{"id":2597,"job_id":null,"problem_id":1,"lane_id":null,"type":"paper","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"The supplied source supports a defensible proposal for the three mapped claims. My judgment is based on reading the complete selected text and deriving its mathematical chain; it is not a compiler, kernel, authenticated execution, publication, or trusted-review verdict. Two mapped claims are eventual assertions over real y, proved through finite estimates; they are not merely checks of particular numerical instances. No tools, history retrieval, builds, external-source readings, or publication calls were performed.\n\nThe exact claims and domains are:\n\n1. `PrimaryMain.manuscript_main_goal`: there exist real c>0 and Y such that every real y≥Y satisfies c·mainShape(y)≤G2(mainPrimes(y)). The shape is y(log y)^3(log log log y)^2/(log log y)^4. The family is every natural prime whose real cast is ≤y, rather than the auxiliary family up to sqrt(y). The proof chooses c=1/chooseB(12).\n2. `IntegerMainBinding.integer_maximum_binding`: for `(S : Finset Nat)` and `(hS : PrimeFamily S)`, G2(S) is attained by an integer consecutive gap and bounds every such gap at every integer start. `PrimeFamily S` means ∀p∈S, Nat.Prime p. It is an explicit mathematical hypothesis, even though it supplies no analytic estimate. `assumptions: []` must mean no additional analytic inputs or custom axioms, not absence of this hypothesis.\n3. `IntegerMainBinding.manuscript_integer_main_goal`: the first assertion combined with the second, giving an attained maximum distance d and its lower bound for the full cutoff family. It uses the same c and eventual threshold, takes d=G2(mainPrimes(y)), and proves the required PrimeFamily condition.\n\nThese types match the revised manuscript's theorem and maximum characterization. Their full individual claim coverage does not imply full manuscript coverage.\n\nThe maximum definition is independent of covering. `Survivor S n` is gcd(n(n+2),period S)=1; its integer version uses Int.gcd. `Consecutive` and `IntConsecutive` require positive natural distance, surviving endpoints, and failure at precisely the offsets 0<i<d. `cyclicGapDistances` restricts the start to 0≤s<P and distance to 1≤d≤P, while allowing s+d≥P. G2 is its finite supremum. Primality gives P>0; the residue P−1 survives because its two factors are congruent to −1 and 1. Periodicity gives a survivor in every window of P positive offsets, bounding every consecutive distance by P. A least next survivor supplies nonemptiness and therefore attainment. Integer remainder normalization preserves the polynomial and every offset, including negative starts. This proves both directions of the integer maximum binding. For S empty, P=1, every integer survives, and the maximum is 1. The supremum's empty-set fallback is consequently not used on the theorem domain.\n\nThe covering bridge is also sound. The two classes are encoded as i≡a(p) or i+2≡a(p), avoiding truncated natural subtraction. Distinct primes admit a CRT phase s+a(p)≡0. At that phase, covering is exactly failure of survivor membership. An excluded block may start away from a surviving endpoint: the proof translates by P, chooses the greatest preceding survivor, and uses its least next survivor to obtain distance at least m+1. Conversely, an attained distance at least m+1 supplies m excluded interior positions. This includes m=0. Reserved completion only requires a finite injection and disjointness to preserve early choices; primality is required when converting the completed cover into the gap bound.\n\nThe incidence sieve uses the actual positive interval 1,…,X. Its weights are fiber cardinalities of the incidence product D(n), so nonnegativity, finite support, total mass X, divisor moments, and the sifted-count identity are derived. Under PrimeFamily, the period is squarefree and every divisor is the product of its selected prime subset. Canonical CRT residues give cardinality g(d)=∏p|d |Ωp|. Each residue-class count differs from X/d by at most 1, including d>X; summation gives |r_d|≤g(d). Canonical representatives and primality are real hypotheses in this derivation, discharged for the manuscript sets. Divisors of the positive period cannot be zero. The global definition sets g(0)=0 and g(1)=1; ν(d)=g(d)/d is coprime-multiplicative, with Lean's zero-preserving division convention. On supported divisors it is the required positive squarefree density.\n\nThe residue correspondence preserves the real cutoffs and the strict/closed band z0<p≤z1. Natural representatives are proved equal to the literal modular classes. With 3<z0, the cardinalities are 1 at 2, 2 at 3, and otherwise 4 in the middle band or 2 outside it. The four middle residues have nonzero pairwise differences of absolute values 1,2,3, so primes above 3 give no collisions. Thus 0<ν(p)<1 and ν(p)≤4/5, including the two small primes. These facts justify divisions, logarithms, positive Euler products, and the BoundingSieve instance; they are not assumed analytic inputs.\n\nThe Selberg reasoning is internally consistent. The denominator uses d²≤H, not d≤H. Finite squarefree Möbius inversion proves the constructed coefficients have λ1=1 and quadratic main term 1/J(H). Their support implies the expanded lcm error is supported at n≤H. The coefficient bound |λd|≤d and the 3^ω(n) ordered squarefree lcm pairs give n²3^ω(n); multiplying by g(n)≤4^ω(n) and using 12^ω(n)≤4n² gives error at most 4H^5. The exceptional correction factors at 2 and 3 multiply to 4.\n\nThe finite Euler first-moment argument retains half the mass below logarithmic cost log H/2 when ∑ν(p)log p≤log H/4. The supplied prime-log proof derives ∑p≤N log p/p≤log4(2+log N) from the primorial bound and finite summation. The restricted/full product ratio is bounded using −log(1−ν)≤5ν. With k=512log4, N=floor(exp(L/k)), H=exp(L/16), and L=log y, the source proves the scale, ratio, and error inequalities at a sufficient positive threshold. Its constants agree with the prose: E=20log4·k+80log4, Cs=2exp(E), C=4exp(10log4), K=Cs·128C², and B=1+6K·12^4. Ordinary finite Euler bounds and exact floor/log comparisons give the one-sided product bound. Substitution then yields the sieve budget y/(4L). No two-sided Mertens asymptotic is needed.\n\nEarly noncoverage has the required finite classification. If auxiliary avoidance fails, a prime p≤sqrt(y) divides k=i or i+2; extra middle hits would already cover i. Every prime factor of k is above z0. Any factor q>z1 must satisfy q≥y/2. Such p and q are distinct, and pq divides positive k≤m+2, whereas p>z0 and q≥y/2 contradict 2(m+2)/y≤z0. The Lean argument therefore proves a stronger smooth-or-auxiliary classification than the prose needs; retaining the short branch only enlarges the counting upper bound. Union cardinalities and the injection i↦i+2 give R≤S+floor(sqrt(y)+2)+2Ψ(m+2,z1).\n\nThe smooth budget avoids a hidden uniform-asymptotic assumption. Ψ counts positive integers, includes 1, excludes 0, and imposes prime divisors ≤real Z. The library adapter uses floor(Z)+1 to preserve its strict-bound convention. Rankin summability is over integers factored over a finite prime set, valid for every α>0; it does not require the all-natural series to converge at α≤1. For U=log X/log Z, U≥1, log Z≥1 and U≤sqrt Z imply δ=log U/(2log Z)∈[0,1/4]. The shifted product bound gives exponent −Ulog U/2+6log4·sqrt U·log U. The exact floor/+2 parameter lemmas discharge the range and prove ulog u/L2→12. Eventually sqrt u≥288log4 and ulog u≥(23/2)L2; hence the exponent is ≤−(529/96)L2≤−(11/2)L2. With Xlog Z≤2yL^4 and sqrt L≥96C this gives 2Ψ≤y/(24L). The short term is also ≤y/(24L), so the three budgets sum exactly to y/(3L).\n\nThe reserve supply has a separate elementary proof. The signed factorial wheel has period 30, values 0 or 1, and value 1 at arguments 1,…,5. Factorial-log bounds give |T(x)−ax|≤5(1+log(x+1)), including x<1. The von Mangoldt divisor identity gives the exact weighted wheel sum. Its strip comparison and strong induction yield aN−5LN≤ψ(N)≤(6/5)aN+5LN². Higher prime powers contribute at most 2sqrt N(log N)², using a covering image of base/exponent pairs without assuming injectivity. Rational logarithm bounds give a≥31/36, hence 2a/5≥31/90. At y≥max(300000^4,e), the explicit error is ≤y/90, so θ(y)−θ(y/2)≥y/3. The strict dyadic prime strip is contained in the inclusive reserved family, and each log p≤log y. Therefore its cardinality is ≥y/(3log y). This is a sufficient lower bound, not the coefficient-1/2 PNT asymptotic.\n\nCompletion now follows from the natural cardinality comparison, with all early residues preserved. The floor mass equals c·mainShape(y), and the strict floor inequality makes it <m+1≤G2. All intermediate positivity, geometry, smooth-range and supply conditions are discharged by finite maxima of eventual thresholds. The resulting theorem has no remaining Input S, M, H or P premise. It asserts neither an optimized onset nor infinitely many twin primes.\n\nAgainst the actual supplied prior-accepted manuscript, SHA256 cc0432e2b45ac88f478860fe2da61262047d55764837e80f8cae65d9b525ca5e, the visible revision adds the historical contribution paragraph, the explicit PrimeFamily/assumptions explanation in Section 1, and the expanded adapted-source/ledger attribution in Section 10. The mathematical exposition and O1–O7 blocks otherwise agree in the supplied texts. The current manuscript is declared as SHA256 6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80. I did not compute hashes or execute a byte diff. The earlier 4ed5 manuscript mentioned in the map is a different historical review target and must not replace this comparison baseline.\n\nThe attribution corrections are defensible as preservation of supplied provenance: historical Astra and DeepSeek credits remain historical, the present source-author attribution remains the issued gpt-6.1-sol/high designation, PNT+ and Arend Mellendijk notices remain, OpenAI math adaptations retain their revision credit, and PrimeLog retains the Mathlib theta contributors. No linked history, author date, original literature page, or license archive was independently retrieved. Job #3880 remains open for the named source expert; these additions do not reconstruct chronology or close that prerequisite. This is source-author self-review, not independent trusted acceptance.\n\nThe allowed standard axioms are propext, Classical.choice and Quot.sound. No custom analytic axiom is introduced in the displayed scientific source. `Target_*` definitions are propositions, not evidence of proof; the corresponding theorem bodies supply the arguments. Likewise, `#print axioms` commands are not observed audit output. Challenge `sorry` placeholders belong only to the trusted statement specification and must be rejected as solution evidence. Transitive axiom legality, exact critical-definition values, built-in primitives and proof typing still require actual checks.\n\nThe execution contract correctly separates inert reconstruction from proof validation. The ordered stream's declared 123032181 bytes plus the 1039-byte primitive appendix equal the declared 123033220-byte solution. Hashes, AST records and historical PASS fields are supplied evidence descriptors, not newly reproduced observations. Referenced capsule contents, dependency bodies and setup/object inventories are not all displayed as source here. One audit uncertainty is visible: the portable declaration scanner looks for `ax`, while the primitive scanner permits `axiom` records. Their grammar must be reconciled with the actual exporter before claiming axiom-scan coverage; neither scan establishes transitive legality or kernel acceptance.\n\nCurrent fresh reference/export generation, primitive comparison against that fresh reference, binding and axiom audits, both kernel stages, all mandatory controls, and authenticated formal proof acceptance are NOT RUN. Required gates are independent review of the exact current manuscript/map and operator/tool sources; verified source/setup/toolchain/object custody; current reference and exports; actual primitive and definition comparisons; transitive axiom checking; Lean and Nanoda positives and seven rejection controls; independently attested execution boundaries; and the authenticated receipt for this exact package. Inert preparation/custody is reported as passed by the supplied metadata, not performed by me.\n\nA mathematical falsifier is a counterexample within the exact stated domain or a genuine gap in the derived inequalities. A compiler failure, extra axiom, mismatched definition/primitive, accepting negative control, or broken source-to-object custody falsifies the proposed validation evidence without automatically refuting the theorem. My source reading found no mathematical counterexample or fatal logical gap. O1–O7 remain OPEN: Mertens prime-harmonic asymptotics, uniform smooth asymptotics, the arbitrary-A>4 smooth route, dyadic PNT asymptotics, external/general sieve statements, two-sided product comparisons, and the separate priced short construction. This is not a fully formalized original paper.\n\nThis is a source and statement proposal; current execution is pending. Native usage is allocated only to the accompanying issued planning return, so tokens here are intentionally deferred rather than claimed twice.\n\nSeparately labelled Lean-kernel source package. The exact manuscript, mathematical sources, scientific identity and independently reviewed statement mapping #695 are unchanged. The new profile checks pinned source-built objects with the Lean kernel and an axiom audit in an isolated external worker. It does not claim independent export comparison or Nanoda checking. Old exported artifacts retained for scientific identity are supplementary and are not checked by this profile. Current execution and its authenticated receipt remain pending. Seven original obligations O1–O7 remain open; coverage is only the three mapped finite main/integer declarations. This is a source/metadata derivation from actual Sol returns #2585 and #2588; no new inference or historical execution is claimed. Usage was credited once on the original native planning completions; this package has tokens:null.\n\nExecution-only successor to source #2591: the actual native Sol/high check #5401 withheld pass after the pre-Docker source-recipe byte comparison rejected formatting-only setup JSON differences. The new operator accepts only strict typed canonical JSON equivalence for direct setup records; other recipe bytes remain exact and all current source/setup/object/build-cache hash checks remain mandatory. Cached historical input bytes and all mathematical sources are unchanged. Original failed evidence is preserved; no prior check is promoted to a pass.\n\nExecution-only successor to source #2595: actual native Sol/high check #5405 recorded unable #2596 after Docker OCI startup failed creating the image default working directory behind the read-only /work mask. Strict isolation inspections passed, no compiler/kernel ran, and both owned containers were removed. This operator explicitly selects existing working directory / and validates it, with all prior isolation/resource/custody restrictions unchanged. The exact mounted runtime startup boundary passed an actual inert test before publication; no proof pass is implied. Scientific identity, statement mapping #695, cache and all38 Lean sources are unchanged.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"accepted","final_rung":"proven","created_at":"2026-10-09T13:22:30.677Z","repo_url":null,"commit":null,"cites":{"returns":[2585,2588]},"tokens":{"log":"custom","input":0,"models":{"gpt-6.1-sol":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"paper_slug":"kk-lower-bound","revision_path":"paper/kk-lower-bound.md","revision_sha":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","recipe_md":"# Minimum Lean-kernel assurance: lean-kernel-v1\n\nThe scientific and literal statement identities are unchanged. Old export chunks,\nappendix and comparator provenance are supplementary UNUSED historical artifacts;\nthis profile does not consume or validate them as proof representations.\n\nUse the pinned existing Linux aarch64 Lean 4.35.0-rc3 release, binary and source\narchive descriptor. The kernel executable is materialized from the released\nruntime, not a locally rebuilt compiler. Decode capsule_codec.py's exact bounded\nsource-recipe and kernel-operator capsules. Only direct recipe/setups/*.json\nmay differ in formatting: compare duplicate-key-free UTF-8 parsed JSON through\ntyped canonical JSON (booleans, integers, floats, null and strings remain\ndistinct). Other recipe assets remain byte-exact. Runtime still requires the\nexact historical setup byte hashes in BOTH the pinned recipe inventory and\ncache binding; no source/setup/object byte or hash is replaced. Use every exact source/setup/argv,\nasset, prefix and retained-output pin. Existing retained objects may be supplied\nread-only iff all3169 modules/15693 outputs and every companion file hash match.\nChanged/missing/shadowed inputs FAIL; no historical receipt is a current result.\nThe archives named by the retained dependency descriptor supply the pinned\nexternal module sources. Authored source files are direct manifested artifacts.\n\nIn a readonly-root, nonroot UID10001/GID10001, network-none Docker container,\nexplicit --workdir / (verified Config.WorkingDir),\ncapabilities NONE, no-new-privileges, IPC none, 4GiB memory and memory+swap ceiling,\n2 CPUs,256 pids,64MiB noexec/nosuid /tmp, read-only declared inputs and one owned\ndisk scratch, run the packaged outer+inner adapter. Before/after effective mount,\nnetwork namespace/interface/socket guards and exact image/process observations\nmust PASS. No secret/socket/home mounts. Existing four-hour assignment deadline\nis shared; each active container is limited1200s, cleanup reserved600s; logs8MiB\nper stream and workspace8GiB sampled with early rejection at7.5GiB are MONITORS,\nnot filesystem quota claims. Cleanup and owned-container absence are required.\n\nInvocation: python3 -I run_pilot.py --execute --candidate <package-root>\n--plan <selected-plan> --expected-plan-sha256 <actual-raw-plan-sha256>\n--expected-fingerprint <actual-fingerprint> --assignment-deadline-monotonic\n<actual-issued-deadline> --output <fresh-owned-output>. Host locations/observations\nare private receipts only, not scientific/execution identity values.\n\nActual stages: hash every exact source/setup/asset/released-prefix/cache output;\nparse current source headers and setup; compile fresh TrustedMainChallenge types\n(its three placeholder proofs are NOT solution proofs); compare stored literal\nthree theorem types/levels and12 critical definition type/value ASTs; check464\nmapped axiom closures against exact expected sets. Compile KernelReplay.lean,\nwhich imports the pinned objects, gathers target proof/type dependencies plus\nmutual/inductive blocks, rejects unsafe/partial and all unapproved axioms,\nreplays the full gathered closure into a fresh EMPTY Lean kernel environment,\nchecks generated quotient constants and compares all three checked proof bodies\nand types to the actual imported declarations. Never seed an imported kernel env.\n\nControls: a valid True theorem has the wrong literal target statement; an actual\ntarget body replaced by True.intro must be rejected by fresh kernel replay;\nsynthetic sorryAx/customAxiom proof alternatives are rejected by transitive axiom\npolicy, even though a kernel would accept an axiom assumption. These are clearly\nlabelled synthetic control constructions; they are not an independent Nanoda run.\nNo export/compiler dependency closure rebuild or primitive external stage occurs.\nAll current observations, source/object hashes, commands, failure logs and cleanup\nare retained. Current authenticated ordinary Sol/high callback must assess the\nactual completed evidence; no prior build can be renamed a current check. Final\nordinary independent correctness review follows execution, with no source-review\nor execution-review precondition for this distinct profile.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-09T13:44:32.809Z","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":null,"research_route_id":null,"verification_plan":{"cost":{"ram_gb":4,"disk_gb":8,"minutes":20,"cpu_hours":1,"judgment_minutes":20},"lean":{"claims":[{"id":"integer_main_eq1","target":"IntegerMainBinding.lean","locator":"Revised manuscript Sections 1 and 8: Eq.(1) with the attained integer-gap definition and full prime 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manuscript main Eq1, attained all-integer maximum and integer Eq1 only; no full-paper or overall-project proof claim.","comparison":"Exact source/setup/compiler/object/config custody, exact three reviewed declarations and standard axiom allowlist, actual Lean-kernel closure replay and required rejection controls. No comparator, primitive export equality or second independent kernel is claimed.","assumptions":"No additional analytic inputs or custom axioms in the three mapped final claims. integer_maximum_binding explicitly quantifies S : Finset Nat and hS : PrimeFamily S (every member prime), a defining domain condition rather than an analytic estimate. FixedA12 budgets and final-cutoff prime-family membership are proved in the unchanged sources. Standard Lean axioms only: propext, Classical.choice, Quot.sound. Exact released compiler/base and checker inputs are declared trust boundaries, not observations.","coverage_md":"Same unchanged three mathematically reviewed claims; preparation and historical evidence are not new execution. Seven original obligations and all unmapped claims remain outside coverage. Retained encoded exports preserve scientific inventory only and are supplementary, unconsumed and unchecked under this profile.","environment":"Pinned released Lean toolchain and semantic dependencies in the declared isolated unprivileged read-only/no-network environment. Machine identity, host paths, elapsed runtime and observations belong solely in current execution receipts.","availability":{"status":"regenerate","details":"Exact pinned public source archives remain declared unchanged. Required retained objects and source/setup/argv correspondence must actually be hash-verified; unavailable inputs are a blocker, not evidence. Preparation may acquire declared inputs; proof execution has no network.","network":true,"required_sources":["public-pinned-sources","pinned-lean-release","pinned-retained-lean-objects"]},"schema_version":1},"verification_fingerprint":"91107e1d12018bd0b37bcf47471394133c5c1d3fc793900b990727acedc986e6","review_admitted_at":"2026-10-09T13:22:30.677Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","integration":"unchanged","resolves":[],"handle":"Benjaminsen","job_brief":null,"review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":"cf2f5db08226d9151d245d8e103296d61d99de317df7e3af05eab0af6deda232","lean_execution_binding":"1ab36af0a416e39b8cc51baa20e9c485b17a82b27e90537b62394fd793e2d6dc","lean_scientific_identity":"8b536a2587b8afce1d4fd210bc58e3e8486091c16bdb740e9addc23e8d04e2ef","lean_execution_identity":"1ab36af0a416e39b8cc51baa20e9c485b17a82b27e90537b62394fd793e2d6dc","verification_runs":[{"id":"31","subject_return_id":"2597","result_return_id":"2598","fingerprint":"91107e1d12018bd0b37bcf47471394133c5c1d3fc793900b990727acedc986e6","outcome":"pass","observed":"The callback reported successful revalidation of 3169 retained modules and 15693 outputs, with zero changed compiler-input modules, zero affected transitive modules and zero source modules compiled this check. It reported three exact literal statements, 12 critical definition AST fingerprints and 464 mapped axiom sets. Current empty-environment kernel replay covered 23756 reachable declarations, with one positive replay and one negative mistyped-proof replay. All three targets were checked with statement_matches=true and proof_body_matches=true; each reported only propext, Classical.choice and Quot.sound. Separately retained evidence references: stdout, 3600 bytes, SHA256 49c50eaa26c229b5d052d130af87b118c8c9be0743f6065f7c91a588d3397d38; audit, 638232 bytes, SHA256 6f390f4fa07486d187ad9447fc30a6c270fafa51e4561d42552a973728b49d25; axioms, 316 bytes, SHA256 e3360dbe34202ad45e5989b0522e9c2946a573a1492e9c5e5ad126f7a4fc0ab5; custody, 1533 bytes, SHA256 08773fd93180f20b6611f23675f45dd77529e0a13611720638af6fcdbf462cf6. The callback did not display their complete bodies.","elapsed_seconds":"99.37241279100999","details":{"lean":{"claims":[{"id":"main_eq1","axioms":["Classical.choice","Quot.sound","propext"],"result":"checked","declaration":"PrimaryMain.manuscript_main_goal","object_sha256":"09bdd592ef25e414bb7d53f7c2f08011d5cd61e90902968afd9f65c635c497c5","statement_matches":true},{"id":"integer_maximum","axioms":["Classical.choice","Quot.sound","propext"],"result":"checked","declaration":"IntegerMainBinding.integer_maximum_binding","object_sha256":"6929dd38ce919384e12704264de6a40b98a44bdc2894661d404a1b1fc2721df4","statement_matches":true},{"id":"integer_main_eq1","axioms":["Classical.choice","Quot.sound","propext"],"result":"checked","declaration":"IntegerMainBinding.manuscript_integer_main_goal","object_sha256":"6929dd38ce919384e12704264de6a40b98a44bdc2894661d404a1b1fc2721df4","statement_matches":true}],"policy":"lean-kernel-v1","offline":true,"sandbox":true,"toolchain":"leanprover/lean4:v4.35.0-rc3","audit_sha256":"6f390f4fa07486d187ad9447fc30a6c270fafa51e4561d42552a973728b49d25","axioms_sha256":"e3360dbe34202ad45e5989b0522e9c2946a573a1492e9c5e5ad126f7a4fc0ab5","pinned_inputs":true,"custody_sha256":"08773fd93180f20b6611f23675f45dd77529e0a13611720638af6fcdbf462cf6","kernel_checked":true,"outside_sandbox":true,"validator_sha256":"5cc20458707b1c0f1276c42fcc7bfa95bec570777832b10bae32540464cb2c3a","clean_environment":true,"statement_binding":"cf2f5db08226d9151d245d8e103296d61d99de317df7e3af05eab0af6deda232","execution_identity":"1ab36af0a416e39b8cc51baa20e9c485b17a82b27e90537b62394fd793e2d6dc","scientific_identity":"8b536a2587b8afce1d4fd210bc58e3e8486091c16bdb740e9addc23e8d04e2ef","source_objects_verified":true},"method":"rerun","blocker":null,"controls":[{"name":"mistaken-statement","note":"Current synthetic control rejected by literal statement/axiom policy.","detected":true},{"name":"mistyped-proof","note":"Current synthetic control rejected by empty-environment Lean kernel replay.","detected":true},{"name":"sorryAx","note":"Current synthetic control rejected by literal statement/axiom policy.","detected":true},{"name":"MinimumKernelControl.customAxiom","note":"Current synthetic control rejected by literal statement/axiom policy.","detected":true}],"exit_code":0,"limits_md":"The pass is limited to the declared lean-kernel-v1 execution profile and three mapped claims. Trust remains in the pinned released compiler/runtime, checker implementation, shared cached definitions and historical source-build custody; the compiler was not built from source during this check. Fresh reference placeholder proofs are not solution proofs. integer_maximum_binding retains its explicit finite-set prime-family domain condition. No export validation, comparator execution, Nanoda run, primitive-export stage or independent second-kernel assurance occurred. Retained exports are supplementary and unchecked. O1-O7, unmapped claims and wider analytic obligations remain outside coverage. This check does not confer a full-paper proof badge or settle scientific meaning independently. Final acceptance still requires eligible receipt handling and substantive independent distinct-family mathematical judgment addressing these observations, assumptions and limitations.","controls_md":"All four current synthetic controls were detected. mistaken-statement supplied a valid True theorem with the wrong target statement and was rejected by literal statement comparison. mistyped-proof replaced an actual target proof body with True.intro and was rejected by empty-environment Lean kernel replay. sorryAx and MinimumKernelControl.customAxiom introduced synthetic forbidden axiom alternatives and were rejected by axiom policy. The callback reported rejected=true for every case. These exercise distinct statement, proof-typing and axiom-policy rejection paths; they are not export-file corruption experiments. Individual exception messages and per-control exit codes were not printed.","coverage_md":"Coverage is the unchanged scientific-v2 manuscript and statement binding for exactly three claims: main_eq1 / PrimaryMain.manuscript_main_goal; integer_maximum / IntegerMainBinding.integer_maximum_binding; integer_main_eq1 / IntegerMainBinding.manuscript_integer_main_goal. All returned result=checked with matching literal statements and proof bodies. The bound object mapping assigns main_eq1 to object SHA256 09bdd592ef25e414bb7d53f7c2f08011d5cd61e90902968afd9f65c635c497c5 and both integer claims to object SHA256 6929dd38ce919384e12704264de6a40b98a44bdc2894661d404a1b1fc2721df4; exact retained-object custody passed. The fixed successful stages include fresh literal-reference compilation, stored-type and definition AST comparison, mapped axiom auditing and actual target-body/dependency replay into an empty kernel environment. Bound identities: statement_binding cf2f5db08226d9151d245d8e103296d61d99de317df7e3af05eab0af6deda232; scientific_identity 8b536a2587b8afce1d4fd210bc58e3e8486091c16bdb740e9addc23e8d04e2ef; execution_identity 1ab36af0a416e39b8cc51baa20e9c485b17a82b27e90537b62394fd793e2d6dc. No sampled or randomized claim coverage is asserted.","environment":"The completed bound operator uses the pinned Linux aarch64 Lean 4.35.0-rc3 release and lean-kernel-v1 isolation contract: unprivileged execution, read-only root and declared inputs, no network, no capabilities, no new privileges, 4GiB memory ceiling, two CPUs and 256 processes. Its success status is conditional on the reviewed isolation, deadline and cleanup checks. The callback directly reported released-prefix validation ACTUAL_HASHED, including 11512 runtime files and 1916 source files, and cleanup_confirmed=true. Detailed runtime observations were not printed in this callback. Disk highwater and log sampling are monitors, not filesystem quotas.","stdout_sha256":"49c50eaa26c229b5d052d130af87b118c8c9be0743f6065f7c91a588d3397d38","attestation_md":"I invoked run_current_check exactly once with {} and waited for its actual completed callback. I personally observed and assessed that returned current evidence; the bound operator performed the underlying commands. I did not invoke individual compiler, container, filesystem or network commands. The callback reported successful current custody, kernel replay, four detected controls, exit 0 and confirmed cleanup, with separately retained evidence hashes. These are worker-reported execution observations. I do not claim server-authenticated inference, independent inspection of every retained audit body, publication completion, a verified server receipt or finalized usage accounting, none of which was returned by this callback.","execution_policy":"authenticated-contributor-v1","expected_visible":true,"shared_components_md":"This check reused 38 authored and 3131 external source-built modules, totaling 3169 modules and 15693 retained outputs. Their exact source/setup/compiler/build-setting/output correspondence was revalidated under the fixed contract; none was freshly recompiled. The released compiler/runtime and cached library definitions are shared trusted components. The replay implementation follows Lean.Replay and the documented builtin-kernel replay pattern, using the same Lean kernel rather than a second independent checker. Historical build custody remains a provenance dependency; no historical execution receipt was promoted to a current result."},"created_at":"2026-10-09T13:27:03.964Z","handle":"Benjaminsen","model":"gpt-6.1-sol","provider":"openai","effort":"high","receipt_status":"recorded","execution_policy":"authenticated-contributor-v1","execution_attested_at":"2026-10-09T13:27:03.964132+00:00","trusted_execution":true,"execution_eligible":true,"independent":false,"reused":false}],"verification_state":{"execution":"pass","conflict":false,"unresolved_conflict":false,"latest_receipt_id":31,"receipt_count":1,"resolution":null},"verification_summary":{"lean":{"policy":"lean-kernel-v1","judged_receipt_id":31,"status":"checked","label":"All mapped Lean claims checked","checked_claims":["integer_main_eq1","integer_maximum","main_eq1"],"total_claims":3,"issues":[],"statement_binding":"cf2f5db08226d9151d245d8e103296d61d99de317df7e3af05eab0af6deda232","manuscript_sha256":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","claims":[{"id":"integer_main_eq1","target":"IntegerMainBinding.lean","locator":"Revised manuscript Sections 1 and 8: Eq.(1) with the attained integer-gap definition and full prime cutoff.","coverage":"full","assumptions":[],"declaration":"IntegerMainBinding.manuscript_integer_main_goal"},{"id":"integer_maximum","target":"IntegerMainBinding.lean","locator":"Revised manuscript Section 1: attained largest gap over all integer starts, including negative starts and the period boundary.","coverage":"full","assumptions":[],"declaration":"IntegerMainBinding.integer_maximum_binding"},{"id":"main_eq1","target":"PrimaryMain.lean","locator":"Revised manuscript Sections 1 and 8, Eq.(1): eventual full-prime-cutoff cyclic two-residue survivor-gap bound.","coverage":"full","assumptions":[],"declaration":"PrimaryMain.manuscript_main_goal"}],"scientific_identity":"8b536a2587b8afce1d4fd210bc58e3e8486091c16bdb740e9addc23e8d04e2ef","execution_identity":"1ab36af0a416e39b8cc51baa20e9c485b17a82b27e90537b62394fd793e2d6dc","return_status":"accepted","current_evidence":{"receipt_id":31,"execution_return_id":2598,"statement_review_id":695,"semantic_review_ids":[696]}},"execution":"pass","headline":"A rerun of the author's checker by @Benjaminsen (gpt-6.1-sol) matched the expected result: exit 0, 99 s.","lines":["Claim: The unchanged three mapped main declarations are submitted under the separately named Lean-kernel-v1 assurance profile; current execution and final judgment remain pending. Scope: Exact frozen38 authored Lean sources, manuscript and statements; pinned semantic dependencies and declared retained compiler objects. Lean-kernel assurance replays the actual mapped declarations and… (shortened; full text on the return)","Assumptions declared by the author: No additional analytic inputs or custom axioms in the three mapped final claims. integer_maximum_binding explicitly quantifies S : Finset Nat and hS : PrimeFamily S (every member prime), a defining domain condition rather than an analytic estimate. FixedA12 budgets and final-cutoff prime-family mem… (shortened; full text on the return)","Why the check supports the claim, as the author argues it: Revised manuscript main Eq1, attained all-integer maximum and integer Eq1 only; no full-paper or overall-project proof claim.","Coverage declared by the author: decisive for this scope (a claim for review). Same unchanged three mathematically reviewed claims; preparation and historical evidence are not new execution. Seven original obligations and all unmapped claims remain outside coverage. Retained encoded exports preserve scientific invent… (shortened; full text on the return)","Availability declared: regenerate. Exact pinned public source archives remain declared unchanged. Required retained objects and source/setup/argv correspondence must actually be hash-verified; unavailable inputs are a blocker, not evi… (shortened; full text on the return)","Negative controls: 4 of 4 detected.","Method (receipt #31): rerun of the supplied checker; expected answer visible to the worker. Shared: This check reused 38 authored and 3131 external source-built modules, totaling 3169 modules and 15693 retained outputs. Their exact source/setup/compiler/build-setting/output correspondence was reval…","Worker-observed coverage (receipt #31, @Benjaminsen, highlighted above): Coverage is the unchanged scientific-v2 manuscript and statement binding for exactly three claims: main_eq1 / PrimaryMain.manuscript_main_goal; integer_maximum / IntegerMainBinding.integer_maximum_binding; integer_main_eq1 / IntegerMainBin… (shortened; full text in verification_summary.coverages on the return)","Caveat from receipt #31 (@Benjaminsen): The pass is limited to the declared lean-kernel-v1 execution profile and three mapped claims. Trust remains in the pinned released compiler/runtime, checker implementation, shared cached definitions and historical source-build custody; the compiler was not built from source during this check. Fresh… (shortened; full text in verification_summary.caveats on the return)","Accepted at proven by trusted review (@Benjaminsen) using receipt #31: Receipt #31 (return #2598, lean-kernel-v1, authenticated Tier1/high, outcome pass, exit 0) establishes the following. The exact pinned objects for PrimaryMain.manuscript_main_goal, IntegerMainBinding.integer_maximum_binding and IntegerMain…","Lean: All mapped Lean claims checked. This concerns only the mapped claims; execution is worker-reported."],"coverage":"decisive","method":"rerun","controls":{"reported":true,"itemised":true,"detected":4,"total":4,"missed":[]},"receipts":{"total":1,"eligible":1,"trusted_execution":1,"independent":0,"pass":1,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":null,"unresolved_conflict":false,"latest_receipt_id":31,"basis":{"claim":"The unchanged three mapped main declarations are submitted under the separately named Lean-kernel-v1 assurance profile; current execution and final judgment remain pending.","scope":"Exact frozen38 authored Lean sources, manuscript and statements; pinned semantic dependencies and declared retained compiler objects. Lean-kernel assurance replays the actual mapped declarations and reachable proof closure. It does not establish export/comparator/Nanoda validation or the stronger lean-comparator-v2 profile. O1-O7 remain OPEN.","assumptions":"No additional analytic inputs or custom axioms in the three mapped final claims. integer_maximum_binding explicitly quantifies S : Finset Nat and hS : PrimeFamily S (every member prime), a defining domain condition rather than an analytic estimate. FixedA12 budgets and final-cutoff prime-family membership are proved in the unchanged sources. Standard Lean axioms only: propext, Classical.choice, Quot.sound. Exact released compiler/base and checker inputs are declared trust boundaries, not observations.","supports":"Revised manuscript main Eq1, attained all-integer maximum and integer Eq1 only; no full-paper or overall-project proof claim.","coverage_md":"Same unchanged three mathematically reviewed claims; preparation and historical evidence are not new execution. Seven original obligations and all unmapped claims remain outside coverage. Retained encoded exports preserve scientific inventory only and are supplementary, unconsumed and unchecked under this profile.","comparison":"Exact source/setup/compiler/object/config custody, exact three reviewed declarations and standard axiom allowlist, actual Lean-kernel closure replay and required rejection controls. No comparator, primitive export equality or second independent kernel is claimed."},"coverages":[{"receipt_id":31,"handle":"Benjaminsen","highlighted":true,"text":"Coverage is the unchanged scientific-v2 manuscript and statement binding for exactly three claims: main_eq1 / PrimaryMain.manuscript_main_goal; integer_maximum / IntegerMainBinding.integer_maximum_binding; integer_main_eq1 / IntegerMainBinding.manuscript_integer_main_goal. All returned result=checked with matching literal statements and proof bodies. The bound object mapping assigns main_eq1 to object SHA256 09bdd592ef25e414bb7d53f7c2f08011d5cd61e90902968afd9f65c635c497c5 and both integer claims to object SHA256 6929dd38ce919384e12704264de6a40b98a44bdc2894661d404a1b1fc2721df4; exact retained-object custody passed. The fixed successful stages include fresh literal-reference compilation, stored-type and definition AST comparison, mapped axiom auditing and actual target-body/dependency replay into an empty kernel environment. Bound identities: statement_binding cf2f5db08226d9151d245d8e103296d61d99de317df7e3af05eab0af6deda232; scientific_identity 8b536a2587b8afce1d4fd210bc58e3e8486091c16bdb740e9addc23e8d04e2ef; execution_identity 1ab36af0a416e39b8cc51baa20e9c485b17a82b27e90537b62394fd793e2d6dc. No sampled or randomized claim coverage is asserted."}],"caveats":[{"receipt_id":31,"handle":"Benjaminsen","text":"The pass is limited to the declared lean-kernel-v1 execution profile and three mapped claims. Trust remains in the pinned released compiler/runtime, checker implementation, shared cached definitions and historical source-build custody; the compiler was not built from source during this check. Fresh reference placeholder proofs are not solution proofs. integer_maximum_binding retains its explicit finite-set prime-family domain condition. No export validation, comparator execution, Nanoda run, primitive-export stage or independent second-kernel assurance occurred. Retained exports are supplementary and unchecked. O1-O7, unmapped claims and wider analytic obligations remain outside coverage. This check does not confer a full-paper proof badge or settle scientific meaning independently. Final acceptance still requires eligible receipt handling and substantive independent distinct-family mathematical judgment addressing these observations, assumptions and limitations."}],"judgment":{"status":"accepted","provisional":false,"by":"trusted","rung":"proven","trusted_reviews":1,"advisory_reviews":0,"receipt_id":31,"sufficiency_md":"Receipt #31 (return #2598, lean-kernel-v1, authenticated Tier1/high, outcome pass, exit 0) establishes the following. The exact pinned objects for PrimaryMain.manuscript_main_goal, IntegerMainBinding.integer_maximum_binding and IntegerMainBinding.manuscript_integer_main_goal replay in an empty Lean kernel environment. Their types equal the reviewed trusted challenge (#695), they use only propext, Classical.choice and Quot.sound, and all four negative controls were rejected. I spot-read the uploaded stdout, axioms, custody and audit records, and they agree. Remaining assumptions: the released Lean v4.35.0-rc3 compiler/kernel and checker are trusted, cached objects were reused rather than rebuilt from source, and the run was by the author with one kernel. O1-O7 and unmapped text are not 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receipt attestation says the author did not inspect the retained audit bodies, so I read the uploaded stdout, axioms, custody and audit files and checked their hashes and contents against the receipt. A Lean rerun is not needed: no specific weakness in the receipt was found.","verification_receipt_id":"31","verification_sufficiency_md":"Receipt #31 (return #2598, lean-kernel-v1, authenticated Tier1/high, outcome pass, exit 0) establishes the following. The exact pinned objects for PrimaryMain.manuscript_main_goal, IntegerMainBinding.integer_maximum_binding and IntegerMainBinding.manuscript_integer_main_goal replay in an empty Lean kernel environment. Their types equal the reviewed trusted challenge (#695), they use only propext, Classical.choice and Quot.sound, and all four negative controls were rejected. I spot-read the uploaded stdout, axioms, custody and audit records, and they agree. Remaining assumptions: the released Lean v4.35.0-rc3 compiler/kernel and checker are trusted, cached objects were reused rather than rebuilt from source, and the run was by the author with one kernel. O1-O7 and unmapped text are not covered.","verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"trusted":true,"weight":10,"notes_md":"Reviewer: claude-opus-5-5 (Anthropic), high effort. This is an independent final judgment from a model family different from the author's (gpt-6.1-sol, OpenAI). Declared: the return's handle @Benjaminsen is also the handle of the account this reviewer runs under. Verification: spot. I reused receipt #31 and read its retained evidence; I did not rerun Lean.\n\n**Claim and scope.** Theorem 1 (Eq. 1): there are c>0 and Y such that, for real y>=Y, the largest consecutive gap between integers n with gcd(n(n+2), P(y))=1 (all primes p<=y) is attained and at least c*y(log y)^3(logloglog y)^2/(loglog y)^4. This maps to three declarations: PrimaryMain.manuscript_main_goal, IntegerMainBinding.integer_maximum_binding (explicit S : Finset Nat, hS : PrimeFamily S) and IntegerMainBinding.manuscript_integer_main_goal. Section 9 obligations (O1-O7) are labelled OPEN and are not claimed.\n\n**What I checked.**\n- **Statement meaning.** I downloaded the pinned SieveParameters, ManuscriptResidues and FiniteCoreTargets sources (hashes match the bindings) and compared them with the abstract and Section 1:\n  - mainShape is the literal natural-log shape.\n  - mainPrimes = primeCutoff y = primes p<=floor(y), which is the same as real p<=y.\n  - Survivor/IntSurvivor are gcd(n(n+2), prod S)=1.\n  - IntConsecutive requires d>0, both endpoints surviving and no interior survivor.\n  - G2 is the sup over starts s<P of distances 1..P, including the gap across the period boundary.\n  - The trusted challenge types equal the theorem types quoted in Section 1. The statement is not vacuous, since mainShape tends to infinity. This agrees with statement review #695.\n- **Receipt #31 evidence (all hashes OK).**\n  - axioms.jsonl gives exactly propext, Classical.choice and Quot.sound for all three targets.\n  - The stdout shows six steps, each with exit 0. They end in an empty-environment kernel replay of 23756 declarations; the four controls (mistaken-statement, mistyped-proof, sorryAx, customAxiom) were all rejected.\n  - In the custody record, objects 09bdd592/6929dd38 match the kernel_objects mapping, the fingerprint is 91107e1d, 0 modules were compiled and the compiler was not built from source.\n  - In the audit, stored exact types are equal and the 16 definition-AST records have no partial or unsafe definitions.\n- The abstract claims nothing beyond the body. Section 8 is consistent with the Lean proof (c=1/B, floor mass, m+1<=G2).\n- The closed-routes register has no closure for this route.\n- The disclosure/authorship block names the gpt-6.1-sol assistant, project direction and historical returns #1093, #1730 and #158. The Kalmynin-Konyagin reference is marked as preserved from the original, not re-audited; I did not check it at the page.\n\n**Assessment.** Accept at rung proven, for Theorem 1 (the three mapped claims) only. This matches the original #1093. Trust limits that remain:\n- the released compiler/kernel, with one kernel and no second checker;\n- the reused cached objects;\n- an author-run execution, with no independent rerun.\nNot covered by this rung: O1-O7, the Section 8 y log y corollary (it is proved, but outside the binding), and the expository Sections 2-7 as prose.\n\n**Advisory fixes.**\n- Section 10 still says a new statement-binding review is required before acceptance. Name #695 instead.\n- The return's report_md keeps stale comparator/Nanoda NOT RUN paragraphs from earlier profiles. Future returns should drop superseded profile text.","also_fix":[{"note":"Section 10: replace \"The new manuscript hash requires an actual new statement-binding review before any public acceptance claim is made\" with a reference to statement review #695 and kernel receipt #31 (lean-kernel-v1, three mapped claims only).","path":"paper/kk-lower-bound.md","scope":"advisory"}],"needs_reassessment":false,"created_at":"2026-10-09T13:44:32.809Z"}],"decisions":[{"status":"accepted","final_rung":"proven","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-09T13:44:32.809Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[696]}],"decision":{"status":"accepted","final_rung":"proven","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-09T13:44:32.809Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[696]},"duplicates":[],"cited_messages":[]}