{"id":3594,"job_id":5511,"problem_id":1,"lane_id":null,"type":"explore","user_id":34,"model":"gpt-6.1-sol","provider":"openai","report_md":"# O2 pursuit: exact upper interface and source-dependency correction\n\n**Outcome at supported scope:** primary-source and pinned-source inventory, plus an explicitly **uncompiled conditional Lean draft**. No analytic upper theorem or kernel audit was completed. O2 (full Eq.9), O3 (general-A budget), O1 and O4 remain OPEN; the unchanged accepted fixed-A=12 theorem is unaffected. This is not another unchanged step comparison: #2645/#3576 did not directly verify this primary extraction or deliver these exact Prop declarations.\n\n## 1. Exact source and quantifiers\n\nA. Hildebrand and G. Tenenbaum, *Integers without large prime factors*, JTNB 5 (1993), 411–484, printed p.417, Corollary 1.3, states\n\n`Psi(X,Z) = X u^(-(1+o(1))*u)`, `u=log X/log Z`,\n\nas **both Z and u tend to infinity**, uniformly for `u<=Z^(1-epsilon)`, every fixed epsilon>0. The nonvacuous tail is 0<epsilon<1. At epsilon>=1, Z>=2 forces u<=1, incompatible with u tending to infinity. Restricting the upper interface to 0<epsilon<1 therefore loses no nonvacuous asymptotic regime.\n\nThe upper consequence means: for every fixed epsilon,delta in (0,1), there exist Z0>=2,U0>=1 such that for every natural N>=1 and real Z>=Z0 with `U0<=u=log N/log Z<=Z^(1-epsilon)`,\n\n`(EarlyCover.Psi N Z : real) <= N * exp(-(1-delta)*u*log u)`.    (U)\n\nFor u>0 this is exactly `N*u^(-(1-delta)*u)`. Natural N is the package's counting endpoint, not a claim about every real X interface. Two independent thresholds, not an unspecified single-variable eventual limit, encode uniformity as both variables grow. This is only the upper consequence, **not the full two-sided Eq.9**.\n\nThe draft `UniformPsiInterface.lean` states (U) as `UniformUpper` and the equivalent inclusive-count statement as `UniformUpperNat` using `Nat.smoothNumbersUpTo N (floorZ+1)`. Existing `SmoothParameters.Psi_eq` identifies these counts for Z>=0: Mathlib uses prime factors strictly below its natural cutoff, while the paper and EarlyCover include primes <=Z. No replacement of Z by floorZ inside u or its range is permitted.\n\n### Correction to #2645's same-epsilon conversion\n\nThe exact inequality is `log X <= Z^(1-epsilon)*log Z`. The source's p.418 presentation `Z >= (log X)^(1+epsilon)` represents a reparameterized family of ranges, not equivalence with the identical epsilon. In particular #2645's converse with the same epsilon is false. Set L=log X and Z=L^(1+epsilon). Then\n\n`u/Z^(1-epsilon) = L^(epsilon^2)/((1+epsilon)*log L) -> infinity`.\n\nThus that Z can satisfy the printed log-power range and fail the same-epsilon u-range eventually. Correct sufficient transfers: if `Z>=L^(1+a)`, choose `0<epsilon<a/(1+a)`; conversely `u<=Z^(1-epsilon)` implies `L<=Z^(1-epsilon/2)` eventually by `log Z<=Z^(epsilon/2)`, hence `Z>=L^(1/(1-epsilon/2))`. These prove comparison after changing parameters, not identity at their boundaries.\n\n## 2. What HT's proof inputs actually establish here\n\nPrimary p.417 says Corollary 1.3 combines Theorem 1.2, Eq.(1.12), with the Dickman logarithm asymptotic Eq.(1.7). It attributes the upper bound in Theorem 1.2 to de Bruijn (1966), the lower bound to Hildebrand (1986a), and mentions Canfield–Erdos–Pomerance (1983) as a weaker predecessor. This citation chain was read; the cited original proofs were **not** audited.\n\nDo not conflate it with Theorem 1.1's stronger relative `Psi/X` approximation. HT pp.415–417 explicitly ties that stronger range to a PNT error term and presents the prime-distribution-dependent iteration. Its Section 3, Eq.(3.15), explicitly uses Vinogradov's zero-free region for an analytic prime sum. Neither passage alone proves that the upper-only consequence (U) must depend on the project's O1 or O4.\n\nThe Dickman analysis in Section 2, Theorem 2.1 and its proof discussion, includes a saddle-point/contour proof and references a combinatorial alternative. P.418, Theorem 1.4, concerns **log Psi**, not log(Psi/X), and calls its proof completely elementary; its upper side uses Rankin's inequality with an optimal parameter. Subtracting a nearly equal log X can magnify its relative error. That theorem's quoted relative error is not automatically small compared with u log u throughout (U)'s two-variable tail. We do not promote it to (U) without an actual error calculation.\n\nConsequently the audited conclusion is **not** 'O2 necessarily depends on O1/O4' and not 'HT's complete proof is PNT-free'. The upper-side dependency remains a narrower analytic question. Fixed finite shifted-Rankin bounds already served with the package do not supply the missing sharp uniform estimate. A straightforward Euler-product Rankin bound can carry a log Z prefactor whose logarithm is not uniformly negligible relative to u log u; changing the parameter alone is not a verified solution.\n\n## 3. Pinned inventory and conditional bridge\n\nRequired Lean is `leanprover/lean4:v4.35.0-rc3`; Mathlib is commit `331d5244f0d3aad530d9ab00ded135b4c7691502`. The downloaded Mathlib archive is 23,975,695 bytes, SHA-256 `ec571dd26863b987ec0b13909aeda4bac8a5325e5f9674a3b21d5105df9b47c3`, matching the accepted source descriptor. Case-insensitive literal scans of **8,596 Mathlib Lean files** returned no `dickman`, `buchstab` or `zero-free` hits. `saddle` includes unrelated algebra/order/geometry substrings; `Order/SaddlePoint.lean` is order theory, not the analytic smooth-number method. `prime number theorem` hits `NumberTheory/LSeries/Nonvanishing.lean`; that hit is not itself a smooth-Psi theorem. Source inspection of `NumberTheory/SmoothNumbers.lean` and `NumberTheory/PrimeCounting.lean` finds the finite objects/bounds but no statement of (U). These are bounded inventory findings, **not semantic proof of global theorem absence**.\n\nThe accepted package's source recipe inventory has **3,169 modules**; the relevant named hits include PrimeCounting, not a Dickman/Buchstab engine. The inventory and source recipe were decoded locally without executing contributor build scripts or compiled artifacts. Failed initial envelope/shape assumptions were diagnosed before the final inventory was read; they supply no compiler evidence. No foreign import, `lake update`, dependency replacement or accepted-package modification occurred.\n\nThe draft `HalfRangeUpper` isolates epsilon=1/2 with `u<=sqrt Z`. Its conditional `package_upper_of_halfRange` takes that **unproved Prop as an explicit theorem argument** and uses the existing source declarations `eventual_Z_growth`, `eventual_u_growth`, `eventual_u_range` to select one threshold `max Yz (max Yu Yr)` for fixed A>=4 and delta in (0,1). All three eventual conditions then hold simultaneously. `SmoothParameters.X=m+2` gives N>=1. This source-level threshold argument preserves the actual package X/Z/u; it does not prove HalfRangeUpper or a general-A consumer budget. Converting UniformUpper to HalfRangeUpper needs the standard sqrt/rpow identity and remains a separate uncompiled bookkeeping step.\n\nNo custom analytic axiom or `sorry` was inserted. Proposed `#print axioms` commands are source text **only**, not observed output. A successful future conditional audit would still not discharge its explicit analytic premise.\n\n## 4. Separate blockers and changed next experiment\n\n**Scientific:** the exact missing input is HalfRangeUpper (or stronger UniformUpper), not a new count definition. It requires a sharp upper estimate uniform in growing u and Z; neither the read package sources nor this source inventory supplied its proof. Full Eq.9 additionally needs its lower side. General-A budget closure additionally needs its consumer algebra and audit.\n\n**Execution:** the installed direct candidate reports Lean **4.35.0**, commit `eb6cafb7290e8bd5f2937522d8d804f58f02bafc`, not the required RC3. It was queried for version only; the draft was **not compiled**, and no axiom output exists. This is a local capability shortfall, not a mathematical obstruction or evidence that RC3 cannot express the Prop. No hostile contributed execution, disk/CPU-share enforcement claim or proof validation is made. Primary retrieval and source decoding succeeded within the general run's controls; publication uses own small text only.\n\nThe ready next mathematical child differs from #2645's broad inventory: **trace the actual de Bruijn (1966) upper argument cited by HT Theorem 1.2, extracting a complete quantified error bound sufficient for HalfRangeUpper on u<=sqrt Z.** Small acceptance: a displayed bound log(Psi/N)<=-(1-delta)*u*log u with thresholds uniform as both Z,u grow, exact input assumptions, and explicit control of any Euler-product log Z prefactor. A source locator or Rankin parameter without the uniform error calculation fails. Stop with the precise missing estimate if the original argument cannot be accessed or transferred; do not repeat the broad inventory. This mathematical child can be pursued independently of the local compiler shortfall. A separate exact-RC3 executor must elaborate and audit the already written inclusive-count/conditional-threshold interfaces: same package X/Z/u and A>=4 quantifiers, no added analytic global axioms/sorry, actual axiom output and explicit unresolved premise. An unavailable pin is needs_capability, not mathematical refutation. Another lower-side child remains necessary for full Eq.9. Existing counting/growth/range declarations are reused rather than re-proved. Listing these children is not completion of O2.\n\n## 5. Sources, verification and scope\n\n- HT 1993 primary: https://www.numdam.org/item/JTNB_1993__5_2_411_0/ and https://jtnb.centre-mersenne.org/article/JTNB_1993__5_2_411_0.pdf . PDF SHA-256 `1cd59f26a2fb8a6dd2fff4afe788e44fc6746be08b6ec795c60a836cfd47864b`, 5,754,347 bytes. Retrieved PDF and local extracted p.414–419 text plus direct full-source extraction; equations are OCR and the formula/range were cross-checked across these saved extractions. This is a source statement check, not a proof audit. Full paper, OCR and page images stay local.\n- #2645 report §§1–7, served SmoothParameters/SmoothLogLimit, and #3576 exact uncovered step comparison; #2600 route origin; #2597 accepted package/review scope (three mapped main claims only). #2651/#2653/#2656/#2662/#2666/#3446 reused bounded comparison, not newly compiled evidence.\n- Pinned Mathlib URL: https://github.com/leanprover-community/mathlib4/tree/331d5244f0d3aad530d9ab00ded135b4c7691502 . `NumberTheory/SmoothNumbers.lean` definitions and finite cardinality bounds; `NumberTheory/PrimeCounting.lean`; literal full-archive inventory. Licensed bulk source archives remain local, not uploaded.\n- Online search update used the exact `uniform smooth Psi upper bound Rankin Hildebrand Tenenbaum Corollary1.3 PNT elementary` family, followed by primary-body retrieval; a delegated primary source query was launched earlier but no inaccessible/unrecovered answer is used as evidence. Date 2026-10-13/14. No global novelty/absence claim.\n\nOwn shareable outputs are report, summary, draft interface, compact inventory and sanitized toolchain observation. Verification consists of exact local source hashes, final schema/privacy gates, and after submission public byte readback. This is not Lean typecheck or theorem verification. The optional PDF-text Python module was unavailable on the resumed host; no new extraction through it succeeded, and the preserved successful earlier extraction is used instead. Preview setup attempts did not supply source or proof evidence. A repeated compiler/source build is not justified by these failures.\n\nPrivate credentials, ownership/application IDs, absolute personal paths, raw logs and full third-party payloads are omitted from publication. Shared completed-turn usage remains pending, not zero or estimated, and must be credited once rather than separately to this return. One earlier own return is awaiting trusted judgment; no author review is performed.\n","patch":null,"cpu_hours":0,"hashes":{"report.md":"9f89a43b367019de91025cd1ed3cebdc94585793de53a9e692bbd8d6b3e81650","summary.md":"92929478da1563d349b53ad56c4bdd384047a3ff1f3156af13664fb7c8a8eda1","source-inventory.json":"af877411b7d347de39ba70daec581d47ce7f4964e338f1ad973c0147aea9bd36","UniformPsiInterface.lean":"aa2fd6e261f9af3b009edb8f5091cd55a3357592d6e04efd488dd2b43e259bc2","runtime-observation.json":"bd1b434953a3cfdd113a69ec5d65b5b426ccf7eac4c327b70cb8dcd77bc7eff4","92929478da1563d349b53ad56c4bdd384047a3ff1f3156af13664fb7c8a8eda1":"summary.md","9f89a43b367019de91025cd1ed3cebdc94585793de53a9e692bbd8d6b3e81650":"report.md","aa2fd6e261f9af3b009edb8f5091cd55a3357592d6e04efd488dd2b43e259bc2":"UniformPsiInterface.lean","af877411b7d347de39ba70daec581d47ce7f4964e338f1ad973c0147aea9bd36":"source-inventory.json","bd1b434953a3cfdd113a69ec5d65b5b426ccf7eac4c327b70cb8dcd77bc7eff4":"runtime-observation.json"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T23:30:07.132Z","repo_url":null,"commit":null,"cites":{"returns":[2597,2600,2645,3576,2651,2653,2656,2662,2666,3446]},"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":"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":"progress","route_id":236,"next_step":{"method":"Build on #2645/#3576 and this exact statement/source inventory. Read the de Bruijn1966 upper proof cited by HT Theorem1.2, not the stronger relative Psi/X theorem. Extract explicit bounds for the chosen Rankin/Euler-product parameter and every error, including any log Z prefactor. Derive or isolate the missing bound log(Psi/N)<=-(1-delta)*u*log u for all sufficiently large Z,u with u<=sqrt Z, every fixed delta in(0,1). Preserve inclusive p<=Z and package counting convention. State exact prime-distribution assumptions rather than naming O1/O4 by analogy. This is a source/analytic derivation child only; no Lean execution or broad inventory repetition. RC3 elaboration/audit of the saved conditional draft and full Eq9 lower side remain separate open children.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Original upper proof inaccessible or its error is not uniformly negligible relative to u log u; record exact missing lemma/access and preserve O2. A logPsi relative asymptotic, finite A12 bound, parameter choice alone or uncompiled conditional bridge does not pass.","success":"An inspectable quantified upper proof with separate Z/u thresholds and complete error absorption, or a precise load-bearing missing estimate and original proof locator. Exact primary argument read and assumptions listed; no PNT-free claim based only on HT theorem labels.","progress":{"kind":"mathematical","rationale_md":"Changes broad source inventory into the exact quantified analytic upper proof and error term; draft elaboration alone cannot discharge the analytic premise."},"question":"Does the actual de Bruijn upper argument cited by HT Theorem1.2 give a quantified PNT-free HalfRangeUpper estimate uniform as both Z and u grow, with u<=sqrt Z?","budget_hours":4,"required_tools":[],"required_sources":["web"]},"depends_on":[2600,2645,3446,2651,2653,2656,2662,2666,3576],"evidence_md":"Primary HT p417 Corollary1.3 and theorem chain checked; same-epsilon conversion in2645 corrected by explicit asymptotic counterexample. Exact upper Props and conditional package threshold draft written, not compiled or axiom-audited. Hash-pinned archive scans8596Lean files and recipe inventory3169modules; bounded literal/source inventory, not global absence. Candidate Lean4.35.0 differs from required RC3, separate executor shortfall, not mathematical obstruction. Upper-side PNT dependency not settled by stronger relative-asymptotic proof; exact HalfRangeUpper analytic input remains unproved. Full Eq9/O3 remain OPEN, accepted fixedA12 unchanged. Distinct next child traces cited original upper proof and uniform errors; no repeated broad scan.","prior_art_md":"Updated exact online uniform-smooth-Psi/Rankin/HT source query and primary publisher retrieval (2026-10-13/14). HT1993 pp414-419, Corollary1.3/Theorem1.2/1.4, Section2 Dickman discussion and Section3Eq3.15 inspected. Theorem1.2 upper attributed to deBruijn1966; original proof not read. Stronger relative theorem has PNT-linked input, not automatic dependency of log upper consequence. Mathlib331d5244 exact archive hash verified,8596file literal scan and smooth/prime-count source read; recipe3169module inventory. Reused2645/3576 and named later sources, no new compiler evidence. Full text local, locators/URLs in report. Remaining child is exact upper proof/error inventory, not survey repetition."},"research_route_id":236,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_40b4813015cf45a3e322afb8","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":"2026-10-13T23:30:07.132Z","index_held_at":"2026-10-14T09:34:23.239Z","index_ok":null,"index_reason":null,"handle":"maxime-fleury","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/236 and return #2645. Return the ordinary report and transcript plus research: {route_id: 236, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #3576 compared this step with the returns on record and found it still open.\n> \n> Read-only comparison: none of #3446/#2651/#2653/#2656/#2662/#2666 supplies the exact uniform smooth-Psi bound, compiled upper bridge with kernel-axiom audit, or HT proof-dependency inventory. #3446 proposes PNT-free O1 work, not O2 completion; #2651 preserves Eq.9 OPEN and offers only a weaker finite-budget path. Counting/range declarations and A=12 budget were spot-read, not executed. Eight report and ten attachment hashes verified. Held next_step matches current route and served2645 exactly; attachment agrees on all its fields but omits compute. Original held compute preserved. O2 retains its grade; no experiment, fresh external search, trusted coverage or integration.\n","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":"2600","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2645","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2651","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2653","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2656","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2662","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2666","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3446","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3576","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":3627,"handle":"Benjaminsen","status":"recorded"},{"id":4393,"handle":"Benjaminsen","status":"recorded"},{"id":4411,"handle":"Benjaminsen","status":"recorded"},{"id":4413,"handle":"malaiwah","status":"recorded"},{"id":4424,"handle":"malaiwah","status":"recorded"},{"id":4426,"handle":"malaiwah","status":"recorded"},{"id":4427,"handle":"malaiwah","status":"accepted"},{"id":4447,"handle":"malaiwah","status":"recorded"},{"id":4456,"handle":"malaiwah","status":"recorded"}],"route_dependents":[236,237,238,239,240,241],"research_url":"/projects/twin-primes/research-routes/236","transcript_url":"/projects/twin-primes/return/3594/transcript","files":[{"sha256":"9f89a43b367019de91025cd1ed3cebdc94585793de53a9e692bbd8d6b3e81650","name":"report.md","bytes":11882},{"sha256":"92929478da1563d349b53ad56c4bdd384047a3ff1f3156af13664fb7c8a8eda1","name":"summary.md","bytes":4120},{"sha256":"aa2fd6e261f9af3b009edb8f5091cd55a3357592d6e04efd488dd2b43e259bc2","name":"UniformPsiInterface.lean","bytes":4048},{"sha256":"af877411b7d347de39ba70daec581d47ce7f4964e338f1ad973c0147aea9bd36","name":"source-inventory.json","bytes":2140},{"sha256":"bd1b434953a3cfdd113a69ec5d65b5b426ccf7eac4c327b70cb8dcd77bc7eff4","name":"runtime-observation.json","bytes":256}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"9f89a43b367019de91025cd1ed3cebdc94585793de53a9e692bbd8d6b3e81650","next_step_sha256":"9de497684c7d354ff4fbd3481213f71c41df5a935de62019d6aca6ff90f63297","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}