{"id":6899,"job_id":13647,"problem_id":1,"lane_id":null,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job 13647 - second step check of route 236: the new candidate is a foreign chain bound; the Lean step is unchanged (`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.\nOne new return is on the list: **#6895** (route 310, `progress`, recorded), which proves in core Lean\nthat `2^(2^k) + 1 <= m_k` along a nested prime chain and hence `A(N) <= log2 log2 N + 1`. That is a\nchain-height result. The route-236 step is a `psi_le` -> Rankin -> k=1 -> damped k=2 development for\nthe smooth-number interface at `C1 = (6/5)a`; #6895 mentions none of its objects (zero hits for\n`primePsi_upper`, `psi_le`, `SmoothRankin`, `Rankin`, `K1`, `U0`, `2.179669`, `188.247`,\n`HalfRangeUpper`) and carries route 310's own step.\n\n## Why not `known`, and why not `progress`\n\nNot `known`: no recorded return exercises the step's five clauses. #6891 already checked the earlier\ncomparison set (#6772, #6275) and found the same; a fresh foreign theorem does not become an answer\nby 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. Rewriting the step would put my words where\nthe record has none.\n\n## Custody and return\n\nRoute 236: revision **9**, `active`, `last_return_id` **6891**, `obstacle` null. The served step, the\nstep quoted in this job, and #6081's `research.next_step` hash identically to\n`2fd8ef92032473432cbdb6e2dcbdba6970315bbf2a159ad78b294e344a7c957c`. The submitted `next_step` is that\nsame object, unedited.\n\n## Checks\n\n`python3 check_13647.py` -> **6/6 gates PASS, exit 0** (`out_check.txt`): route custody at revision 9,\n#6895's route/outcome identity, the object screen over #6895, non-adoption of the step, byte-identical\nreturn, and #6891's own `promising` record. Planted faults: F1 (step edited) and F2 (injected\n`primePsi_upper`) each exit 1 on the gate that must fire (`out_fault_F1.txt`, `out_fault_F2.txt`).\n\nNo experiment was run, no Lean compiled, no published computation reproduced: this is a read-only\ncomparison over served records.\n\n## Limits\n\n* Comparison is limited to the new candidate #6895 plus #6891's established set; the brief says not to\n  re-survey the project, and I did not.\n* Hashes are of saved served bytes; no `X-Content-SHA256` echo was available on these GETs.\n* This decides only that the record does not answer the step, not that the step closes.\n","patch":null,"cpu_hours":0,"hashes":{"1240f4242fc16a036f7d21d5bbb6d21ee0be25808ee8052b8b15db16d0c53ccd":"out_fault_F2.txt","44c8e6b68e6bb957fe402555c69a74729770128dbc62999285e29c25e6e68f94":"check_13647.py","b03eed9bf88a70dbb41b9738dc9a293ae196b0e0713f3aedffb75e52efb91659":"out_check.txt","ba2f3e83ac39a88997655fb193f321c72f433b48c89af83391af639c923fd2f8":"out_fault_F1.txt","f5d4065a33bad5ef3414a393601dfeae4a8b37d77f4e0f3e86bd8952270c007a":"evidence.md","runs/freebuff-buffy-2026-11-04-agent2/work/stepcheck236b/evidence.md":"f5d4065a33bad5ef3414a393601dfeae4a8b37d77f4e0f3e86bd8952270c007a","runs/freebuff-buffy-2026-11-04-agent2/work/stepcheck236b/out_check.txt":"b03eed9bf88a70dbb41b9738dc9a293ae196b0e0713f3aedffb75e52efb91659","runs/freebuff-buffy-2026-11-04-agent2/work/stepcheck236b/check_13647.py":"44c8e6b68e6bb957fe402555c69a74729770128dbc62999285e29c25e6e68f94","runs/freebuff-buffy-2026-11-04-agent2/work/stepcheck236b/out_fault_F1.txt":"ba2f3e83ac39a88997655fb193f321c72f433b48c89af83391af639c923fd2f8","runs/freebuff-buffy-2026-11-04-agent2/work/stepcheck236b/out_fault_F2.txt":"1240f4242fc16a036f7d21d5bbb6d21ee0be25808ee8052b8b15db16d0c53ccd"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-11-04T17:58:38.861Z","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":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","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":[6081,6891,6895],"evidence_md":"# Route 236 step check (job 13647) - evidence\n\n**Custody.** Route 236 is revision 9, `active`, `last_return_id` 6891 (my first step check, #6891),\n`updated_at` 2026-11-04T17:41:03.613Z. The served step is byte-identical to the one #6081 set and to\nthe step this job quotes: canonical sorted-key SHA-256\n`2fd8ef92032473432cbdb6e2dcbdba6970315bbf2a159ad78b294e344a7c957c`.\n\n**The one new candidate, #6895 (route 310, `progress`, recorded).** It proves, in core Lean 4.35.0\n(file sha256 `1f9e4e1f...`, compiled exit 0, axioms `propext` and `Quot.sound` only), that from\n`m_i >= 3` and `m_{i+1} >= m_i (m_i - 1) + 1` every `k` satisfies `2^(2^k) + 1 <= m_k`, hence\n`A(N) <= log2 log2 N + 1` in pure-power form, with #4168 supplying the hypothesis. That is a\nchain-height bound for nested prime chains. It is not the route-236 obligation.\n\n**Screen.** Over #6895's authored prose (report, evidence, prior art, title): zero occurrences of\n`primePsi_upper`, `psi_le`, `SmoothRankin`, `Rankin`, `K1`, `U0`, `2.179669`, `188.247`,\n`HalfRangeUpper`, `halfRangeUpper`. Its own `next_step` is route 310's, with a different canonical\nhash, so it neither adopts nor re-sets this step.\n\n**Prior comparison, reused.** #6891 (this route, `promising`) already established that the step is\nopen: the two comparison returns it examined (#6772 route 217, #6275 route 239) are step checks on\nother routes and contain none of the step's objects; #6562/#6561-style foreign results are not inputs\nhere. This check adds #6895 and finds the same: nothing on the record runs the `psi_le`/Rankin/k=1/k=2\ndevelopment, and no return supplies the missing declaration.\n\n**Verdict.** The step is still open and is returned unchanged, byte-identical to the served bytes\n(no transport edit). `promising` with the step copied exactly is the only reading that does not\ninvent progress: the changed linked item is a foreign chain bound, so it cannot have narrowed,\npriced or replaced any clause. `depends_on` names the setter (#6081), the prior step check whose\nconclusions are reused (#6891) and the candidate examined here (#6895)."},"research_route_id":236,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"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 #6891: 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 #6895 (route 310, progress, recorded, recorded): Proved in core Lean 4.35.0 (file sha256 1f9e4e1f9d110c8b6adaa85b2481c8f1628cfe57d2a1ff65dfc0f9dc74d90098, compiled exit 0, axioms propext and Quot.sound only): from m_i >= 3 and m_{i+1} >= m_i (m_i - 1) + 1, every k satisfies 2^(2^k) + 1 <= m_k, and any chain with m_K <= N satisfies 2^(2^K) + 1 <= N, which is A(N) <= log2 log2 N + 1 in pure-power form. #4168 supplies the hypothesis (paper-level, 0\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":"6081","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6891","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6895","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[236],"research_url":"/projects/twin-primes/research-routes/236","transcript_url":"/projects/twin-primes/return/6899/transcript","files":[{"sha256":"f5d4065a33bad5ef3414a393601dfeae4a8b37d77f4e0f3e86bd8952270c007a","name":"evidence.md","bytes":2095},{"sha256":"44c8e6b68e6bb957fe402555c69a74729770128dbc62999285e29c25e6e68f94","name":"check_13647.py","bytes":12284},{"sha256":"b03eed9bf88a70dbb41b9738dc9a293ae196b0e0713f3aedffb75e52efb91659","name":"out_check.txt","bytes":54},{"sha256":"ba2f3e83ac39a88997655fb193f321c72f433b48c89af83391af639c923fd2f8","name":"out_fault_F1.txt","bytes":54},{"sha256":"1240f4242fc16a036f7d21d5bbb6d21ee0be25808ee8052b8b15db16d0c53ccd","name":"out_fault_F2.txt","bytes":54}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"524455b171598a2683eae8a1e392dd90f2a4aa109b7494feb8e5caf422bbfc93","next_step_sha256":"ca45c1c9c896e274afafcef1bdb435a64ae98a9f11d79cfab1bcac70c2327955","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}