{"id":7052,"job_id":13919,"problem_id":1,"lane_id":null,"type":"explore","user_id":121,"model":"gpt-6-astra","provider":"openai","report_md":"# Route237 step check: A-general completion and the analytic input remain open\n\nOutcome **promising**. Copy the held next_step exactly. The11 assigned later returns do not provide a compiled A-general PrimaryMain completion, a proof of InputH9Upper(1/2), or a new machine-checked reduction to an identified external analytic theorem. This is a bounded read-and-compare judgment, with no experiment or previous computation rerun.\n\n## Step custody and scope\nThe route was active at revision9, last_return5728, obstacle null. The route step, the job's step and5728.research.next_step are equal. Canonical SHA256: fe3abd935dafe849d41be7fcc17fc2e6a8884cd85b7d6b83d13a3bcf149ce070, using sorted keys, compact UTF8 JSON and ensure_ascii=False. The setter's served next_step_sha256 agrees. Route associations are empty. All six setter cited_by records were inspected at report scope:5736,5742,5749,5751,6003,6010. This checks the direct association/citation index rather than assuming the route event list is complete; it is not a whole-project search.\n\n## Existing result reused, with its status preserved\nReturn5728 is recorded, author_rung proven, with zero reviews in the fetched snapshot. We independently fetched nine immutable source/log artifacts from5728 and2597 and checked their byte hashes, but did not independently compile them. The report's historical compilation and axiom audit are author-run evidence, not a new verification by this check.\n\nSmoothBudgetA.lean (0ad2ee9758587795dfceb9476eda18a1066b9adcc955c6f097ce2a48a361c64d) supplies eventual smooth and uncovered budgets for A>240/23. It also defines InputH9Upper and supplies the conditional budgets for every A>4. The H premise remains explicit in the source and recorded audit. The corrected unconditional cost uses a fourth power of log(y), requiring decay exponent greater than5; the InputH branch uses a third power and requires exponent greater than4. The smaller-range conditional result must not be incorrectly reduced to A>5.\n\nThe original PrimaryMain.lean from accepted2597 (cff1bb8609e7ca834419684da3c81bb8630f85a7730ea5b0dd3018f24504ed68) still defines primaryCoefficient=1/chooseB12 and proves eventual_actual_completion at that fixed parameter. Its proof visibly composes the fixed-A12 budget with PrimeReserveBudget.actual_reserved_supply and ReserveBudget.actual_completion_gap. The latter ingredients are already available and parameter-compatible, but that fact is not an already-published compiled A-general theorem. No such final composition appears among5728's new declarations.\n\n## The two closest later candidates\nReturn6081 explicitly claims no new proof. Its package psi bound includes a positive additive logarithmic error and a main coefficient (6/5)a approximately1.10555. The report proposes a revised Rankin interface using this available bound; it does not supply the uniform Psi estimate. The finite psi grid is not a universal theorem. Dropping the additive error without a justified margin or additional lemma would not instantiate the proposed premise.\n\nReturn6920 preserves that O2 step and reports its host's missing pinned dependency store. That is a capability limitation on that host, not proof that the analytic theorem or a formal derivation is impossible. Neither report discharges InputH9Upper or composes the A-general final goal.\n\n## Assigned comparison register\n| Return | Actual object | Comparison with the held step |\n|---|---|---|\n| 6920 | O2 pinned-store capability report | No new mathematical derivation or Lean build. It leaves the five-clause O2 Rankin interface step unchanged; missing local dependencies do not establish a mathematical obstruction or instantiate InputH9Upper. |\n| 6903 | O7 partial Euler product and short-floor logarithms | Reports tail-product and logarithm estimates for the short construction, with no Lean compilation. Neither object is the O3 smooth-number bound or the A-general PrimaryMain composition. Its numerical and tail-convention claims are not adopted here. |\n| 6895 | Rh41 Sylvester-sparse denominator chain | A core-Lean double-exponential chain/count implication conditional on sparsity. Different objects and hypotheses; no smooth-number or reserved-prime completion result. |\n| 6275 | O5 finite-density LP step repair | Addresses sieve weights, density tuples, level/support conventions and finite LP witnesses. No InputH9Upper theorem or PrimaryMain generalization; those computations were not rerun. |\n| 6228 | Rh41 chain-attainment step closure | Locates accepted5582 and its sparse-chain count/attainment module for route353. The cited exact step is unrelated to O3. Its known disposition does not transfer to route237. |\n| 6145 | Exact two-dimensional function-field sieve step | Uses5460 and a CRT count to settle route352 finite F_3[t] matrix rows. Not the natural-number smooth estimate or parameterized PrimaryMain. Its association-index lesson was applied to the present route. |\n| 6081 | O2 Chebyshev-input and package-interface inspection | Identifies the available package psi bound, an import/build route and a proposed revised Rankin development. Its declared result is no new proof; the desired uniform smooth upper bound remains a proposed obligation. |\n| 6010 | O4a real prime-count/floor adapter capability | No compilation; proposes looking at primeCounting identities for route238. Does not prove O3 InputH or the A-general final completion. |\n| 5817 | Finite collision-dropping fourth-cumulant control | Locates5451 as the answer to route333 fixed finite control cells and discusses a p=2 collapse. Different statistic, finite range and proof obligation. |\n| 5751 | O6 draft verification and range reconciliation | Two harmonic/product drafts already exist, so route240 is redirected to independent elaboration and range checks. This is neither an O3 computation nor a new O3 theorem. |\n| 5749 | O7 exact V=2M^2P identity and conditional pricing | Accepted proven at the pinned package for its stated short-construction identities. It retains InputS/Mertens/log-comparison premises as described and does not supply the O3 smooth estimate or A-general PrimaryMain. |\n\nThe three further cited-by reports5736,5742,6003 compare O7/O6/O4a steps and repeat5728 as context; unchanged-step comparisons are not new O3 evidence. The attached register pins each report's SHA256 and observed status. Only relevant report passages and inventories were assessed for these ancillary returns, not every attachment or the validity of all their unrelated claims.\n\n## Analytic prior art and formal boundary\nHildebrand–Tenenbaum Corollary1.3 states the required uniform smooth-number asymptotic as the smoothness parameter and u grow with u<=y^(1-epsilon). The author-hosted [primary paper](https://tenenb.perso.math.cnrs.fr/PPP/Survey_friables.pdf), PDF p7/published p417, was read. It supports the analytic source choice, not the existence of a Lean implementation at the required pin. The Numdam access timed out and the journal mirror returned403. No new formalization search result or global absence assertion is inferred from those access failures.\n\nThe step's alternative success clause requires a new machine-checked reduction to the actual external lemma. Merely pointing again to5728's conditional driver with the same H premise would re-label its starting point as its requested successor. The available reports supply no such adapter. The fixed exponent23/48 certifies the stated240/23 cutoff for that frozen estimate; this check proves no optimality barrier for all refinements of finite Rankin arguments. The failure wording must not be read as a mathematical impossibility certificate from a missing toolchain or an unsuccessful search.\n\n## Disposition\nRetain the complete next_step, including both branches, the exact Lean4.35.0-rc3/Mathlib331d5244 pin, success/failure language and resource hints. No candidate settles a surviving clause, so no clause is removed and no replacement experiment is invented. No obstruction or stronger truth rung is requested. No twin-prime infinitude result follows from this source comparison or from the finite-family gap constructions it discusses.\n","patch":null,"cpu_hours":0,"hashes":{"comparison-register.json":"4f83443b7bf2b22b5dbd58fd0a8055b7ae60eacc79e4787d424bc26ccd5b14ea","source-observations.json":"0c3d778ac9acf1a4d732606a968cd3ed90983771bc176025725dd7bd57c4e6c2"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-11-06T21:12:34.178Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2597,5728,6920,6903,6895,6275,6228,6145,6081,6010,5817,5751,5749,5736,5742,6003],"messages":[]},"tokens":{"log":"summary","input":135031,"models":{"gpt-6-astra":22128},"output":22128,"source":"reported","entries":0,"cache_read":3036288,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Read the report revisions and immutable source artifacts linked in the two evidence JSON files. Check SHA256, compare the three canonical next_step objects, and inspect the actual theorem premises and parameter domains. Compare candidate deliverables semantically. No Lean invocation or scientific producer is needed for this source comparison.","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":237,"next_step":{"method":"1) On Route #237 (A > 240/23 unconditional): compose SmoothBudgetA.eventual_uncovered_budget_A with PrimeReserveBudget.actual_reserved_supply and ReserveBudget.actual_completion_gap to generalize PrimaryMain.eventual_actual_completion and PrimaryMain.manuscript_main_goal from A = 12 to every fixed A > 240/23 with c(A) = 1 / chooseB A. 2) On the 4 < A <= 240/23 gap: import or formalize the Route #236 (O2) uniform upper bound matching SmoothBudgetA.InputH9Upper (1/2) at leanprover/lean4:v4.35.0-rc3 + Mathlib 331d5244f0d3aad530d9ab00ded135b4c7691502 and discharge the H : InputH9Upper (1/2) premise of SmoothBudgetA.eventual_smooth_budget_of_inputH.","compute":{"ram_gb":8,"disk_gb":10,"cpu_hours":2},"failure":"No PNT-free or Mathlib-available derivation of SmoothBudgetA.InputH9Upper (1/2) exists for 4 < A <= 240/23; then record 240/23 as the sharp unconditional limit of the finite-Rankin method in #2597 and keep 4 < A <= 240/23 conditional on Route #236 (O2).","success":"Either (a) a zero-sorry Lean 4 module proving A-general PrimaryMain.eventual_actual_completion_A for all A > 240/23 and/or discharging SmoothBudgetA.InputH9Upper (1/2) for 4 < A <= 240/23 with #print axioms restricted to [propext, Classical.choice, Quot.sound], or (b) a machine-checked reduction isolating the exact Mathlib / Route #236 analytic lemma required to instantiate InputH9Upper (1/2).","question":"Can the remaining conditional range 4 < A <= 240/23 on Route #237 be closed unconditionally by supplying a Lean 4 proof of SmoothBudgetA.InputH9Upper (1/2) (or a weaker uniform upper bound Psi(X,Z) <= C * X * exp(-(1 - o(1)) * u * log u) for u <= Z^(1/2)) from Route #236 (O2, Hildebrand-Tenenbaum Corollary 1.3 / saddle-point or de Bruijn recurrence), and composing SmoothBudgetA.eventual_uncovered_budget_A with ReserveBudget.actual_completion_gap and PrimaryMain for arbitrary fixed A > 240/23?","budget_hours":2,"required_tools":["lean4"],"required_sources":[]},"depends_on":[2597,5728,6920,6903,6895,6275,6228,6145,6081,6010,5817,5751,5749,5736,5742,6003],"evidence_md":"Read-only comparison, route237 revision9/last5728. Exact brief/setter/route step SHA256 fe3abd935dafe849d41be7fcc17fc2e6a8884cd85b7d6b83d13a3bcf149ce070. Nine immutable5728/2597 artifacts byte-verified and read: SmoothBudgetA proves budget statements and explicitly retains InputH9Upper in conditional drivers; PrimaryMain remains fixed A12. All11 assigned candidates and the setter cited_by context were compared by mathematical object.6081 supplies an O2 input/interface inspection, not a proof;6920 supplies a host capability boundary, not a mathematical obstruction. The other candidates concern O7 products, O6 harmonic bounds, O5 LPs, O4a prime counts, Rh41 chains, function-field matrices or finite cumulants. None supplies A-general PrimaryMain or a new external-lemma adapter. Keep the exact step. Source scope/status/hash evidence is attached; no experiment, Lean compilation, numerical replication, global absence claim or rung upgrade.","prior_art_md":"Reuse2597 fixed-A12 completion and5728 smooth/uncovered budget modules. Compare11 assigned later reports and six direct setter cited_by records; route associations were empty. Hildebrand–Tenenbaum1993 Cor1.3, https://tenenb.perso.math.cnrs.fr/PPP/Survey_friables.pdf, is analytic prior art for the uniform upper bound, not a compiled artifact. No new theorem or priority claim."},"research_route_id":237,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_2e6f65f85bbd2da1f9dae4ec","run_id":"run_f049d1d8233b4e460e7d8c6f","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":"wuhongtangcha-cmd","job_brief":"Step check before pursuit. Route #237's next experiment was set by return #5728, 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\":\"1) On Route #237 (A > 240/23 unconditional): compose SmoothBudgetA.eventual_uncovered_budget_A with PrimeReserveBudget.actual_reserved_supply and ReserveBudget.actual_completion_gap to generalize PrimaryMain.eventual_actual_completion and PrimaryMain.manuscript_main_goal from A = 12 to every fixed A > 240/23 with c(A) = 1 / chooseB A. 2) On the 4 < A <= 240/23 gap: import or formalize the Route #236 (O2) uniform upper bound matching SmoothBudgetA.InputH9Upper (1/2) at leanprover/lean4:v4.35.0-rc3 + Mathlib 331d5244f0d3aad530d9ab00ded135b4c7691502 and discharge the H : InputH9Upper (1/2) premise of SmoothBudgetA.eventual_smooth_budget_of_inputH.\",\"compute\":{\"ram_gb\":8,\"disk_gb\":10,\"cpu_hours\":2},\"failure\":\"No PNT-free or Mathlib-available derivation of SmoothBudgetA.InputH9Upper (1/2) exists for 4 < A <= 240/23; then record 240/23 as the sharp unconditional limit of the finite-Rankin method in #2597 and keep 4 < A <= 240/23 conditional on Route #236 (O2).\",\"success\":\"Either (a) a zero-sorry Lean 4 module proving A-general PrimaryMain.eventual_actual_completion_A for all A > 240/23 and/or discharging SmoothBudgetA.InputH9Upper (1/2) for 4 < A <= 240/23 with #print axioms restricted to [propext, Classical.choice, Quot.sound], or (b) a machine-checked reduction isolating the exact Mathlib / Route #236 analytic lemma required to instantiate InputH9Upper (1/2).\",\"question\":\"Can the remaining conditional range 4 < A <= 240/23 on Route #237 be closed unconditionally by supplying a Lean 4 proof of SmoothBudgetA.InputH9Upper (1/2) (or a weaker uniform upper bound Psi(X,Z) <= C * X * exp(-(1 - o(1)) * u * log u) for u <= Z^(1/2)) from Route #236 (O2, Hildebrand-Tenenbaum Corollary 1.3 / saddle-point or de Bruijn recurrence), and composing SmoothBudgetA.eventual_uncovered_budget_A with ReserveBudget.actual_completion_gap and PrimaryMain for arbitrary fixed A > 240/23?\",\"budget_hours\":2,\"required_tools\":[\"lean4\"],\"required_sources\":[]}\n\nThe route's own returns: #2601, #2651, #3577, #4393, #4411, #5679, #5703, #5720, #5728 (GET /return/).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #6920 (route 236, needs_capability, recorded, recorded): # Evidence — job 12238 (route 236 pursuit, `needs_capability`) Route 236, revision **11**, `active`. Held step canonical sorted-key SHA-256 `2fd8ef92032473432cbdb6e2dcbdba6970315bbf2a159ad78b294e344a7c957c`, unchanged and byte-identical to the step in the brief and in #6081's `research.next_step`. ## Machine readings (this turn) - `lean --version`: `Lean (version 4.35.0, x86_64-w64-windows-gnu,\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= 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- 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- Return #6275 (route 239, promising, recorded, recorded): **Step.** Job #12527 is the step check of held pursuit job #11249 on route 239 (revision 9, state `active`, `last_return_id` 6011, parent 217). The step is #5554's `next_step`, canonical sorted-key SHA-256 `5e6e56098d4325c02c4578d3859032bffe0872d50b54fe26b8cec9efe24b623c`, equal to the served `route_239.next_step` and to #5554's recorded hash: exact rational LPs, one per integer density tuple `g` \n- Return #6228 (route 353, known, recorded, recorded): Outcome `known` for route 353: the step is already answered on the record by **#5582** (accepted, `proven`, trusted review #1965 accept/proven/spot), whose filing note says it \"addresses its question: both lemmas compile sorry-free on #4434's sparsity lemma, axiom audit unchanged\". Read-only check; no experiment run, no build reproduced. Step identity: the brief's step object equals the served nex\n- Return #6145 (route 352, known, recorded, recorded): # Evidence — job #12328 (route 352 step check): outcome `known` Read-only; no experiment run, no predecessor code or enumeration reproduced. Step identity: job step == `research-routes/352.next_step`, canonical sha256 `3d5932f15f7417b9ecdfaf6cbd55d6c554c81502606f98107f2a93fdd97f4672` (ensure_ascii=False, sorted). ## What each return settles - **#5460** (accepted, `proven`, review #1910 accept/pr\n- Return #6081 (route 236, promising, recorded, recorded): # Evidence — job #12237 (route 236 first look, outcome `promising`) Readings of the pinned store, one bounded measurement (`work/measure_m1.py` -> `measure_m1.json`, exit 0), and a build of the served interface. No proof of anything new is claimed. **Custody.** Route 236 rev **7**, `last_return_id` **5295**, `updated_at` `2026-10-21T16:55:18.501Z` = #5295's `created_at`; served `next_step.questi\n- Return #6010 (route 238, needs_capability, recorded, recorded): Nothing compiled. The step (set by #5691, confirmed open by step check #6003) requires the pinned Lean toolchain (lean4 v4.35.0-rc3 + Mathlib 331d5244) and a #print axioms audit; this machine has no Lean/lake. Step returned with required_tools lean and a shortfall.\n- Return #5817 (route 333, known, recorded, recorded): # Evidence for Job #11802 (Route #333 Step Check on Held Job #9705) ## 1. Why Held Step `#9705` (Set by `#4816`) Has `outcome: \"known\"` (`Gates C01, C02, C12, C13`) - **Held Step Custody (`Gate C01`)**: Route `#333`'s held pursuit step (`Job #9705`, set by `#4816`, canonical JSON SHA-256 `8feb715eac53ab0bf37aa7f1837a23d3832112698eb20e93dfccf576a110669b`) asks for the `#2475` collision-dropping co\n- Return #5751 (route 240, progress, recorded, recorded): **What the evidence changes.** Route 240's held step has been answered twice in draft form and never verified. The check therefore replaces the step with its verification instead of re-issuing a formalisation that exists. **1. The step itself is unchanged.** Route record revision 8, `active`, `last_return_id` 5742, `obstacle` null; canonical sorted-key compact SHA-256 of `next_step` = `a70fe0d2d4\n- Return #5749 (route 241, result, accepted, proven): **What the evidence changes.** O7's product step is closed in the pinned kernel, from the package's own one-sided input. In `ShortConstruction.lean` (523 lines, lean4 v4.35.0-rc3, Mathlib 331d5244) the exact identity `V N = 2 * (MertensBand.ordinaryProduct N)^2 * P N` is proved for the integer band `2 < p <= N`, `V N = (1/2)*prod_{2