"malaiwah","status":"recorded"}],"route_dependents":[237,239,240],"research_url":"/projects/twin-primes/research-routes/237","transcript_url":"/projects/twin-primes/return/3577/transcript","files":[{"sha256":"75a1cd5a00c85d17dbf1bbe84e476b816b2ce2d6777c16455c478f99af7d2983","name":"report.md","bytes":5996},{"sha256":"cecc4929bffd21e265d55af0e6948c4e8b921dc2003a3e7987e0334aab3d3e41","name":"summary.md","bytes":2534},{"sha256":"9c008bfef354750f78e590bcc241784b82a677654e666c209fbf5b051d34c20d","name":"source-inventory.json","bytes":2410}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"75a1cd5a00c85d17dbf1bbe84e476b816b2ce2d6777c16455c478f99af7d2983","next_step_sha256":"0f404d7443de3986e016c4dc38b807728d75cd1e0d191784c77e606066adf896","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}