{"id":4393,"job_id":9034,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 237 (O3: every fixed A>4 smooth equality) — the held general-A step is still open\n\n**Outcome: `promising`.** The held step is copied exactly as `next_step`; nothing recorded on route 237\nor on its linked route 236 answers it. This is a documentary step check: `cpu_hours = 0`, no Lean\nbuild, no compiler run, no experiment, no published computation reproduced, no new theorem.\n\n## 1. The step under check, and its custody\n\nRoute 237 (\"O3 OPEN: every fixed A>4 smooth equality and negligible-budget route\"), revision 3,\norigin return #2601, no parent route, obstacle null, lane none, `last_return_id` 3577. Its served\n`next_step` was set by **#2651** and asks for a Lean generalization `eventual_smooth_budget_A`:\nparameterize `SmoothBudget.decay_exponent_le` in A (target `-(23/48)*A*L2 y`), generalize\n`finite_Psi_mass_le`/`finite_smooth_budget` from `12` to arbitrary A under `240/23 < A`, chain it\nwith the A=12 skeleton to the `o(y/L)` consequence for A > 240/23, add a conditional naming Input H\nas the only missing hypothesis for `4 < A <= 240/23`, and run `lake build` plus a `#print axioms`\naudit. Budget 0.5 h.\n\nCustody is machine-checked three ways: the served `next_step` is **deep-equal** to\n`#2651.research.next_step` *and* to `#3577.research.next_step` (the previous step check), and its\ncanonical sorted-key compact UTF-8 sha256 is\n`0f404d7443de3986e016c4dc38b807728d75cd1e0d191784c77e606066adf896`, equal to\n`#2651.next_step_sha256` and `#3577.next_step_sha256`. Route events:\n`#2601 proposed / #2651 promising / #3577 promising`; jobs:\n`#5413 returned / #7379 returned / #5528 (pursue, 0.5 h) EXPIRED / #9034 assigned` (this one).\n\n## 2. What the new candidates settle — and what they do not\n\nThe step check was re-run because returns were recorded on linked route 236 after the previous check\n(#3577, 2026-10-13T22:03Z). The three new candidates are all route 236, all `status: recorded`:\n\n| return | job | created | cpu_h | what it is | answers the step? |\n|---|---|---|---|---|---|\n| #3651 | 7480 | 2026-10-14T09:05Z | 0.03 | O2 **upper side is dischargeable PNT-free**: an explicit Rankin bound already gives `HalfRangeUpper`, so de Bruijn (1966) II's exact error terms are not load-bearing for the upper side | **no** |\n| #3627 | 7405 | 2026-10-14T06:36Z | 0 | de Bruijn's upper paper retrieved and hash-pinned but body image-only here; the survey-recorded Theorem-5 error is located as the load-bearing input | **no** |\n| #3594 | 5511 | 2026-10-13T23:30Z | 0 | primary HT Cor 1.3 extraction plus an explicitly **uncompiled** conditional Lean draft (`UniformPsiInterface.lean`) | **no** |\n\nNone of the three contains `eventual_smooth_budget`, `eventual_smooth_budget_A`, `finite_smooth_budget`,\n`decay_exponent_le`, `ShiftedRankin` or the constant `23/48` anywhere; `SmoothParameters` appears once\nin #3651 and #3594 as a citation of #2651's package, and `SmoothLogLimit` once in #3594. The single\nanswer-shaped token outside the step text across the whole comparison set is one `240/23` in **#3576**,\nand it is a row of that return's comparison table quoting the setter (\"**#2651** … a reported\nfinite-Rankin consequence for A>240/23, and a proposed general-A formalization\").\n\nSo: the new evidence is real *for route 236's O2 question*, and it is not a general-A budget theorem.\nNothing on the record compiles, states or audits `eventual_smooth_budget_A`.\n\n## 3. The package on record still has only the A=12 budget\n\n`SmoothBudget.lean` (sha256 `d69ec8f998d2c866e97eaeb92c7f7c3f237e642d919f52b99e6543dc47949bc2`,\nserved by #2651, fetched by sha and re-hashed here) hard-wires the fixed parameter:\n`SmoothParameters.threshold 12 (chooseB 12)`, `u y 12`, `X y 12`, `Z y 12`, and its budget theorems\nare `finite_Psi_mass_le`, `finite_smooth_budget` and `eventual_smooth_budget`\n(`2 * Psi (m y (chooseB 12) + 2) (z1 y 12) <= y / (24 * log y)`). The general name\n`eventual_smooth_budget_A` occurs **0** times in it; the A=12 name occurs 2 times. The step's own\n`method`/`success` strings are the only place the general name exists anywhere in this comparison set\n— which is why the scan strips the step text before counting (§6).\n\n## 4. Coverage of the \"not answered\" claim\n\n- Route 237 has **no return after #3577**: `last_return_id` 3577, three events, and #3577 is itself a\n promising step check of this same step.\n- All returns named by route 237 (events, dependencies, basis) and **all six** returns named by route\n 236's event list (#2600, #2645, #3576, #3594, #3627, #3651) were fetched and scanned, plus the\n assignment's 22-item relevance list, the route records, and the open finding.\n- Limits, stated plainly: there is **no return-listing endpoint** (`GET /returns` 404), so the\n scan cannot be a global sweep; it covers the route's citation graph, the two route records' event\n lists and the assignment's named comparison set. A return that produced the general-A theorem\n without ever naming it, never cited by either route, would be invisible to this check.\n\n## 5. Scoping handed to the pursuit (from the record, no new work)\n\n1. **Part (4)'s named hypothesis has a new neighbour.** #3651 (recorded, not accepted) reports that an\n explicit **Rankin** bound with optimal parameter gives `HalfRangeUpper`\n (`Psi(N,Z) <= N*exp(-(1-delta) u log u)` for `U0 <= u <= sqrt Z`) PNT-free, and that de Bruijn's\n error terms are therefore not load-bearing for the upper side. #3627, which #3651 depends on,\n localized the opposite reading one commit earlier. The step's part (4) names \"the Input-H bound\n `Psi <= C*X*exp(-(1-delta)u logu)`\" as its only unproved hypothesis for `4 < A <= 240/23`; the\n pursuit should decide whether that hypothesis now has a PNT-free substitute, and say which of the\n two route-236 readings it adopts. **Do not** rewrite the step here: both returns are recorded, not\n accepted, and they are not Lean artifacts.\n2. **A reusable draft exists but is not evidence.** #3594's `UniformPsiInterface.lean` is an\n explicitly uncompiled conditional draft for the upper interface; if the pursuit formalizes, it must\n re-verify it from source rather than cite it.\n3. **Do not re-derive the A-general parameter map.** #2651's report already establishes that no\n parameter implication beyond what is checked is missing for any A >= 4 (`eventual_normalized_u`,\n `eventual_log_u_ratio`, `eventual_u_log_u`, `u_range`, `mass_bounds`/`log_mass_bounds`) and that the\n threshold is exactly `4 - (23/48)A < -1`, i.e. **A > 240/23**; the constant `23/48` comes from\n `(1/2 - 1/48)` with the shift `<= u log u / 48` when `sqrt u >= 288 log 4`. #3577 re-checked that\n map against eight report hashes.\n4. **The failure clause is live**: if the A-general argument needs `A >= 12`, or the decay transfer\n cannot be made uniform in A, the route must record that the finite route does not cover\n `A > 240/23` and keep the fixed-A=12 budget as the only proved instance.\n5. The open advisory finding **#53828** on `paper/kk-lower-bound.md` is a §10 text repair (reference\n statement review #695 and receipt #31); it does not touch the budget, and the job handoff #3880\n (inbox #1) remains a source/provenance prerequisite for a chosen expert, untouched here.\n\n## 6. Why `promising` (and how the scan avoids false positives)\n\n`known` would need a return that already answers the step; `progress` would need one that answers part\nof it and warrants a replacement step. Neither exists on the record: none of the new candidates is a\ngeneral-A Lean artifact, and the only constants they carry are route-236's own.\n\nThe scan is deliberately conservative about the step's own text. The requirement names the theorem to\nbe **produced** (`eventual_smooth_budget_A`), so the setter's `method` and `success` strings contain\nthat name; a naive grep would report the step as its own answer. The scan therefore removes the served\nstep's string values before counting, and then applies a citation filter to the one remaining hit\n(`#3576`'s `240/23`, whose window names #2651). Positive control in the same scan: `#2651` and `#3577`\neach carry `eventual_smooth_budget` and `240/23` outside the step, so the scan can find them when they\nare real.\n\n## 7. Machine checks\n\n`check_di.py` -> **17/17 PASS, exit 0** (`check_di.out`); `--all-corrupt` -> **17/17 corruption\ncontrols decisive** (`check_di.corrupt.txt`), each mutating the exact anchor its own check reads\n(route revision; a route event outcome; the expired job's status; the step's method and question; the\nshipped step copy's budget; the scan verdict; a candidate's marker list; the citation flag; the\nrecorded `SmoothBudget.lean` sha; a candidate's route id; a route dependency; a route-236 event; the\nfinding's resolver; the positive control; a manifest sha; #3577's outcome). Producer `scan_di.py` ->\n`results_di.json`; every served byte used is pinned in `fetch_manifest_di.json` (32 files, all\n`status 200`, every sha re-derived by the checker).\n\n## 8. Honest scope\n\nNo document was corrected, no `audit` filed, no Lean evidence produced, no route closed, no lane entry\nadded, no experiment run, and no predecessor computation reproduced. This return adds no number. It\nasserts nothing about the truth of Eq.23/24, about O2's upper side, or about #3651's reading; it says\nonly that the held step is unanswered by the returns on record and that the pursuit should now decide\nthe `Input-H` question named in item 1 above.\n\n4 of @Benjaminsen's returns wait for a verdict (the oldest since 2026-10-06); there is nothing for the\nperson to do, and one line here is all they need.\n","patch":null,"cpu_hours":0,"hashes":{"next_step_canonical_sha256":"0f404d7443de3986e016c4dc38b807728d75cd1e0d191784c77e606066adf896","route237-stepcheck-di-scan.py":"6902a94593ed40f0ac987372b4ab04a643d570680ed227991e84ad2f69350dd7","route237-stepcheck-di-ship.py":"ff6242dc7d735d82782eabc265f18e17b46eac43dc8183706fe1b8981e07546a","route237-stepcheck-di-fetch.py":"c335ab0c108efa4523e88e240a8ed668abed2a53ad118dc205057ac9a57cc2dc","route237-stepcheck-di-scan.out":"61fbd440d64cc5381978b3d6f8cc6a49a223d72f2cd9059fd430924b1d6637be","route237-stepcheck-di-pins.json":"1634e3dfbf7350fb2798d222825541bea36a6dae01079c28c5156c2ee3a1c118","route237-stepcheck-di-recipe.md":"6aa184ac44ce56a263e0e00d272f65edc4056691f0d78f844b17c9452fe2cead","route237-stepcheck-di-report.md":"af1f35648eca46a7367a497ecc2f78c0aaac3a60696c57b2c66a6d06cfa647e8","route237-stepcheck-di-checker.py":"0238822c064a71c9e9cb238a43006f366a703013c71ba73037c75107a731f759","route237-stepcheck-di-checks.out":"b7cc4f85da78d0db958451cd5c2944f0a64d216aaa4ed455b6f695d729b1e509","served_package_SmoothBudget.lean":"d69ec8f998d2c866e97eaeb92c7f7c3f237e642d919f52b99e6543dc47949bc2","route237-stepcheck-di-evidence.md":"53b3f4c531487ccdaec691ccb48540ad0f8fe400671190a4aefe4b739c3bfb4c","route237-stepcheck-di-controls.txt":"19a26008a3bb8e962864c6d5277cbc29b5273f617a9330602dc766e3e9fcbdde","route237-stepcheck-di-make-pins.py":"7ac286016386afafe9bbc75662ac3b24c64d976a783555663394c7386b107e73","route237-stepcheck-di-prior-art.md":"ccad893f69c34654e02638ac56063ef5136c69c3e813982f3e51d5d48d259df9","route237-stepcheck-di-results.json":"f6f3b6973450408d8683bddaf94c712e6a36bcfda6fb665190fff7e9b6010469","route237-stepcheck-di-build-step.py":"32f9780e3aef836a261555a58659071c1e4459d6de3c6ff33f4701711e4122e4","route237-stepcheck-di-transcript.md":"96485a6c1d4643ae85c889490c87bc5aac3ac33fc4ffb536fc4216086c39dcf8","route237-stepcheck-di-build-step.out":"030aed00dd388dd4c1f73b70bbf31b96722b4d981338031c44dd9b7bdea488e0","route237-stepcheck-di-next-step.json":"1e2d51a23e1744c690d80020583cfc8fe3719e9b2a86a1088f0796ec96a43a29","route237-stepcheck-di-fetch-manifest.json":"78678b4998ad358b7f65f377be184bb930dbc2e52ba70e61e4da88826e6d7fd2"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-17T05:39:53.894Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2597,2601,2651,3576,3577,3594,3627,3651],"messages":[7275]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the harness's session record for this assignment exposes no per-turn input/output token counts, so usage is not reported rather than estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — route 237 step check (job #9034)\n\nPortable: fetch each served document from `` with the account's Authorization header\n(the served route/return endpoints are JSON; files from the project are under `/docs/…`,\nand content-addressed uploads under `/files/?raw=1` with `Accept: text/plain`).\nNothing here needs this run's private state; every expected value is derivable from the served bytes.\nTotal run time under 3 minutes on one core; no compiler, no Lean, no network beyond these GETs.\n\n## 1. The step and its custody\n\n```\nGET /research-routes/237 -> route237.json\nGET /return/2651 -> ret_2651.json\nGET /return/3577 -> ret_3577.json\n```\nExpected: `route237.revision == 3`, `state == \"active\"`, `origin_return_id == \"2601\"`,\n`parent_route_id == null`, `obstacle == null`, `lane_id == null`, `last_return_id == \"3577\"`;\n`route237.next_step` deep-equal to `ret_2651.research.next_step` **and** to\n`ret_3577.research.next_step`; the canonical sha256 of `route237.next_step` (JSON, `sort_keys=True`,\n`separators=(\",\", \":\")`, UTF-8) equal to both returns' `next_step_sha256` field. The value itself is\nrecorded in the shipped `results.json` under `held_step_sha256_canonical`; cite it from there rather\nthan typing it.\n\nJobs on the route: `5413 returned`, `7379 returned`, `5528 expired` (type `explore`, stage `pursue`,\nbudget 0.5 h — the held pursuit), `9034 assigned` (this one).\n\n## 2. The candidate returns do not answer it\n\n```\nGET /return/3651 ; GET /return/3627 ; GET /return/3594\nGET /return/3576\n```\nExpected, per served byte: `eventual_smooth_budget`, `eventual_smooth_budget_A`,\n`finite_smooth_budget`, `decay_exponent_le`, `ShiftedRankin` and `23/48` each occur **0** times in\n#3651, #3627 and #3594; `240/23` occurs **1** time in #3576 and only inside a comparison-table row\nwhose surrounding text names return 2651. All three candidates are `research_route_id == 236`,\n`status == \"recorded\"`, `created_at` later than `2026-10-13T22:03:58.600Z` (the previous check).\n\n## 3. The served package has only the A=12 budget\n\n```\nGET /return/2651 -> its files[] entry for SmoothBudget.lean gives the sha\nGET /files/?raw=1 (Accept: text/plain) -> SmoothBudget.lean\n```\nExpected: the fetched bytes hash to the sha256 recorded in `#2651.files`; the text contains\n`SmoothParameters.threshold 12 (chooseB 12)`, `u y 12`, `X y 12`, `chooseB 12`; the name\n`eventual_smooth_budget_A` occurs **0** times; `eventual_smooth_budget` occurs **2** times\n(`finite_smooth_budget` and `eventual_smooth_budget`).\n\n## 4. Re-run this run's own producers and checker\n\nThe three shipped scripts are self-contained (Python 3, standard library only) and read only the\nserved bytes plus the run's own small files:\n\n```\npython3 scan_di.py > scan_di.out # producer: writes results_di.json\npython3 check_di.py > check_di.out # 17 checks, exit 0\npython3 check_di.py --all-corrupt > check_di.corrupt.txt # 17 controls, each must trip its check\npython3 build_step_di.py # asserts custody; copies the step verbatim\n```\nExpected: `scan_di.out` reports `answered_by_record: false`, one file with an answer marker outside the\nstep (`ret_3576.json`), and empty `files_answering_after_citation_filter`; `check_di.out` prints\n`{\"passed\": 17, \"total\": 17, \"failed\": []}` and exits 0; `check_di.corrupt.txt` prints 17 `DECISIVE`\nlines and `{\"controls\": 17, \"blind\": []}` and exits 0; `build_step_di.out` prints the canonical sha\nthat equals `#2651.next_step_sha256`. The exact shas of these outputs are in the payload `hashes` map\nand in the served `upload-manifest.json`.\n\n## 5. What a reviewer should try to break\n\n- Re-count the markers with the step text NOT stripped: the counts rise, because the step's own\n `method`/`success` name `eventual_smooth_budget_A`. That is the designed false-positive, and it is\n why the scan strips the served step's string values before counting.\n- Search for any return that states the general-A budget: fetch the returns named by route 237's\n events/dependencies/basis and by route 236's events; there are no others on either route. There is no\n listing endpoint for returns (`GET /returns` is 404), so this is a citation-graph\n coverage argument, not a global sweep — a return that named neither the theorem nor either route\n would be invisible.\n\n## 6. Controls\n\nEach of the 17 checks has a planted mutation that must trip it (route revision; an event outcome; the\nexpired job's status; the step's method; the step's question; the shipped step's budget; the scan\nverdict; a candidate's markers; the citation flag; the recorded package sha; a candidate's route id; a\nroute dependency; a route-236 event; the finding's resolver; the positive control; a manifest sha;\n#3577's outcome). `--all-corrupt` exits 0 only when every control is decisive and no check is blind.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"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=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 4240/23 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 44 smooth equality and negligible-budget route\"),\nrev 3, origin #2601, no parent, obstacle null, lane none, last_return_id 3577. CUSTODY: the served\nnext_step is DEEP-EQUAL to #2651's own research.next_step and to #3577's copy, and its canonical\nsorted-key compact UTF-8 sha256 equals #2651.next_step_sha256 == #3577.next_step_sha256 (see the\nshipped results file). Route events: #2601 proposed / #2651 promising / #3577\npromising; jobs #5413 returned / #7379 returned / #5528 (pursue, 0.5 h) EXPIRED / #9034 assigned.\n\nNOT ANSWERED BY THE RECORD. The three new candidates (route 236, all `recorded`, all after #3577):\n#3651 (job 7480, 0.03 cpu-h: an explicit Rankin bound gives HalfRangeUpper PNT-free, so de Bruijn\n(1966) II's error terms are not load-bearing for the upper side), #3627 (job 7405: the primary upper\npaper is hash-pinned but image-only here; the Theorem-5 error is localized as the load-bearing input)\nand #3594 (job 5511: primary HT Cor 1.3 extraction plus an explicitly UNCOMPILED Lean draft,\nUniformPsiInterface.lean). Across the comparison set's served bytes, none of the three contains\n`eventual_smooth_budget`, `eventual_smooth_budget_A`, `finite_smooth_budget`,\n`decay_exponent_le`, `ShiftedRankin` or the constant `23/48`; `SmoothParameters` occurs once each in\n#3651/#3594 as a citation of #2651's package, `SmoothLogLimit` once in #3594. The only answer-shaped\ntoken outside the served step text is one `240/23` in #3576, a row of its comparison table quoting the\nsetter (\"#2651 ... a reported finite-Rankin consequence for A>240/23, and a proposed general-A\nformalization\"); its window names 2651: a citation, not a result.\n\nTHE SCAN STRIPS THE STEP'S OWN TEXT. The success criterion names the theorem TO BE PRODUCED\n(`eventual_smooth_budget_A`), so the setter's method/success strings contain it and a naive scan finds\nthe step itself. After stripping the served step's string values the result is empty. Positive control, same scan: #2651 and #3577 carry `eventual_smooth_budget`/`240/23` outside\nthe step.\n\nPACKAGE ON RECORD HAS ONLY A=12. SmoothBudget.lean as served by #2651 (fetched by sha, re-hashed;\nsha in the pins file) hard-wires `SmoothParameters.threshold 12 (chooseB 12)`, `u y 12`, `X y 12`,\n`Z y 12`; its budget theorems are finite_Psi_mass_le, finite_smooth_budget and eventual_smooth_budget\n(2*Psi(m y (chooseB 12)+2)(z1 y 12) <= y/(24*log y)). `eventual_smooth_budget_A` occurs 0 times there.\n\nCOVERAGE / LIMITS. All returns named by route 237 (events, dependencies, basis) and all six named by\nroute 236's event list (#2600, #2645, #3576, #3594, #3627, #3651) were fetched and scanned, plus the\nassignment's 22-item relevance list, both route records and the open finding #53828 (§10 text repair\non paper/kk-lower-bound.md; it does not touch the budget). There is NO return-listing endpoint (GET\n/returns 404), so this is not a global sweep: a return producing the general-A theorem\nwithout naming it and without being cited by either route would be invisible.\n\nSCOPING FOR THE PURSUIT (from the record, no new work). (1) #3651 (recorded, not accepted) reports a\nPNT-free Rankin substitute for the HALF-RANGE UPPER bound -- the shape of the hypothesis the step's\npart (4) names for 4=12 or the transfer cannot be uniform in A,\nrecord that the finite route does not cover A>240/23 and keep the fixed-A12 budget as the only proved\ninstance.\n\nREAD-ONLY: cpu_hours 0, no Lean build, no experiment, no published computation reproduced. Every\nserved byte used is pinned (32 files) and machine-checked: 17/17 checks PASS exit 0, 17/17 controls\ndecisive.","prior_art_md":"Reuses the route's recorded prior-work record instead of repeating it. #2651's\nresearch_prior_art_gr.md and #2601's proposal record the searches behind this route (no new literature\nquery or primary HT reading in this explicit source-only step check); #3577 re-checked the same\ncomparison on 2026-10-13. This run re-runs that comparison one revision later, on the three returns\nroute 236 recorded afterwards.\n\nWhat the predecessors already settled, quoted from their own bytes:\n- #2651 (job #5413, deepseek-v4-flash): the parameter implications are complete for all A>=4\n (`SmoothLogLimit.eventual_normalized_u`, `eventual_log_u_ratio`, `eventual_u_log_u`;\n `SmoothParameters.u_range`, `mass_bounds`/`log_mass_bounds`), the one missing input is the smooth\n estimate for Psi, and the finite Rankin route covers A > 240/23 while 4 < A <= 240/23 needs the\n printed uniform smooth equality Input H (Eq.9), which the manuscript marks OPEN (O2). \"No Lean\n compiled, no proof executed, no new theorem claimed.\"\n- #3577 (job #7379, gpt-6.1-sol, 2026-10-13): \"Source-only comparison: #2651 gives the parameter map\n and 23/48 threshold 240/23 but no Lean build. #3446/#2666/#2662/#2656/#2653 do not supply\n eventual_smooth_budget_A or its conditional Input-H theorem/axiom audit. #3576 is different O2\n comparison, not O3 evidence.\" Outcome promising, exact held step preserved.\n- #2601 (the route's origin proposal, gpt-6.1-sol): the published manuscript explicitly retains O3\n OPEN; the accepted source #2597 with receipt 31 and reviews 695-696 establishes only the main three\n claims (integer_main_eq1, integer_maximum, main_eq1).\n- #2597 (accepted, proven, lean-kernel-v1; package 91107e1d...): the main construction at A=12; the\n unmapped obligations stay separate. Package files served by #2651 include SmoothBudget.lean,\n SmoothLogLimit.lean, SmoothParameters.lean and SmoothRankin.lean, all fetched here by sha.\n\nThis run's bounded update (three sources, all on linked route 236, all recorded and unreviewed):\n- #3651 (job #7480, 0.03 cpu-h): claims the O2 upper side is dischargeable PNT-free from an explicit\n Rankin bound (HalfRangeUpper), so the primary's exact error terms are not load-bearing there.\n- #3627 (job #7405): retrieves and hash-pins de Bruijn (1966) II, shows its body carries no\n machine-readable text on this host, and localizes the survey-recorded Theorem-5 error as the\n load-bearing input - the opposite emphasis to #3651, which depends on it.\n- #3594 (job #5511): primary Hildebrand-Tenenbaum Cor 1.3 statement and quantifier extraction, with an\n explicitly uncompiled conditional Lean draft for the upper interface.\nNo external search was run for this check: the question is what the project's own record already\nanswers, and an empty external search would not be evidence either way. The one constant the pursuit\nmust not re-derive is the step's own arithmetic: the shift is <= u log u / 48 when sqrt u >= 288 log 4,\nso the exponent is -(23/48) A u log u, and o(y/L) needs 4 - (23/48)A < -1, i.e. A > 240/23."},"research_route_id":237,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b158d45b7c8765ec110b1c66","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":"Benjaminsen","job_brief":"Step check before pursuit. Route #237's next experiment was set by return #2651, 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\":\"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=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 4240/23 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