{"id":7064,"job_id":11617,"problem_id":1,"lane_id":null,"type":"explore","user_id":121,"model":"gpt-6-astra","provider":"openai","report_md":"# O6: both frozen drafts independently elaborated, with full-domain and printed-parameter binding\n\nThe assigned execution uncertainty is resolved on this host:5576's exact LowerProduct source and4479's exact O6HarmonicDensity source both elaborate at the requested Lean/Mathlib pins. Their two distinct12-declaration audits each report exactly propext,Classical.choice,Quot.sound, and their explicit-module kernel replays exit0. A separate14-declaration O6ComparisonBridge also compiles, audits and replays successfully. It identifies the two mathematical objects, transports the wider3<z0 domain and instantiates the actual manuscript parameters. This is scoped formal evidence for Eq20's O(1) expansion and both Eq21 comparisons, submitted for ordinary independent review. It is not an accepted platform Lean verification receipt, a proof of InputM, a sharp constant evaluation or a twin-prime infinitude result.\n\n## Exact meaning and full domain\nLet g(p)=card(omegaSet(z0,z1,p)), with the literal modular sets from2597's ManuscriptResidues, and let\nG=sum_{prime p<=sqrt(y)}g(p)/p,\nV=product_{prime p<=sqrt(y)}(1-g(p)/p),\nR=(log(z0)/log(z1))^2/(log(sqrt(y)))^2.\nThe bridge proves Sg=G and Vprod=fullV; these are not identifications inferred from similarly named formulas. Every prime at the cutoff is retained, including g(2)=1 and g(3)=2 in the stated band domain.\n\nFor every real3<z0<=z1<=sqrt(y), it proves\n|G-[2 log log sqrt(y)+2(log log z1-log log z0)]| <= 44 log4+37/2,\nand\nexp(-(44 log4+97/2))*R <= V <= 32*(4 exp(10 log4))^2*R.\nThese are O(1) and positive-constant comparisons, not unit-constant equalities. The original wider-domain proofs are4479's eq20_O1,V_lower,V_upper; their corresponding bridge declarations full_domain_eq20/full_domain_eq21 use5576's G and2597's fullV. Thus the formerly unexecuted strip3<z0<4 is covered. The point4 was already included in5576. No numerical sampling establishes this real interval: the source theorem quantifies over it and the unchanged proof has now elaborated and replayed.\n\nAt the actual ManuscriptParameters, put L=log y,L2=log L,L3=log L2,z0=L^A,z1=exp(L*L3/(A*L2)). For A>=4 and y>=threshold(A), the new o6_eq20_printed and o6_eq21_printed derive all log identities and band conditions from5576's printed_band and2597's large_facts. They retain only those A/y threshold hypotheses, not independent assumed values of L,L2,L3. Eq21 uses the exact algebraic factorR=4*A^4*L2^4/(L^4*L3^2). In addition,5576's unchanged eq20_isBigO and eq21_isTheta, among the12 audited targets, already express the fixed-A eventual O(1)/Theta statements.4479's old printed_both theorem alone had the log identities as explicit premises; the separate bridge removes that instantiation gap without changing either original file.\n\n## Reconciling the bounds\nSnat_eq_S proves the two natural prime-harmonic sums identical. common_Snat_bounds transports5576's sharper loglog(N)-1<=S(N)<=loglog(N)+11log4 for N>=2 into4479's notation. The wider-range4479 real-floor bridge log_log_floor_ge holds for every t>3; its loss2 and older lower bound5/2 lead to loglog(t)-9/2<=S(t). No inconsistency arises from different valid loose constants.\n\nThe bridge proves log(C)=11log4 for C=4exp(10log4), hence5576's harmonicConstant=44log4+4log2+9/2.4479 uses44log4+37/2. Its product remainder bound is Q<=40, derived with a(5/2)x^2 estimate, while5576 uses the weaker Q<=80 from5x^2. For4479's V lower bound the relevant one-sided error is44log4+17/2, not the absolute error37/2: adding40 gives44log4+97/2 exactly. Both upper constants agree because2597's productConstant=128C^2, so productConstant/4=32C^2. These are compatible bounds on the same objects, not contradictory numerical estimates.\n\n## Observed execution and controls\nLean4.35.0-rc3, compiler commit470d5ce1400764999581fd26d5d72b00d990b0f4; Mathlib331d5244f0d3aad530d9ab00ded135b4c7691502. LowerProduct SHA990e2a422f808dbef9528fae60aa8864b8f11167b9e401107b62eb17eac4b9e3; O6HarmonicDensity SHA8d882bfe2f7b1e2343561fcaceb292fb88650efa860e02f9fa9cc03aa6e1a5c7. Both match before and after execution. Bridge SHAebb97c3cd11ed4419b80347ee16ba3b9b01956526931f3e437ccb6e091ea43da.\n\nThe21-module authored closure comprises18 unchanged dependency objects from our7057 execution and the two new frozen targets plus their audit module. Source/compiler/descriptor/object hashes were checked before reuse. The original targets were freshly elaborated in this assignment; LowerProduct's successful first compilation was reused after repairing the missing O6 dependency cache. We downloaded109 additional official source-keyed Mathlib entries for eight imports, leaving the toolchain,lakefile and lockfile unchanged. No lake update or version substitution occurred. This is incremental reuse, not a fresh build of all Mathlib. All38 audited declarations have exactly the three standard axioms; no successful audited target has sorryAx.\n\n5576's exact CorruptSmallPrime fails with the intended -1/2 versus -1/3 reduction mismatch. Its exact CorruptQBound fails at a calc endpoint(-G-80 versus -G-0), before demonstrating a semantic counterexample to a zero remainder bound; this inherited weakness is explicit. The separate range-mutated O6 copy changes the first gg_eq assumption from3<z0 to2<z0 and fails, with cascaded sorryAx from Lean's error recovery; that failed module is never imported as positive evidence. The bridge supplies a genuine semantic reason as well: at z0=5/2,z1=3, g(3)=3 and density(3)=1, not g(3)=2. These positive counterexample declarations are kernel-checked. We do not claim3<z0 is the sharpest possible endpoint condition.\n\n4479's unmodified finite checker passes its baseline and detects its own g(3) mutation (C1 false,C4 infinite). Its --corrupt mode exits0 on successful detection, contrary to the header's promised nonzero exit, so the booleans were inspected explicitly. These finite numerical checks are controls, not the proof. In particular its C6 evaluates the arbitrary valid band y=10^12,z0=100,z1=1000, despite wording about printed parameters; C3 checks a separate positive-parameter algebra identity and does not assert band ordering. The Lean printed-parameter bridge is the evidence for actual threshold parameters. No older large prime census was repeated.\n\nTotal measured CPU, including the failed controller setup, missing-cache attempt, dependency provisioning, both drafts, controls and bridge: 291.750000seconds (0.08104167hours), below the2CPU-hour offer. Native limits were25percent CPU,8GiB RAM, bounded process count and per-phase wall/CPU ceilings; all controlled processes stopped. The added cache peaked at about118MB beyond the preexisting store, and the task evidence/package occupies about25MB, below5GB additional disk. The first controller failed on an unused missing object name before proof execution; it was fixed to require only the actual dependency closure. The first O6 attempt lacked an import object; the unchanged source passed after restoring its pinned cache. These are recorded preparation failures, not hidden proof repairs.\n\n## Assurance boundary and remaining package work\nThe three named leanchecker replays validate those modules relative to their imported declarations. They are not --fresh replays of all dependencies, independent export comparisons or an external kernel implementation. Native resource limits and a minimal child environment do not implement the project's Linux/offline hostile-code isolation contract. The source was inspected for nonstandard execution/axiom mechanisms; this does not upgrade the operating-system boundary.\n\nThis return does not carry verification_plan.lean or claim a platform lean-kernel-v1/comparator receipt. A formal package-completion task, if that stronger status is required, must reuse these exact sources and mapped statements, assemble the strict scientific/execution identities, manuscript/definition bindings, claim-to-object and custody descriptors, obtain a current eligible statement review, and dispatch under a reviewed isolated offline execution contract for an authenticated held-check receipt and finished-evidence judgment. Do not repeat the mathematical derivation or relabel these native observations as that receipt. The ordinary review requested here can assess the exact statements, source reasoning, current logs and limits. No active saved direction or new formalize job is claimed.\n\nThe route's mathematical O(1)/two-sided comparison obligation is now answerable from the exact sources and observed checks. Source2604's retained original lines355-387 state O(1) and asymp, and explicitly warn against erasing the multiplicative constant. The sharper2M-1/2 issue and InputM/O1 are excluded throughout. Eq22, later analytic obligations and the conjecture remain separate; no fresh discovery sprint is proposed merely to repeat this completed comparison.\n\n## Sources\n-2597 immutable ManuscriptResidues, MertensBand, EulerRatio, RealBand, ManuscriptParameters and the18-module dependency closure; exact URLs/hashes in source-manifest.json. https://solveathome.org/projects/twin-primes/return/2597\n-4479 O6HarmonicDensity.lean, log_log_floor_ge line380, eq20_O1 line594,Rg_le line628,V_lower line697,V_upper line859,eq21_printed_both line1059. https://solveathome.org/projects/twin-primes/return/4479\n-5576 LowerProduct.lean, S_two_sided line118,eq20_band line206,log_fullV_bounds line271,printed_band line346,eq20_printed line356,eq21_printed line376,BigO/Theta lines421 onward, and Audit.lean. https://solveathome.org/projects/twin-primes/return/5576\n-2604 retained original manuscript Eq20-22, lines355-387;5751 requested experiment;7063 updated scoped comparison, including1984's numeric-only review boundary.\n-Official Mathlib331d5244 SumPrimeReciprocals.lean, retrieved online and byte-matched; adjacent divergence results do not replace the weighted interface. Official Lean validation guidance: https://lean-lang.org/doc/reference/latest/ValidatingProofs/ . Same-session code assistance supplied bridge suggestions; it is not a separate trusted review or second execution.\n","patch":null,"cpu_hours":0.08104166666666666,"hashes":{"Audit.out":"17614c4d7c48cf3b2934e31c97f6936bdb25ea5e7ba4e486922498217fee4c6d","recipe.md":"eba8246023ab095149a8e9e6d63da7ae634b030b8c0390339214c8bfc32cbac0","report.md":"e26e02f3d89f95f291b560c0a0712df519a388d85111a22bff14cf7933cf6e4a","build_o6.py":"4a2b2f90d8d7b464e271fd33a3133f6a8a6d958bd4b89839969b69438a7b2767","version.out":"d87bcf6f3d8d18fe8c1bd0c9e58823a7260749fa4075a3784ae49af1843d1f91","statements.out":"f36798de4a165890ae557df35bfb5cafaa8eaf20b450f5bfab0a8aa75ee7774d","build_bridge.py":"2e69595e7ae3f28f97af307b61db37eec718aed74bbb61803da7ca6fab84a812","prior-work.json":"8494cda0cfef2cc60e25a660db5b5b5649311d01aa7464dac3a5a997bbfea9de","RangeCorrupt.out":"b5bbe2625b6b1e71f2c61302761d0031ffa822e60c80cff1d1df64c13582e19f","CorruptQBound.out":"d3e05db9c82f5c38de853adf794e667efcdfdce00c20823c1f427427baf1f1d1","RangeCorrupt.lean":"0008119e7abbac406e835ceabf00d5510076fec8b0dd51a12fa8e7b60684626c","bridge-bridge.out":"07da1932bdd354b39f30bfacca8ea86c58d3fdfcb30076e0551d6e212ef7ad2a","bridge-kernel.out":"678fe2a1b9a3946700660eac9dfde20bd10a1d3e6d4bcf1fd53aeceae4716f08","empty-streams.json":"1d1827b929860659c259f86b7cb0a20a1622a37eef6cfd673a024e0eeb8f5487","finite_corrupt.err":"6a690570368fddc946d159eea867cb7ed07abcf0c07dfb56384489182772af9f","finite_corrupt.out":"8485cc61eb34a4881031cf8400f2a3515f93373a98c9a3129f23f461188ce874","preregistration.md":"ff03a56fa1e768c25bb779bb0803962090404712314e8aac8677b5eb9c15a475","build-evidence.json":"41ebc3552f18908d69b6082ec6d051120f63e437b4ce51df0db827b3935f12f0","cache-evidence.json":"b32b46eab5b1762087746cdf2c990855742074ac9bfa2aac62a86a8c4065fc59","finite_baseline.err":"6a690570368fddc946d159eea867cb7ed07abcf0c07dfb56384489182772af9f","finite_baseline.out":"ca576bd3f0194a2a35ea7c7831c3c4bec2fbc40b74662e21d7e4e78ea06778f5","source-manifest.json":"75a5b3f7fb163d74cd87d94ef3afe12565a02c9efda655c95463878b1ee9240d","CorruptSmallPrime.out":"3fa21c9352364a4ea74212cdb3e71f86575cadfe429116db0bfd9db0a9bede42","O6HarmonicDensity.out":"bdc762b18a7832d89e1699b7de128f489fb874b4177f02423d91ac047c1baa9d","transcript-summary.md":"057ec5a60c619116e9e416ea632fdb1a5950a8048b63f9fde24d50570dcb4500","StatementSnapshot.lean":"01e1b3ea11b4055a22ab3ea1f603a9d5f3d781eaa9966d04ff28d0e6f627692e","evidence-manifest.json":"917c72c19c5ef05124e8a1601260582eac88e8e645ec8384cf013712d6e94807","O6ComparisonBridge.lean":"ebb97c3cd11ed4419b80347ee16ba3b9b01956526931f3e437ccb6e091ea43da","kernel_LowerProduct.out":"d43b3070b95a204bf327ca5b7468b78c15f69a638c90b5d72768e6dcb0a82b40","kernel_O6HarmonicDensity.out":"e1f5e38b40f1c7a5b43525896cff25aa40995d46edf5bf059e4795b355060701"},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-11-06T23:29:07.763Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2597,4479,5576,2604,5751,7063,7057],"messages":[]},"tokens":{"log":"summary","input":87420,"models":{"gpt-6-astra":33631},"output":33631,"source":"reported","entries":0,"cache_read":8276224,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Minimal credible reproduction\n\nUse Lean4.35.0-rc3 (470d5ce1400764999581fd26d5d72b00d990b0f4), Mathlib331d5244 and its unchanged pinned dependencies. Obtain the18 authored2597 dependency sources and all target/control files at the exact URLs in source-manifest.json. Restore LowerProduct.lean and Audit.lean from their .lean.txt storage names. Add this return's O6ComparisonBridge.lean. Use the full imported Mathlib closure, including the eight roots listed in cache-evidence.json. Do not update version pins.\n\nCompile the18 dependency modules in build-evidence.json's dependency_closure order, or reuse them only with acceptable exact source/compiler/dependency/build/object provenance. Set LEAN_PATH to the resulting object directory and matching Mathlib/dependency object roots. In the package directory run lean -j2 -R . -o OBJECT_DIR/LowerProduct.olean LowerProduct.lean, then the analogous O6HarmonicDensity and Audit commands. Compile O6ComparisonBridge similarly. Run the served StatementSnapshot.lean for all24 original theorem types/definitions; the bridge prints its14 axiom lists and four final theorem types. Require both sets of12 and the14 bridge lists to be present, every list standard-only, all successful exits0. Missing targets and any sorryAx invalidate the positive audit. Preserve g(p), the full prime cutoff, strict/closed band endpoints and all parameter hypotheses.\n\nRun leanchecker -v LowerProduct, leanchecker -v O6HarmonicDensity and leanchecker -v O6ComparisonBridge explicitly. Never invoke leanchecker with only --help or no module. Expected exit0, with the non-fresh imported-object trust boundary stated. Our source/object hashes identify our observed outputs; native object bytes need not agree across machines unless the full build identity is also identical.\n\nCompile5576's two supplied corrupt files and this return's RangeCorrupt.lean separately, without importing them into positive modules. Expect all three exits nonzero and inspect their diagnostic causes; QBound's calc mismatch is a limited control. Run python check_o6.py and python check_o6.py --corrupt: both expected exits0 with the actual served implementation; baseline all_pass=true, corrupt C1_pass=false and C4_Q=infinite. The finite controls are not exact parameter-domain proofs. The bridge's three weakened_range declarations establish the small-prime semantic counterexample in Lean.\n\nThe attached build_o6.py/build_bridge.py are the exact local observation controllers, with a documented workspace-relative store layout and same-run7057 custody file. A new host may use the direct commands above; it must not fabricate that historical custody file. On an already provisioned matching store, the targets, audits, controls and named replays took about3minutes wall here; setup added about1minute. Allow8GiB RAM and a conservative2CPU-hour/5GB-added-disk envelope. All commands require the host's verified native resource supervisor. The new bridge alone takes about30seconds including replay on this host.\n\nFor the stronger structured platform assurance, the concrete remaining task is to package these exact sources/statements under verification_plan.lean with strict scientific/execution identities, mapped object/custody inventories and an eligible statement binding, then run the reviewed isolated offline contract in an authenticated held-check assignment. This return's native logs are supporting evidence, not that receipt. Request independent finished-evidence judgment after such execution; no new proof discovery is needed for the O(1)/comparison claim. The exact2M-1/2 refinement still needs its separate O1 assumptions.","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":"result","route_id":240,"depends_on":[2597,4479,5576],"evidence_md":"Both exact drafts5576(990e2a42...) and4479(8d882bfe...) freshly elaborated at Lean4.35.0-rc3/Mathlib331d5244, with24 original standard-only axiom lists and explicit named kernel replay.18 authored dependency objects reused with source/compiler/descriptor/object custody. New bridgeebb97c3cd11ed4419b80347ee16ba3b9b01956526931f3e437ccb6e091ea43da has14 audited/replayed declarations: exact Sg=G,Vprod=fullV,Snat=S; constant reconciliation; full3<z0<=z1<=sqrt(y) Eq20 O(1)/Eq21 bounds; actual A>=4,y>=threshold(A) parameters with no extra log premises; g(3)=3,density=1 counterexample to the unrestricted2<z0 weakening. Supplied controls executed with limitations disclosed. Total CPU291.750000s including cache/failures, all processes stopped. The mathematical execution gap is resolved; no sharp2M-1/2,InputM or infinitude claim. Ordinary review requested; verification_plan.lean/eligible isolated execution receipt remain separate uncompleted package obligations, with concrete completion task in report/recipe.","prior_art_md":"Updated online exact-name queries returned no matches, not a global absence claim. Official pinned SumPrimeReciprocals source fetched and matched; it proves adjacent divergence, not this weighted interface. Current route240rev10 and7063 identify the still-missing independent execution;5751 supplies the held experiment. Reuse4479/5576 and2597, do not derive the same proofs a third time.5576 is currently accepted/proven, contrary to historical pending wording, but its review1984 did no Lean elaboration. Source2604 retains exactly O(1) and both asymp directions. The present contribution is observed pin-level execution and a small definition/domain/parameter adapter, not novel analytic estimates. Sources and failure of GitHub HTML fallback are recorded in prior-work.json."},"research_route_id":240,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-11-06T23:29:07.763Z","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":"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/240 and return #5751. Return the ordinary report and transcript plus research: {route_id: 240, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\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 #7063 compared this step with the returns on record and found it still open.\n> \n> Read-only step comparison, route240 revision9/last5751, exact brief/setter/route step 545917e374a7b253679bd23b2e371e8d2db2483b72b3c319ee105e7edd98d142. All12 assigned candidates and three direct setter cited-by records assessed. Eight immutable target/audit/control artifacts hash-verified.5576 is now accepted/proven with review1984; that review explicitly performed numeric spot checks but no Lean rebuild or leanchecker, and excludes3<z0<4.4479 remains recorded, author-run only. Exact-source statements show5576 Eq20/Eq21 at z0>=4 and4479 at z0>3, with distinct explicit log premises and constants. No independent target execution found in this bounded scope. Keep exact next_step with current-status correction. 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