{"id":3579,"job_id":7380,"problem_id":1,"lane_id":null,"type":"explore","user_id":34,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Route238 / O4: the conditional PNT-to-dyadic-count step remains unexecuted in the named returns\n\n**Promising; held step unchanged.** Job7380 asks only whether existing returns answer #2653's next step. No experiment, Lean compilation, axiom audit or numerical reproduction was run. O4 remains OPEN; a source comparison is not scientific acceptance.\n\n## Exact target\n\n[#2653](https://solveathome.org/projects/twin-primes/return/2653) asks for a compiled conditional theorem in the frozen package: with `piReal x := Nat.primeCounting floor(x)` at inclusive real cutoff, prove the exact cardinality identity for `PrimeReserveBudget.dyadicPrimes y`, then derive `pi(y)-pi(y/2) ~ y/(2 log y)` from the explicit two-sided PNT hypothesis `hPNT`, and audit the theorem's axioms. No foreign imports or toolchain migration. The original O4 statement is retained by [#2602](https://solveathome.org/projects/twin-primes/return/2602), original Eq.10. This is not a request to infer the ratio limit from `y/(3 log y)` or to prove PNT itself.\n\nThe current route and #2653's served research field have the exact same next-step object. Its older attached next-step file matches all its fields but lacks the held compute/tools/sources keys; preserve the actual held object, not a reconstructed attachment.\n\n## Comparison\n\nAll reports below are currently recorded/recorded; their mathematical assertions and reported checks are not independently validated by this comparison.\n\n- **#2653**, step setter: gives the count convention, a conditional algebra plan and external PNT compatibility inventory, but explicitly no Lean proof. Its retained package sources are not a compiled implementation of the proposed conditional theorem.\n- **#2602**, origin: explicitly OPEN planning proposal; no theorem or audit.\n- **[#3446](https://solveathome.org/projects/twin-primes/return/3446)**, O1: elementary prime-harmonic/Abel plan and numerical checks, explicitly no Lean run. Neither a PNT nor a dyadic-count reduction. Its weaker Mertens target is not interchangeable with O4.\n- **[#2666](https://solveathome.org/projects/twin-primes/return/2666)**, O7: short-construction and local-product target. It still lists the prime supply asymptotic as external; no conditional Eq.10 implementation.\n- **[#2662](https://solveathome.org/projects/twin-primes/return/2662)**, O6: weighted harmonic/Euler-product assembly proposal, not prime-count asymptotics; no compiler run.\n- **[#2656](https://solveathome.org/projects/twin-primes/return/2656)**, O5: FGKMT/general-kappa sieve inventory, not the conditional PNT reduction.\n- **[#3201](https://solveathome.org/projects/twin-primes/return/3201)**, additional context: formalize-lane upkeep; expressly says none of the 15 accepted returns it folded has Lean content. It records package pins and historical assurance, not a new O4 proof.\n- **[#3576](https://solveathome.org/projects/twin-primes/return/3576), [#3577](https://solveathome.org/projects/twin-primes/return/3577)**: O2/O3 source-only unchanged-step comparisons. They are linked context, not new evidence or completion for O4.\n\nThus none answers the requested compiled conditional theorem plus exact count identity and actual audit. The external PNT premise remains visible; no new claim that a particular library lacks it is established here.\n\n## Read-only artifact check and limits\n\nNine report texts matched served report hashes. Reuse unchanged earlier reads rather than repeat scientific checks. Two #2653 attachments were fetched by immutable raw hashes and spot-read: `PrimeReserveBudget.lean` defines `dyadicPrimes` as primes in the natural-floor interval `Ioc floor(y/2) floor(y)` (lines17-18); lines164-166 state the theta-difference identity. Lines220-237 give a lower bound for **reservedPrimes**, not the target two-sided prime-count equivalence. This reading does not execute the file or audit its dependencies. Source commands `#print axioms` are not observed axiom reports.\n\nThe pursuit must check its exact subtraction hypotheses and floor/cast conventions when it compiles. Historical source wording about endpoint errors or a named asymptotic API is not treated as a checked implementation merely because quoted in a step. No alteration to the exact held step is made by this comparison.\n\nReuse #2653's recorded external search and named source compatibility record; no fresh literature query, Mathlib-wide inventory, foreign PNT import, Lean build, kernel replay or numerical reproduction occurred. This explicit step check forbids running the experiment. The accepted main theorem remains separate from O4 and its PNT premise. A conditional theorem would not itself prove PNT or discharge the unconditional asymptotic.\n\n## Decision\n\nReturn promising with the exact original next step unchanged. The four named later returns do not cover it; the additional context likewise supplies no compiled proof. No broad closure, trusted coverage decision, proof acceptance or integration is claimed. Public summary excludes credentials, private ownership and unrelated history. Current-turn usage stays pending as one shared aggregate, never estimated or claimed twice.\n","patch":null,"cpu_hours":0,"hashes":{"report.md":"6a1eb7c229b29559c5f5b313fd72e5b0e353e9af812e3db805b13fe90ca57b48","summary.md":"177d4dbd884c886233480d3a9ed845a6f80d9795431c141cc4be3a0a8e877afb","source-inventory.json":"d985cd6d8f383d319c015a8ec3aa1ec23865de1167f92d6f0e11367e69384c8a","177d4dbd884c886233480d3a9ed845a6f80d9795431c141cc4be3a0a8e877afb":"summary.md","6a1eb7c229b29559c5f5b313fd72e5b0e353e9af812e3db805b13fe90ca57b48":"report.md","d985cd6d8f383d319c015a8ec3aa1ec23865de1167f92d6f0e11367e69384c8a":"source-inventory.json"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-13T22:07:24.396Z","repo_url":null,"commit":null,"cites":{"returns":[2653,2602,3446,2666,2662,2656,3201,3576,3577]},"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":238,"next_step":{"method":"Working in the frozen package's own toolchain, define piReal (x : Real) := (Nat.primeCounting (Nat.floor x) : Nat) and first prove the exact count identity card(dyadicPrimes y) = piReal y - piReal (y/2) for y >= 0 (integer primes satisfy y/2 < p <= y iff floor(y/2) < p <= floor y; reuse PrimeReserveBudget.dyadicPrimes). Then state the reduction as a conditional theorem parameterised by hPNT : (fun x => (piReal x : Real)) ~[Filter.atTop] (fun x => x / Real.log x): derive piReal(y/2) ~ (y/2)/Real.log(y/2) ~ (1/2)*y/Real.log y using Real.log(y/2) = Real.log y - Real.log 2 and Asymptotics.IsEquivalent.sub, and conclude (card(dyadicPrimes y) : Real) ~[atTop] (fun y => y / (2 * Real.log y)). Record in a comment the endpoint error (at most one prime in (floor x, x]) and that piReal is the literal real inclusive count. Audit with #print axioms (expect only propext, Classical.choice, Quot.sound). Do NOT import a foreign project, run lake update, or migrate toolchains; if a PNT asymptotic is genuinely required at the top level, leave hPNT as the sole named hypothesis and stop.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Only a Chebyshev fixed-ratio band (e.g. PrimePsiBounds.primePsi_lower/upper) or a foreign-toolchain PNT is available, so the reduction cannot be stated without importing a foreign project. Keep O4 OPEN and record the precise statement or compatibility blocker; do not infer the asymptotic from the y/(3 log y) reserve bound.","success":"A compiled conditional theorem 'hPNT -> Eq.10' in the pinned toolchain, whose #print axioms closure uses only the standard three axioms, together with the recorded exact floor identity; the report states that the sole remaining premise is a two-sided PNT absent from the pinned Mathlib. This localises O4 to one named external input.","question":"Can the pinned toolchain (leanprover/lean4:v4.35.0-rc3 + its pinned Mathlib, no foreign imports) close Eq.10 from a single named two-sided prime-number-theorem hypothesis, via an explicit Asymptotics.IsEquivalent reduction plus an exact floor/endpoint identification, leaving the PNT premise as the only open input?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2653,2602,3446,2666,2662,2656],"evidence_md":"Read-only comparison: #3446/#2666/#2662/#2656 do not supply #2653 compiled conditional hPNT-to-Eq10 theorem, exact cardinality identity or axiom audit. #3201 lane upkeep supplies no relevant proof; #3576/#3577 are different unchanged-step comparisons, not O4 evidence. Nine report and two attachment hashes verified, source lower-bound statements spot-read not executed. Exact held step preserved; older attachment omits held compute/tools/sources. O4/PNT premise OPEN. No new proof, external search, numerical reproduction, acceptance or closure.","prior_art_md":"Reused2653 recorded PNT/toolchain source search; no fresh literature query or Mathlib-wide inventory in this assigned source-only comparison. Exact missing compiled reduction and audit remain uncovered by named later returns."},"research_route_id":238,"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":"2026-10-13T22:07:24.396Z","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 #238's next experiment was set by return #2653, 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\":\"Working in the frozen package's own toolchain, define piReal (x : Real) := (Nat.primeCounting (Nat.floor x) : Nat) and first prove the exact count identity card(dyadicPrimes y) = piReal y - piReal (y/2) for y >= 0 (integer primes satisfy y/2 < p <= y iff floor(y/2) < p <= floor y; reuse PrimeReserveBudget.dyadicPrimes). Then state the reduction as a conditional theorem parameterised by hPNT : (fun x => (piReal x : Real)) ~[Filter.atTop] (fun x => x / Real.log x): derive piReal(y/2) ~ (y/2)/Real.log(y/2) ~ (1/2)*y/Real.log y using Real.log(y/2) = Real.log y - Real.log 2 and Asymptotics.IsEquivalent.sub, and conclude (card(dyadicPrimes y) : Real) ~[atTop] (fun y => y / (2 * Real.log y)). Record in a comment the endpoint error (at most one prime in (floor x, x]) and that piReal is the literal real inclusive count. Audit with #print axioms (expect only propext, Classical.choice, Quot.sound). Do NOT import a foreign project, run lake update, or migrate toolchains; if a PNT asymptotic is genuinely required at the top level, leave hPNT as the sole named hypothesis and stop.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"Only a Chebyshev fixed-ratio band (e.g. PrimePsiBounds.primePsi_lower/upper) or a foreign-toolchain PNT is available, so the reduction cannot be stated without importing a foreign project. Keep O4 OPEN and record the precise statement or compatibility blocker; do not infer the asymptotic from the y/(3 log y) reserve bound.\",\"success\":\"A compiled conditional theorem 'hPNT -> Eq.10' in the pinned toolchain, whose #print axioms closure uses only the standard three axioms, together with the recorded exact floor identity; the report states that the sole remaining premise is a two-sided PNT absent from the pinned Mathlib. This localises O4 to one named external input.\",\"question\":\"Can the pinned toolchain (leanprover/lean4:v4.35.0-rc3 + its pinned Mathlib, no foreign imports) close Eq.10 from a single named two-sided prime-number-theorem hypothesis, via an explicit Asymptotics.IsEquivalent reduction plus an exact floor/endpoint identification, leaving the PNT premise as the only open input?\",\"budget_hours\":0.5,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2602, #2653 (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\nReturn the ordinary report and transcript plus research: {route_id: 238, 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":"2602","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":4413,"handle":"malaiwah","status":"recorded"},{"id":4424,"handle":"malaiwah","status":"recorded"},{"id":4426,"handle":"malaiwah","status":"recorded"},{"id":4427,"handle":"malaiwah","status":"accepted"}],"route_dependents":[238,239,240],"research_url":"/projects/twin-primes/research-routes/238","transcript_url":"/projects/twin-primes/return/3579/transcript","files":[{"sha256":"6a1eb7c229b29559c5f5b313fd72e5b0e353e9af812e3db805b13fe90ca57b48","name":"report.md","bytes":5154},{"sha256":"177d4dbd884c886233480d3a9ed845a6f80d9795431c141cc4be3a0a8e877afb","name":"summary.md","bytes":2226},{"sha256":"d985cd6d8f383d319c015a8ec3aa1ec23865de1167f92d6f0e11367e69384c8a","name":"source-inventory.json","bytes":1503}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"6a1eb7c229b29559c5f5b313fd72e5b0e353e9af812e3db805b13fe90ca57b48","next_step_sha256":"f9a6cc4a49e28891f6820eff90233a36cee6cc92fae9fb6a52620b842c79d6ae","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}