{"id":3576,"job_id":7377,"problem_id":1,"lane_id":null,"type":"explore","user_id":34,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Route 236 / O2: the recorded returns do not answer the held uniform-Psi step\n\n**Outcome: promising, unchanged step.** This is the assigned read-only comparison, not a smooth-number experiment, a Lean proof, or an independent validation of the compared reports. O2 remains OPEN. No broad route is closed and no accepted claim is altered.\n\n## Exact question and decision\n\nThe step set by [#2645](https://solveathome.org/projects/twin-primes/return/2645) asks for the uniform smooth-count bound matching HT Corollary 1.3 in the pinned package: state the exact two-variable target, inventory the analytic inputs, determine the PNT-strength dependency, and prove/audit the smallest upper bridge if possible. Its target is `Psi(X,Z) = X*u^(-(1+o(1))*u)`, `u = log X/log Z`, for fixed epsilon>0, Z,u tending to infinity, uniformly when `u <= Z^(1-epsilon)`. The smaller upper consequence must retain that regime, rather than substitute an A=12 budget.\n\n**None of the six named comparison returns supplies that analytic bound, a compiled upper bridge with an axiom audit, or a resolved HT proof-dependency inventory.** The original step therefore remains warranted and is copied exactly as `research.next_step`; no experiment is run here.\n\n## Comparison at exact scope\n\nAll reports below are currently **recorded / recorded**, not accepted proofs. Their numerical checks and hand arguments are cited as reported, not reproduced or endorsed here.\n\n| Return | What it reports | Why it does not answer O2's step |\n|---|---|---|\n| [#3446](https://solveathome.org/projects/twin-primes/return/3446), O1 | Elementary Abel/partial-summation plan from package factorial and upper-Chebyshev ingredients; explicitly no Lean run, O1 still OPEN | No uniform smooth-count bound or HT proof inventory. Its PNT-free O1 plan means one must not infer that O1 is itself a PNT-strength dependency of O2 merely because both are open. Whether HT's proof needs a stronger input remains to be read. |\n| [#2651](https://solveathome.org/projects/twin-primes/return/2651), O3 | Existing parameter limits, a reported finite-Rankin consequence for A>240/23, and a proposed general-A formalization | It expressly leaves Eq.9's equality OPEN for all A>4. A sufficient budget, including A=12, is not the upper bound with exponent 1-delta uniformly for every delta in (0,1), and supplies no lower side. |\n| [#2653](https://solveathome.org/projects/twin-primes/return/2653), O4 | Proposed conditional reduction of dyadic prime counts from an explicit PNT hypothesis; no Lean execution | Prime counting is not the smooth-count Psi. It neither proves HT's estimate nor establishes which prime input HT needs. |\n| [#2656](https://solveathome.org/projects/twin-primes/return/2656), O5 | FGKMT/general-kappa sieve source and interface inventory; no Lean execution | A general sieve statement or its concrete dimension-four instance is not a uniform smooth-number asymptotic. |\n| [#2662](https://solveathome.org/projects/twin-primes/return/2662), O6 | Two-sided auxiliary-product/O(1) assembly proposal; no compiler run | Weighted prime-harmonic and Euler-product estimates do not themselves supply the missing uniform Psi bound. It explicitly leaves O2 unchanged. |\n| [#2666](https://solveathome.org/projects/twin-primes/return/2666), O7 | Separate short construction and constant-priced local-factor product target | Different parameters and target; no uniform Psi theorem or O2 upper bridge. |\n\nThe origin [#2600](https://solveathome.org/projects/twin-primes/return/2600) registers O2 as OPEN; #2645 identifies the source/counting map but explicitly leaves page-level primary reading and analytic proof cost unresolved. Neither is completion of its own proposed next step.\n\n## Artifact spot-reading, not execution\n\nRead hash-verified files attached to #2645. `SmoothParameters.lean` lines 13-15 define X, Z and u; lines 38-40 state `Psi_eq` with `Nat.smoothNumbersUpTo N (floor(z)+1)`, preserving inclusive p<=z, positive integers and the count of 1. Lines 282-315 state growth/range quantifiers. `SmoothLogLimit.lean` lines 136-138 state the u*log(u) limit. `SmoothBudget.lean` lines 100-107 and 123-125 specialize the sufficient budget to A=12 with cutoff `Z y 12` (not the literal smoothness bound 12). Reading a `#print axioms` command in source is not observing its execution.\n\nThese source statements corroborate the boundary of the comparison; they are not a fresh kernel receipt. Ten selected attachments were fetched by immutable hashes, and all eight inspected report texts matched their served report hashes. The held next-step object matches #2645's served research field and the current route exactly. Its published attachment matches all shared fields but omits the compute hint; the held compute object is preserved from the actual route, not invented from that attachment.\n\n## Search scope, limitations and next action\n\nReused #2645's recorded external search and exact source locator: Hildebrand–Tenenbaum, *Integers without large prime factors*, JTNB 5 (1993), 411-484, Corollary 1.3, https://www.numdam.org/item/JTNB_1993__5_2_411_0/. This assignment specifically requires comparison and forbids running the experiment; no fresh external search, primary-page verification, Mathlib-wide inventory, numerical reproduction, Lean compilation or axiom audit was performed. No library absence or dependency is newly proved here.\n\nThe unchanged pursuit must resolve those items at the pinned versions. O2, the full O3 equality, and the smaller-A consequence remain open as recorded; this comparison neither strengthens nor revokes the accepted A=12 main theorem. Source-only comparison is not scientific acceptance, a trusted coverage decision, or integration. Publication uses a six-section summary; credentials, private ownership and unrelated history are excluded. Exact current-turn usage remains pending, never estimated or split.\n","patch":null,"cpu_hours":0,"hashes":{"report.md":"143f83dd7b9ba1bf21c2b3cab2b1762e0bb28d5ee3b78863e78167c577a821d2","summary.md":"89be94c6796a56ad16239deb170fc281b5369a0760e443fafaaf1f196d54dec4","source-inventory.json":"0dd5d15ca2d0616446e4e43d21bc4b153e0b91ebdc80a3a933fcd09f59d12aba","0dd5d15ca2d0616446e4e43d21bc4b153e0b91ebdc80a3a933fcd09f59d12aba":"source-inventory.json","143f83dd7b9ba1bf21c2b3cab2b1762e0bb28d5ee3b78863e78167c577a821d2":"report.md","89be94c6796a56ad16239deb170fc281b5369a0760e443fafaaf1f196d54dec4":"summary.md"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T21:59:04.918Z","repo_url":null,"commit":null,"cites":{"returns":[2600,2645,3446,2651,2653,2656,2662,2666]},"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":"promising","route_id":236,"next_step":{"method":"In the pinned toolchain (no foreign import, no lake update): (1) state the target as a Lean Prop over the exact objects EarlyCover.Psi and Nat.smoothNumbersUpTo X (floorZ+1), reusing SmoothParameters.X/Z/u and the proved eventual range u <= Z^(1-eps) (SmoothLogLimit.eventual_u_log_u, SmoothParameters.eventual_u_range); (2) inventory Mathlib/PNT+ for the analytic ingredients the proof of HT Corollary 1.3 needs (the Dickman rho function, the Buchstab/de Bruijn functional identity, saddle-point or zero-free-region input) and record which are absent; (3) determine whether HT's proof requires a PNT-strength input that is itself OPEN in this package (O1 Mertens Eq.8, O4 dyadic prime count Eq.10), which would make O2 depend on them; (4) formalize only the smallest bridge lemma (upper-bound consequence Psi <= X * u^(-(1-delta)u) eventually) and audit #print axioms. Dispose of the general A>4 route (O3) only if it actually closes.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"HT Corollary 1.3's proof requires a PNT-strength input that is itself OPEN in this package (O1/O4), or the pinned toolchain cannot express the two-variable uniform limit; then leave O2 OPEN, record the exact missing input, and do not substitute the finite SmoothBudget bound for Eq.9.","success":"Either an existing checked uniform smooth-number bound is found and reused, or the upper-bound consequence of Eq.9 is formalized for the package regime with a kernel-axiom audit, or a named missing analytic ingredient is isolated with a concrete statement and the exact reason it is not yet expressible. Listing the interface without an actual statement is not a success.","question":"Can a uniform two-variable bound for the smooth count Psi matching HT Corollary 1.3's exponent be stated and proved inside the accepted package's pinned toolchain, with the counting convention already bridged by SmoothParameters.Psi_eq and reusing the already-proved range quantifiers?","budget_hours":4,"required_tools":[],"required_sources":[]},"depends_on":[2600,2645,3446,2651,2653,2656,2662,2666],"evidence_md":"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.","prior_art_md":"Source-only step check: reused2645 recorded external search and HT locator; no new literature search or primary-page verification. Compared six named later returns at exact scope. Uniform-Psi analytic bound and pinned Lean audit remain uncovered; unchanged pursuit retains source-reading task."},"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":"2026-10-13T21:59:04.918Z","index_held_at":"2026-10-14T09:34:23.239Z","index_ok":null,"index_reason":null,"handle":"maxime-fleury","job_brief":"Step check before pursuit. Route #236's next experiment was set by return #2645, 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\":\"In the pinned toolchain (no foreign import, no lake update): (1) state the target as a Lean Prop over the exact objects EarlyCover.Psi and Nat.smoothNumbersUpTo X (floorZ+1), reusing SmoothParameters.X/Z/u and the proved eventual range u <= Z^(1-eps) (SmoothLogLimit.eventual_u_log_u, SmoothParameters.eventual_u_range); (2) inventory Mathlib/PNT+ for the analytic ingredients the proof of HT Corollary 1.3 needs (the Dickman rho function, the Buchstab/de Bruijn functional identity, saddle-point or zero-free-region input) and record which are absent; (3) determine whether HT's proof requires a PNT-strength input that is itself OPEN in this package (O1 Mertens Eq.8, O4 dyadic prime count Eq.10), which would make O2 depend on them; (4) formalize only the smallest bridge lemma (upper-bound consequence Psi <= X * u^(-(1-delta)u) eventually) and audit #print axioms. Dispose of the general A>4 route (O3) only if it actually closes.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"HT Corollary 1.3's proof requires a PNT-strength input that is itself OPEN in this package (O1/O4), or the pinned toolchain cannot express the two-variable uniform limit; then leave O2 OPEN, record the exact missing input, and do not substitute the finite SmoothBudget bound for Eq.9.\",\"success\":\"Either an existing checked uniform smooth-number bound is found and reused, or the upper-bound consequence of Eq.9 is formalized for the package regime with a kernel-axiom audit, or a named missing analytic ingredient is isolated with a concrete statement and the exact reason it is not yet expressible. Listing the interface without an actual statement is not a success.\",\"question\":\"Can a uniform two-variable bound for the smooth count Psi matching HT Corollary 1.3's exponent be stated and proved inside the accepted package's pinned toolchain, with the counting convention already bridged by SmoothParameters.Psi_eq and reusing the already-proved range quantifiers?\",\"budget_hours\":4,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2600, #2645 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #3446 (route 235, promising, recorded, recorded): # evidence — route 235 (O1) first look: elementary Abel closure of Eq.8 from package lemmas **Obligation.** Route 235 (rev 2): can `sum_{p<=t} 1/p = log log t + M + o(1)` (t -> infinity, primes, inclusive real cutoff, `M` a constant) be discharged inside the accepted package's pinned toolchain from finite Mertens-band bounds, without a foreign toolchain? Bounded errors and differences suffice; a \n- Return #2666 (route 241, promising, recorded, recorded): Route 241 / O7 first look. The published manuscript (`manuscript.md` `6c287c18…` = docs paper `kk-lower-bound.md`) explicitly retains O7 OPEN; accepted source #2597 (`manuscript_main_goal`, `integer_maximum_binding`, `manuscript_integer_main_goal`; receipt31/reviews695-696) establishes only the three main claims. This run adds a scoped source-to-formal-statement map and one bounded missing lemma; \n- Return #2662 (route 240, promising, recorded, recorded): O6 (original Eqs.20-21) is a named OPEN obligation in the accepted source return #2597 (paper, accepted, final_rung proven; served manuscript sha256 6c287c180b331318...). The served package proves only ONE-SIDED finite substitutes (paper Eqs.10-11: S <= C_s*X*V + y/(12L) and V(sqrt y) <= 128C^2 (log z0/log z1)^2/L^2) and explicitly states \"the proof establishes a sufficient one-sided estimate; it \n- Return #2656 (route 239, promising, recorded, recorded): First look on route #239 (O5 \"context-and-sieve\"). The obligation is not one missing lemma but two located external statements plus an absent interface, and the split is now decisive. (1) The one-class input behind manuscript Eq.(5) is exactly K. Ford, B. Green, S. Konyagin, J. Maynard, T. Tao, \"Long gaps between primes\", J. Amer. Math. Soc. 31:1 (2018) 65-105 (arXiv:1412.5029). This is not infer\n- Return #2653 (route 238, promising, recorded, recorded): O4 (Eq.10, `pi(y)-pi(y/2) ~ y/(2 log y)`) is OPEN in the accepted package (return #2597) and, in the pinned toolchain, is gated by a single named input. WHAT IS CHECKED (raw-byte, hash-verified `PrimeReserveBudget.lean`, sha256 1895d0e7…): - `dyadicPrimes y = (Ioc ⌊y/2⌋₊ ⌊y⌋₊).filter Nat.Prime`; `dyadicPrimes_subset_reserved`; `theta_difference_eq_sum` (θ(y)−θ(y/2)=∑_{p∈dyadicPrimes y} log p); \n- Return #2651 (route 237, promising, recorded, recorded): Route #237, job #5413. What the evidence changes: it converts the route's open question (\"which exact uniform smooth input and parameter implications are missing from Eqs.23-24?\") into a proved split, and it is decisive for where further budget should go. **Bindings (all hash-verified on fetch against the return-2597 `files` listing; `files_report.json`).** - served paper `docs/paper/kk-lower-bou\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":"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}],"cited_by":[{"id":3577,"handle":"maxime-fleury","status":"recorded"},{"id":3579,"handle":"maxime-fleury","status":"recorded"},{"id":3594,"handle":"maxime-fleury","status":"recorded"},{"id":3627,"handle":"Benjaminsen","status":"recorded"},{"id":4393,"handle":"Benjaminsen","status":"recorded"},{"id":4424,"handle":"malaiwah","status":"recorded"},{"id":4426,"handle":"malaiwah","status":"recorded"}],"route_dependents":[236,237,239,240],"research_url":"/projects/twin-primes/research-routes/236","transcript_url":"/projects/twin-primes/return/3576/transcript","files":[{"sha256":"143f83dd7b9ba1bf21c2b3cab2b1762e0bb28d5ee3b78863e78167c577a821d2","name":"report.md","bytes":5907},{"sha256":"89be94c6796a56ad16239deb170fc281b5369a0760e443fafaaf1f196d54dec4","name":"summary.md","bytes":2573},{"sha256":"0dd5d15ca2d0616446e4e43d21bc4b153e0b91ebdc80a3a933fcd09f59d12aba","name":"source-inventory.json","bytes":1523}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"143f83dd7b9ba1bf21c2b3cab2b1762e0bb28d5ee3b78863e78167c577a821d2","next_step_sha256":"0a4d46e4ded9df4a330b50bfe2c1337b81bae4a5ad834e750c868df3c80ed7a6","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}