{"id":5720,"job_id":11526,"problem_id":1,"lane_id":null,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Route #237 (O3: every fixed A > 4 smooth equality and negligible-budget route) — Step Check Before Pursuit on Return #5707 (Reusing Step Check #5703)\n\n**Outcome: `promising` (`author_rung: \"measured\"`).** Route `#237`'s held `next_step` (set by Return `#4411`; canonical sorted-key compact UTF-8 SHA-256 `3cbe12d028547c07a01deef7477f014985d92df7f93763c1302bc6a1524a4172`) is **not answered** by the single new comparison candidate Return `#5707` on linked Route `#310` (nor by anything else recorded since Step Check `#5703`). The held step is copied byte-for-byte as `next_step` (same canonical SHA-256), so held pursuit `job #9088` is released with this note and `#5707` never holds it again.\n\n## 1. Route custody at revision 7\n\nServed `route237.json` (revision 7, state `active`, origin `#2601`, `last_return_id: 5703`, `updated_at: 2026-10-24T16:14:30.549Z` — exactly Step Check `#5703` creation time) carries 7 events: `#2601`, `#2651`, `#3577`, `#4393`, `#4411`, `#5679`, `#5703`. The served `next_step` is deep-equal to the step preserved by `#5679` and `#5703`; its canonical digest recomputes to `3cbe12d028547c07a01deef7477f014985d92df7f93763c1302bc6a1524a4172` (check `C02`), matching the value both predecessors bind. No newer return exists on Route `#237` itself: `last_return_id` is still `5703`.\n\n## 2. Audit of the new candidate Return #5707\n\n`#5707` (`job #9129`, Route `#310`, `status: recorded`, `final_rung: recorded`, created `2026-10-24T16:35:53.308Z` — after `#5703` at `16:14:30.549Z`) assembles `Rh41.TakeCount` (`#4434`) as a `lean-kernel-v1` **statement proposal**: byte-exact sources (`Rh41/GreedyResidual.lean` `4d5dc128…`, `Rh41/TakeCount.lean` `52fde740…`), a fresh pinned-executor rebuild (`leanprover/lean4:v4.35.0-rc3` commit `470d5ce1…` + Mathlib `331d5244…`), statement bundle, verification plan draft, kernel object/custody records, axiom audit and negative controls. It claims no new mathematics.\n\nChecks `C03`–`C05`, `C08` (instrument `check_step11526.py`):\n\n1. All **23** served attachments re-hash to their `return_5707.json` digests (`C04`).\n2. Across the return JSON (its `job_brief` field excluded) and all 23 attachments, with the served step text stripped, there are **0 occurrences** of the four target declarations (`decay_exponent_le_A`, `finite_Psi_mass_le_A`, `finite_smooth_budget_A`, `eventual_smooth_budget_A`), the conditional declaration (`eventual_smooth_budget_of_inputH`), the patch specification (`proposal_smooth_budget_A_ec`), the sharp rationals (`23/48`, `240/23`, `503/46`, `529/96`), and additionally 0 occurrences of `SmoothBudget`, `ShiftedRankin`, `SmoothRankin`, `SmoothLogLimit`, `SmoothParameters`, `eventual_smooth_budget` (`C05`).\n3. `#5707` never imports or compiles `SmoothBudget.lean`; its package contains only the two `Rh41` modules, their audit/statement modules and build descriptors.\n4. Reusable operational note (not O3 evidence): the served `lean-toolchain.txt` (`bc84812c…`), `dependencies-mathlib-lakefile.lean` (`7a0f6151…`) and `lake-manifest.json` (`479029d8…`) equal `#2597`'s executor pins (`C08`), so `#5707` adds a fourth independent confirmation (with `#4427`, `#4461`, `#5691`) that the pinned `v4.35.0-rc3` + Mathlib `331d5244` executor rebuilds `#2597`-lineage modules cleanly.\n\n**Positive control (`C07`):** the identical scanner detects all 10 target tokens in the reused `#5703` corpus (`eventual_smooth_budget_A=49`, `proposal_smooth_budget_A_ec=27`, `23/48=31`, `240/23=31`, `503/46=23`, `529/96=24`, and the rest), so a zero on `#5707` is a scanner reading, not a scanner defect.\n\n## 3. Reused conclusions of Step Check #5703 (unchanged scope)\n\n- The eight prior comparison returns (`#5554`, `#5528`, `#5295`, `#4926`, `#4492`, `#4472`, `#4461`, `#4427`) and the context returns `#5683`, `#5691`, `#5699` carry **0** O3 target-token occurrences outside `job_brief`; none answers the step.\n- The exact rational invariants stand: `1/2 - 1/48 = 23/48`; crossover `4 - (23/48)A < -1 <=> A > 240/23`; the `A=12` fixed-margin re-impositions in `SmoothBudget.lean` (`d69ec8f9…`, 10,322 B, 10 `chooseB 12`) are `hut : (23/2)*T <= U log U` and the rounded target `529/96 = 11/2 + 1/96`; free-eps repair `0 < eps < A - 240/23`, `A' := A - eps`; the printed Eq. 9 conditional closes `4 < A <= 240/23` where any fixed-`(1-delta)` Input H paraphrase caps at `A > 5/(1-delta) > 5`.\n- Open findings `#53828`, `#120070`, `#143613`, `#146586` do not touch `SmoothBudget.lean`.\n\n## 4. Pending out-of-route candidate #5581 (fresh status)\n\nReturn `#5581` (`scope r237-a-general`) is still `status: \"pending\"`, `final_rung: null`, `research_route_id: null`, no review ids; its 16 listed attachment names are unchanged from `#5703`'s verified audit (`C06`). Pending unreviewed work is not settled evidence and cannot close the step; its author-side logs (an `AGeneral` build and audit) remain exactly what the pursuit executor is asked to inspect and verify on the pinned executor.\n\n## 5. Why the route is not `known`\n\nThe held step demands a compilation experiment: restore the A=12 package from the served 63 files of `#2597`, confirm the pre-state build and clean `#print axioms`, then compile the four free-eps A-general declarations plus the printed Eq. 9 conditional. No return on record runs that experiment; the only on-record artifact of that shape (`#5581`) is pending and unreviewed. The step therefore stays open and is copied byte-for-byte as `next_step`.\n\n## Verification suite and scope limits\n\n- `python3 check_step11526.py` (stdlib-only, deterministic stdout, relative paths): **8/8 PASS, exit 0** (`check_step11526.out`); structured results in `analysis_11526.json`.\n- `python3 check_step11526.py --corrupt`: **6/6 single-field negative controls decisive, exit 1** (`check_step11526.corrupt.out`); each control mutates exactly one expected constant (step digest, route last-return id, one attachment digest, the pending status, the positive-control threshold, the candidate timestamp) and trips exactly its target check.\n- Re-verified here: the served route record, the served step digest, 28 attachment digests (23 of `#5707`, 5 of `#5703`) and the `#5581` return JSON. `#5703`'s 138-attachment digest audit and its positive controls on `#4411`/`#5679` are reused, not repeated.\n- Per the step-check contract this check is read-only: `cpu_hours 0`, no Lean build, no experiment, no computation reproduced. The scan strips the step text and the `job_brief` field before matching (otherwise the step self-hits); the positive control covers scanner liveness. O3 remains OPEN.\n","patch":null,"cpu_hours":0,"hashes":{"analysis_11526.json":"9d5c248194e9c6ea7b78d7213fb8437efcf300ff5814323c3cf747b35f5b0346","check_step11526.out":"824b5e9e0159ea45218f179870120f8a93512e10e1c0e5751768c70c14c37397","check_step11526.corrupt.out":"e32a05be5fc6e41d014f449dc52e8afcacf3909a1aa2f358ce559fe39c62c456"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-24T18:06:44.148Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5707,5703,5679,4411,5581,2651,2597],"messages":[]},"tokens":{"log":"summary","input":163737,"models":{"glm-5.3-flash":88241},"output":88241,"source":"reported","entries":0,"cache_read":17867795,"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":"xhigh","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":"Restore the A=12 package from the served bytes (63 files of #2597) at the pinned toolchain and a recorded Mathlib revision, confirm lake build and a clean #print axioms for SmoothBudget.eventual_smooth_budget (this is the pre-state, not to be re-derived). Then apply the four declarations in proposal_smooth_budget_A_ec.md: decay_exponent_le_A (sharp -(23/48)A'T from correction_le, hypothesis A'T <= U log U), finite_Psi_mass_le_A and finite_smooth_budget_A at parameter A, and the driver eventual_smooth_budget_A choosing eps with 0 < eps < A - 240/23 so that (23/48)(A-eps) > 5. Supply the one missing comparison t^(4-c) <= 1/24 for c > 5 via Mathlib's eventual power growth, in place of the A=12 closed form 1/(t*sqrt t). Add the conditional branch keyed to printed Eq. 9 (Psi = X u^{-(1+o(1))u}), not to a fixed 1-delta. Do NOT re-derive the A-general parameter map, the Eq.23 limits or the 23/48 / 240/23 arithmetic (#2651 re-checked by #3577), and do not rebuild the A=12 chain.","compute":{"ram_gb":8,"disk_gb":10,"cpu_hours":4},"failure":"The argument still needs A >= 12 (e.g. the fixed-margin hypothesis is kept), or only a fixed-threshold-in-A result can be compiled, or the t^(4-c) comparison cannot be closed inside the pinned Mathlib API, or the A=12 package does not rebuild at the pinned toolchain so no pre-state can be certified. Then record that the finite route does not cover A > 240/23 uniformly, keep fixed-A12 as the only proved instance, and leave Eq.24 OPEN for all A > 4 pending Input H.","success":"lake build succeeds with no sorry; #print axioms on decay_exponent_le_A, finite_Psi_mass_le_A, finite_smooth_budget_A and eventual_smooth_budget_A lists only propext, Classical.choice, Quot.sound; the statement of eventual_smooth_budget_A is forall A, 240/23 < A -> eventually 2 Psi(X y A)(Z y A) <= y/(24 log y); the conditional theorem for 4 < A has Eq.9 as its only unproved hypothesis; and a reviewer can read 23/48 and 240/23 off the statements.","question":"On a host with Lean at leo4 v4.35.0-rc3 and a pinned Mathlib, does the served A=12 package still build byte-identically, and does the free-epsilon A-general patch (target -(23/48)*A'*T with A' := A - eps and 0 < eps < A - 240/23) compile into a declaration whose statement exhibits 23/48 and 240/23 and whose #print axioms lists only propext, Classical.choice and Quot.sound?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[5703,5679,5691,5707,4411,4393,3577,2651,2601,2597,5554,5528,5295,4926,4492,4472,4461,4427],"evidence_md":"STEP CHECK on Route #237 (\"O3 OPEN: every fixed A>4 smooth equality and negligible-budget route\"), rev 7, active, origin #2601, last_return_id 5703, reusing Step Check #5703 and comparing new candidate #5707. Outcome: `promising` (`author_rung: \"measured\"`). The held step set by #4411 and preserved by #5679/#5703 is copied byte-for-byte as `next_step` (canonical sorted-key compact UTF-8 SHA-256 `3cbe12d028547c07a01deef7477f014985d92df7f93763c1302bc6a1524a4172`, recomputed from the served `route237.json[\"next_step\"]`).\n\n1) NOT ANSWERED BY NEW CANDIDATE #5707 (C03-C05, C08). Return #5707 (Route 310, job #9129, `recorded`/`recorded`, created 2026-10-24T16:35:53.308Z, after #5703 at 16:14:30.549Z) packages `Rh41.TakeCount` (#4434) as a `lean-kernel-v1` statement proposal: byte-exact sources (`Rh41/GreedyResidual.lean` `4d5dc128...`, `Rh41/TakeCount.lean` `52fde740...`), a fresh pinned-executor rebuild (`leanprover/lean4:v4.35.0-rc3` + Mathlib `331d5244...`), statement bundle, drafted verification plan, kernel object/custody records, axiom audit and controls; it claims no new mathematics. All 23 served attachments re-hash to their return digests (C04). Across the return JSON (its `job_brief` field excluded) and all 23 attachments, with the served step text stripped, there are **0 occurrences** of `decay_exponent_le_A`, `finite_Psi_mass_le_A`, `finite_smooth_budget_A`, `eventual_smooth_budget_A`, `eventual_smooth_budget_of_inputH`, `proposal_smooth_budget_A_ec`, `23/48`, `240/23`, `503/46`, `529/96`, and also 0 of `SmoothBudget`, `ShiftedRankin`, `SmoothRankin`, `SmoothLogLimit`, `SmoothParameters`, `eventual_smooth_budget` (C05); #5707 never imports or compiles `SmoothBudget.lean`. Positive control (C07): the identical scanner detects all 10 target tokens in the reused #5703 corpus (counts 20-49 per token).\n\n2) REUSED INVARIANTS AND CURRENT STATUS (C01, C02, C06). Route custody at revision 7: state `active`, origin #2601, `last_return_id: 5703`, `updated_at: 2026-10-24T16:14:30.549Z` (exactly #5703 creation), 7 events; no newer return exists on Route #237 itself. #5703's audit is reused unchanged (0 O3 token occurrences outside `job_brief` across #5691, #5683, #5699 and the 8 comparison returns; exact rationals `1/2 - 1/48 = 23/48`, crossover `4 - (23/48)A < -1 <=> A > 240/23`, A=12 fixed margin `529/96 = 11/2 + 1/96` (`503/46` if `eps=1/2` kept), free-`eps` recovery for all `A > 240/23`, fixed-`(1-delta)` Input H limit `5/(1-delta) > 5` vs printed Eq. 9 closing all `A > 4`; open findings #53828, #120070, #143613, #146586 do not touch `SmoothBudget.lean`). Fresh fetch of #5581 (`scope: r237-a-general`): still `status: \"pending\"`, `final_rung: null`, `research_route_id: null`, no review ids, 16 attachment names unchanged from #5703's verified audit (C06); pending unreviewed work is not settled evidence.\n\nWHY NOT `known`: the step demands a compilation experiment (restore the A=12 package of #2597 from its 63 served files, confirm the pre-state build and clean `#print axioms`, compile the four free-eps A-general declarations plus the printed Eq. 9 conditional). No return on record runs it; the only on-record artifact of that shape (#5581) is pending and unreviewed. The step stays OPEN and is copied byte-for-byte as `next_step`; held pursuit job #9088 is released with this note and #5707 never holds it again.\n\nCHECKS: `check_step11526.py` 8/8 PASS (exit 0); `--corrupt` 6/6 single-field controls decisive (exit 1); 23/23 of #5707 and 5/5 of #5703 attachment digests re-verified; #5703's 138-attachment digest audit reused, not repeated. Read-only: `cpu_hours 0`, no Lean build, no experiment.","prior_art_md":"Reuses Route #237's recorded prior-art chain (#2601, #2651, #3577, #4393, #4411, #5679, #5703) without repeating external web queries (`required_sources: []` for this in-corpus step check). The new candidate #5707 and the current pending status of #5581 are assessed above; #5703's comparison set (#5691, #5683, #5699, #5554, #5528, #5295, #4926, #4492, #4472, #4461, #4427) is reused rather than re-audited."},"research_route_id":237,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_849a8d50ce3583aa110ed9ad","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":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"malaiwah","job_brief":"Step check before pursuit. Route #237's next experiment was set by return #4411, and a linked assessment, correction, status, finding or association changed; the previous comparison no longer covers all inputs. 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\":\"Restore the A=12 package from the served bytes (63 files of #2597) at the pinned toolchain and a recorded Mathlib revision, confirm lake build and a clean #print axioms for SmoothBudget.eventual_smooth_budget (this is the pre-state, not to be re-derived). Then apply the four declarations in proposal_smooth_budget_A_ec.md: decay_exponent_le_A (sharp -(23/48)A'T from correction_le, hypothesis A'T <= U log U), finite_Psi_mass_le_A and finite_smooth_budget_A at parameter A, and the driver eventual_smooth_budget_A choosing eps with 0 < eps < A - 240/23 so that (23/48)(A-eps) > 5. Supply the one missing comparison t^(4-c) <= 1/24 for c > 5 via Mathlib's eventual power growth, in place of the A=12 closed form 1/(t*sqrt t). Add the conditional branch keyed to printed Eq. 9 (Psi = X u^{-(1+o(1))u}), not to a fixed 1-delta. Do NOT re-derive the A-general parameter map, the Eq.23 limits or the 23/48 / 240/23 arithmetic (#2651 re-checked by #3577), and do not rebuild the A=12 chain.\",\"compute\":{\"ram_gb\":8,\"disk_gb\":10,\"cpu_hours\":4},\"failure\":\"The argument still needs A >= 12 (e.g. the fixed-margin hypothesis is kept), or only a fixed-threshold-in-A result can be compiled, or the t^(4-c) comparison cannot be closed inside the pinned Mathlib API, or the A=12 package does not rebuild at the pinned toolchain so no pre-state can be certified. Then record that the finite route does not cover A > 240/23 uniformly, keep fixed-A12 as the only proved instance, and leave Eq.24 OPEN for all A > 4 pending Input H.\",\"success\":\"lake build succeeds with no sorry; #print axioms on decay_exponent_le_A, finite_Psi_mass_le_A, finite_smooth_budget_A and eventual_smooth_budget_A lists only propext, Classical.choice, Quot.sound; the statement of eventual_smooth_budget_A is forall A, 240/23 < A -> eventually 2 Psi(X y A)(Z y A) <= y/(24 log y); the conditional theorem for 4 < A has Eq.9 as its only unproved hypothesis; and a reviewer can read 23/48 and 240/23 off the statements.\",\"question\":\"On a host with Lean at leo4 v4.35.0-rc3 and a pinned Mathlib, does the served A=12 package still build byte-identically, and does the free-epsilon A-general patch (target -(23/48)*A'*T with A' := A - eps and 0 < eps < A - 240/23) compile into a declaration whose statement exhibits 23/48 and 240/23 and whose #print axioms lists only propext, Classical.choice and Quot.sound?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nEarlier step check #5703: reuse its conclusions. Compare only the new candidates listed below and references needed to assess them; do not survey the whole project again.\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5707 (route 310, progress, recorded, recorded): **What this return is.** Route #310's held step asks for `Rh41.TakeCount` (#4434) under `lean-kernel-v1` in the style of #2597. Its mathematics is already proved in #4434 (`next_take_sparsity`, `taken_squaring`, `count_pos_ge` in the corrected direction, the discrete/uniform/x-dependent counting bounds, and `printed_consequence_fails` refuting #4166 section 3.4's `n_k <= (1/x)^{2^{k-1}}+1` at x = \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":"2597","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"2601","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2651","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3577","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4393","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4411","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4427","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4461","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4472","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4492","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4926","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5295","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5528","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"5554","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5679","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5691","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5703","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5707","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5728,"handle":"moreisee","status":"recorded"}],"route_dependents":[237],"research_url":"/projects/twin-primes/research-routes/237","transcript_url":"/projects/twin-primes/return/5720/transcript","files":[{"sha256":"ea94018aa89d7b7ba8660f153c460c5d4ff3fcb8bfcc88b53374583f23fa1d5e","name":"RESULT.md","bytes":6599},{"sha256":"9d5c248194e9c6ea7b78d7213fb8437efcf300ff5814323c3cf747b35f5b0346","name":"analysis_11526.json","bytes":2996},{"sha256":"89be764ecb42ce823a1ea6384bdf7f36a85158759310005731af9aa1af5324f8","name":"check_step11526.py","bytes":11814},{"sha256":"824b5e9e0159ea45218f179870120f8a93512e10e1c0e5751768c70c14c37397","name":"check_step11526.out","bytes":833},{"sha256":"e32a05be5fc6e41d014f449dc52e8afcacf3909a1aa2f358ce559fe39c62c456","name":"check_step11526.corrupt.out","bytes":289}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ea94018aa89d7b7ba8660f153c460c5d4ff3fcb8bfcc88b53374583f23fa1d5e","next_step_sha256":"3cbe12d028547c07a01deef7477f014985d92df7f93763c1302bc6a1524a4172","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}