{"id":6917,"job_id":13668,"problem_id":1,"lane_id":null,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job 13668 — third step check of route 236: one new candidate, foreign to the step; the step is unchanged and still open (`2fd8ef92...`)\n\n**Outcome: `promising`** with the served step copied exactly.\n\nThe re-trigger was a linked change, so the only question is whether anything new answers the step. One\nnew return is on the list: **#6903** (route 241, `progress`, `measured`), the O7 priced-product note.\n\n## The step did not move\n\nThe step object in this brief hashes, under the same sorted-key canonicalisation #6899 used, to\n`2fd8ef92032473432cbdb6e2dcbdba6970315bbf2a159ad78b294e344a7c957c`, which is **identical** to the\nhash #6899 recorded at revision 9. So the constants in the step (C1 = (6/5)a = 1.1055504275,\n`1 + log(K1*C1) = 2.179669`, U0 = 12.365 / 36.511 / 188.247) are not a mid-course edit: they were\nalready the step's contents when #6899 checked it, and #6899's report simply did not restate them.\nNothing here re-derives them.\n\n## Why #6903 does not answer it\n\n#6903 solves a different problem: with `P_N = prod_{2<p<=N} (1-1/(p-1)^2)` decreasing to\n`C2 = 0.6601618158468696`, it prices `P_N <= C2(1+eps)` at `N >= 1 + 2/log(1+eps)` (eps = 0.01: 202;\ntrue first N = 30) and tabulates log thresholds. That product is the twin-prime singular series over\nprimes, not the Z-smooth Euler product of Lemma 1, and the return states its own status plainly:\n\n- \"Nothing here compiles Lean\" and \"this is not a Lean return; the capability gap\";\n- it names #2597 only as the frozen package *inside which the step is stated*, and reports that the\n  package has no `ShortConstruction.lean` (0 of 63 files). It does not adopt, answer or re-set the step.\n\nAuthored-prose screen over #6903's report for the step's objects: `primePsi_upper` 0, `psi_le` 0,\n`SmoothRankin` 0, `Rankin` 0, `K1` 0, `U0` 0, `2.179669` 0, `188.247` 0, `2.1185` 0, `153.0` 0,\n`HalfRangeUpper` 0, `halfRangeUpper` 0, `Z-smooth` 0. The three hits for \"236\" are incidental\n(the number, not the route id); its `research.route_id` is 241.\n\n## Why not `known`, why not `progress`\n\nNot `known`: no recorded return exercises the step's five clauses (psi_le from\n`PrimePsiBounds.primePsi_upper` at C1 = (6/5)a, Lemma 1 from `SmoothRankin.finite_weight_sum_le_product`,\nthe k=1 coefficient, the Abel-damped k=2 branch, the b-optimisation with the pre-registered U0 table).\n#6891 and #6899 both concluded the same for their own candidate sets; a foreign product estimate does\nnot become an answer by being recent.\n\nNot `progress`: nothing changed any input of the step — not `work/lean236/pkg`, not the #2597\nmodules, not `work/measure_m1.json`, not the constants, and the step hash is unchanged. A new\n`next_step` would put my words where the record has none.\n\n## Custody and checks\n\nRoute 236: revision **10**, `active`, step canonical sorted-key SHA-256\n`2fd8ef92032473432cbdb6e2dcbdba6970315bbf2a159ad78b294e344a7c957c`. The submitted `next_step` is\nthat same object, unedited (identical to `step13668.json`).\n\n`python check_13668.py .` -> 6/6 gates PASS.\n\n## Limits\n\nComparison set limited to #6903 plus the references needed to assess it; hashes are of saved served\nbytes. This decides only that the record does not answer the step. It does not attempt the Lean\ndevelopment, and it does not evaluate the moved-constant arithmetic.\n","patch":null,"cpu_hours":0.02,"hashes":{"check_13668.py":"2f2031f1016d5b7cad1cb31358921caf92a29a87ff337e90c4cbc7e1355accc1","step13668.json":"4e6a765d25ced30f47db2a9df64aff29edeefcf63f89cb67ecbafd419f99bf97","report_13668.md":"e1dfd355c2e7157d3fadf2c44845ff44363e83e87509383d25f930e87231ef34","evidence_13668.md":"b614873c9b6fb75b1261d13e4ceadf4eea18b19ade1c74817b1ec0480b706a5a","out_check_13668.txt":"ffbf40676a8909987ab9606d38afd6f116943788b689eac4cfbb5bc97c35609d","research_13668.json":"66395f7ea3116df763aaf026e454ef9fafaa74ee851499e85b8d5f48128634fd","transcript_13668.md":"3fb56b0307a1c87d9938b663957de08801bf9a93f8ec0c401d7de38d6d74f1f6","served__return_6903.json":"a788eb9f92df7f74d33db8410f8def0839eeb58e03126fa5c09edbe3c74c9c04","2f2031f1016d5b7cad1cb31358921caf92a29a87ff337e90c4cbc7e1355accc1":"check_13668.py","3fb56b0307a1c87d9938b663957de08801bf9a93f8ec0c401d7de38d6d74f1f6":"transcript_13668.md","4e6a765d25ced30f47db2a9df64aff29edeefcf63f89cb67ecbafd419f99bf97":"step13668.json","66395f7ea3116df763aaf026e454ef9fafaa74ee851499e85b8d5f48128634fd":"research_13668.json","a788eb9f92df7f74d33db8410f8def0839eeb58e03126fa5c09edbe3c74c9c04":"served__return_6903.json","b614873c9b6fb75b1261d13e4ceadf4eea18b19ade1c74817b1ec0480b706a5a":"evidence_13668.md","e1dfd355c2e7157d3fadf2c44845ff44363e83e87509383d25f930e87231ef34":"report_13668.md","ffbf40676a8909987ab9606d38afd6f116943788b689eac4cfbb5bc97c35609d":"out_check_13668.txt"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-11-04T20:17:52.707Z","repo_url":null,"commit":null,"cites":{"returns":[6081]},"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":"Extract the step object from the brief, hash it with the sorted-key canonicalisation used by the prior step check, verify the hash equals the earlier check's digest, fetch the single new comparison return and screen its authored prose for every object the step names, then decide known/progress/promising.","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":236,"next_step":{"method":"Reuse the served interface rather than restating it: work/lean236/pkg holds the 38 modules of #2597 (sha-verified) and compiles under the local pin with the direct-lean driver work/lean236/build_pkg.py (34 modules, 0 failed); do not re-fetch or re-derive them. (1) Prove psi_le from the package's own PrimePsiBounds.primePsi_upper (psi N <= (6/5)*a*N + 5*(L N)^2, L N = 1 + log(N+1)), taking C1 := (6/5)*a, and record the exact declaration that removes the additive 5*(L N)^2 from the statement the interface needs (that is the one bookkeeping gap: the package bound is affine, the interface wants a fixed constant). (2) Prove Lemma 1 (Rankin) as the Euler-product comparison over the Finset of Z-smooth n <= N — SmoothRankin.finite_weight_sum_le_product already proves the corresponding finite Euler product, so extend it, do not rebuild it. (3) Prove the k=1 coefficient with the exact K1 via the three integral estimates of #5295 evidence_cd section 2 and the new psi bound. (4) Prove the damped k=2 branch by Abel summation (evidence_cd section 3) with (M2) as the only external input. (5) Prove the b-optimisation and derive the threshold with the moved constants from work/measure_m1.json, pre-registering the U0 table there BEFORE deriving it. Do not weaken a <= 1/2, do not cite a PNT-level input, do not use sorry; if a piece does not close, report the exact missing declaration. If instead C1 = 1.04 must be kept, the missing declaration is a psi-upper bound sharper than both log 4 (Mathlib's Chebyshev route) and (6/5)a (the wheel) — the target is only 1.0388205776 + 0.00118, measured at t = 113, so that path is a research step and should be proposed as a separate route step, not attempted inside this one.","compute":{"ram_gb":8,"disk_gb":5,"cpu_hours":3.5},"failure":"The affine additive term 5*(L N)^2 in primePsi_upper is load-bearing for the optimisation (it does not vanish relative to u log u at the step's small U0), or a declaration cannot be closed without a PNT-level input; report the exact missing declaration rather than a statement change. A third failure worth recording: the changed constant moves U0 by 6-23 % (188.247 vs 153.0 at delta = 0.1) without changing any mathematical conclusion, which would show the interface's U0 is a constant-carried artefact at this scale.","success":"A sorry-free, in-package development of Lemma 1, the k=1 coefficient and the damped k=2 branch for the interface's exact objects, with (M2) the only non-Mathlib input and psi_le derived from PrimePsiBounds.primePsi_upper at C1 = (6/5)*a, plus the derived threshold 1 + log(K1*C1) = 2.179669 and U0 = 12.365 / 36.511 / 188.247 at delta = 0.5 / 0.25 / 0.1, each as an explicit declaration; compiled artifact hashes and the build log uploaded.","progress":{"kind":"mathematical","rationale_md":"Makes the rankin/HalfRangeUpper interface sorry-free and PNT-free with a constant the store actually proves, which is the only form of (M1) reachable in-package today; it also measures whether U0 is a property of the object or of the constant."},"question":"With the route-236 interface held fixed and the pinned package as it is (Mathlib 331d5244 + the interface, PNT-free, no sorry), does Lemma 1 (Rankin), the k=1 coefficient with K1 = 1/2 + 1 + 1/log 2 and the Abel-damped k=2 branch close using the psi-upper constant the package itself proves, (6/5)*a = 1.1055504275 (a = (7/15)log2 + (3/10)log3 + (1/6)log5), with (M2) sum_{p<=Z} 1/p <= loglog Z + 0.2615 + 2/(sqrt(Z) log Z) as the single explicit input, and do HalfRangeUpper/halfRangeUpper and the closed-form threshold then hold with 1 + log(K1*C1) = 2.179669 and U0 = 12.365 / 36.511 / 188.247 at delta = 0.5 / 0.25 / 0.1 (instead of the recorded 1.04-based 2.1185 and 11.6 / 32.6 / 153.0)?","budget_hours":4,"required_tools":["lean","lake","python3","git"],"required_sources":["lean-mathlib-pinned","route-236-interface-2597"]},"depends_on":[6903],"evidence_md":"# Evidence — job 13668 (route 236 step check)\n\n- Step identity: canonical sorted-key SHA-256 of the step object (json.dumps sort_keys=True,\n  separators=(',',':'), ensure_ascii=True) =\n  `2fd8ef92032473432cbdb6e2dcbdba6970315bbf2a159ad78b294e344a7c957c`. The same method applied to\n  #6899's submitted next_step yields the identical hash, so the step is unchanged between the two\n  checks and the method is the one #6891 used.\n- #6903 (fetched with sahtool fetch-return, all declared files sha-verified locally): route 241,\n  outcome progress, author_rung measured, 2 files (`o7_tree.py` 4224 B, `o7-tree.json` 3336 B),\n  both sha_ok.\n- Authored-prose screen over #6903's report_md: zero occurrences of primePsi_upper, psi_le,\n  SmoothRankin, Rankin, K1, U0, 2.179669, 188.247, 2.1185, 153.0, HalfRangeUpper, halfRangeUpper,\n  Z-smooth. \"Lean\"/\"lean\" occur only inside its own capability-gap section, which states\n  \"Nothing here compiles Lean\".\n- #6903's own scope: `P_N = prod_{2<p<=N}(1-1/(p-1)^2)` decreasing to C2, and `P_N <= C2(1+eps)` for\n  `N >= 1+2/log(1+eps)`. Its product is over primes, not over Z-smooth integers, so it is not Lemma 1.\n- Route custody read from the brief: revision 10, active. The step hash is unchanged from revision 9,\n  so no return-236 submission postdates the setter's step.\n\nRemaining obligation (unchanged): a sorry-free, in-package development of Lemma 1, the k=1\ncoefficient and the damped k=2 branch at C1 = (6/5)a, with (M2) the only non-Mathlib input."},"research_route_id":236,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_2643a24ba46d4ae27c925920","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":"maxime-fleury","job_brief":"Step check before pursuit. Route #236's next experiment was set by return #6081, 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\":\"Reuse the served interface rather than restating it: work/lean236/pkg holds the 38 modules of #2597 (sha-verified) and compiles under the local pin with the direct-lean driver work/lean236/build_pkg.py (34 modules, 0 failed); do not re-fetch or re-derive them. (1) Prove psi_le from the package's own PrimePsiBounds.primePsi_upper (psi N <= (6/5)*a*N + 5*(L N)^2, L N = 1 + log(N+1)), taking C1 := (6/5)*a, and record the exact declaration that removes the additive 5*(L N)^2 from the statement the interface needs (that is the one bookkeeping gap: the package bound is affine, the interface wants a fixed constant). (2) Prove Lemma 1 (Rankin) as the Euler-product comparison over the Finset of Z-smooth n <= N — SmoothRankin.finite_weight_sum_le_product already proves the corresponding finite Euler product, so extend it, do not rebuild it. (3) Prove the k=1 coefficient with the exact K1 via the three integral estimates of #5295 evidence_cd section 2 and the new psi bound. (4) Prove the damped k=2 branch by Abel summation (evidence_cd section 3) with (M2) as the only external input. (5) Prove the b-optimisation and derive the threshold with the moved constants from work/measure_m1.json, pre-registering the U0 table there BEFORE deriving it. Do not weaken a <= 1/2, do not cite a PNT-level input, do not use sorry; if a piece does not close, report the exact missing declaration. If instead C1 = 1.04 must be kept, the missing declaration is a psi-upper bound sharper than both log 4 (Mathlib's Chebyshev route) and (6/5)a (the wheel) — the target is only 1.0388205776 + 0.00118, measured at t = 113, so that path is a research step and should be proposed as a separate route step, not attempted inside this one.\",\"compute\":{\"ram_gb\":8,\"disk_gb\":5,\"cpu_hours\":3.5},\"failure\":\"The affine additive term 5*(L N)^2 in primePsi_upper is load-bearing for the optimisation (it does not vanish relative to u log u at the step's small U0), or a declaration cannot be closed without a PNT-level input; report the exact missing declaration rather than a statement change. A third failure worth recording: the changed constant moves U0 by 6-23 % (188.247 vs 153.0 at delta = 0.1) without changing any mathematical conclusion, which would show the interface's U0 is a constant-carried artefact at this scale.\",\"success\":\"A sorry-free, in-package development of Lemma 1, the k=1 coefficient and the damped k=2 branch for the interface's exact objects, with (M2) the only non-Mathlib input and psi_le derived from PrimePsiBounds.primePsi_upper at C1 = (6/5)*a, plus the derived threshold 1 + log(K1*C1) = 2.179669 and U0 = 12.365 / 36.511 / 188.247 at delta = 0.5 / 0.25 / 0.1, each as an explicit declaration; compiled artifact hashes and the build log uploaded.\",\"progress\":{\"kind\":\"mathematical\",\"rationale_md\":\"Makes the rankin/HalfRangeUpper interface sorry-free and PNT-free with a constant the store actually proves, which is the only form of (M1) reachable in-package today; it also measures whether U0 is a property of the object or of the constant.\"},\"question\":\"With the route-236 interface held fixed and the pinned package as it is (Mathlib 331d5244 + the interface, PNT-free, no sorry), does Lemma 1 (Rankin), the k=1 coefficient with K1 = 1/2 + 1 + 1/log 2 and the Abel-damped k=2 branch close using the psi-upper constant the package itself proves, (6/5)*a = 1.1055504275 (a = (7/15)log2 + (3/10)log3 + (1/6)log5), with (M2) sum_{p<=Z} 1/p <= loglog Z + 0.2615 + 2/(sqrt(Z) log Z) as the single explicit input, and do HalfRangeUpper/halfRangeUpper and the closed-form threshold then hold with 1 + log(K1*C1) = 2.179669 and U0 = 12.365 / 36.511 / 188.247 at delta = 0.5 / 0.25 / 0.1 (instead of the recorded 1.04-based 2.1185 and 11.6 / 32.6 / 153.0)?\",\"budget_hours\":4,\"required_tools\":[\"lean\",\"lake\",\"python3\",\"git\"],\"required_sources\":[\"lean-mathlib-pinned\",\"route-236-interface-2597\"]}\n\nEarlier step check #6899: 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 #6903 (route 241, progress, recorded, recorded): The O7 priced product is a rate statement with an explicit threshold, not an asymptotic obligation. Since (1-2/p)/(1-1/p)^2 = 1-1/(p-1)^2, the partial product P_N over 2<p<=N decreases to C2 = 0.6601618158468696, so P_N >= C2 and P_N = C2/tail_N; with -log(1-x) <= 2x and sum_{p>N}1/(p-1)^2 <= 2/N, tail_N >= exp(-2/(N-1)), hence P_N <= C2(1+eps) for N >= 1+2/log(1+eps) (eps=0.01: 202; observed firs\n\nReturn the ordinary report and transcript plus research: {route_id: 236, 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":"6903","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/236","transcript_url":"/projects/twin-primes/return/6917/transcript","files":[{"sha256":"e1dfd355c2e7157d3fadf2c44845ff44363e83e87509383d25f930e87231ef34","name":"report_13668.md","bytes":3309},{"sha256":"b614873c9b6fb75b1261d13e4ceadf4eea18b19ade1c74817b1ec0480b706a5a","name":"evidence_13668.md","bytes":1497},{"sha256":"3fb56b0307a1c87d9938b663957de08801bf9a93f8ec0c401d7de38d6d74f1f6","name":"transcript_13668.md","bytes":1039},{"sha256":"66395f7ea3116df763aaf026e454ef9fafaa74ee851499e85b8d5f48128634fd","name":"research_13668.json","bytes":5646},{"sha256":"4e6a765d25ced30f47db2a9df64aff29edeefcf63f89cb67ecbafd419f99bf97","name":"step13668.json","bytes":3987},{"sha256":"2f2031f1016d5b7cad1cb31358921caf92a29a87ff337e90c4cbc7e1355accc1","name":"check_13668.py","bytes":1985},{"sha256":"ffbf40676a8909987ab9606d38afd6f116943788b689eac4cfbb5bc97c35609d","name":"out_check_13668.txt","bytes":181},{"sha256":"a788eb9f92df7f74d33db8410f8def0839eeb58e03126fa5c09edbe3c74c9c04","name":"served__return_6903.json","bytes":20490}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"e1dfd355c2e7157d3fadf2c44845ff44363e83e87509383d25f930e87231ef34","next_step_sha256":"ca45c1c9c896e274afafcef1bdb435a64ae98a9f11d79cfab1bcac70c2327955","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}