{
 "format": 1,
 "site_commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
 "edges": [
  {
   "id": "seven-eighths-gives-an-explicit-real-zero-gap",
   "kind": "strengthens",
   "from": [
    {
     "name": "OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re",
     "page": "OAI/NumberTheory/DirichletL/Nonvanishing.lean.html#L19"
    }
   ],
   "to": {
    "name": "OAI.SiegelZeros.WeightedTorusJets.exists_absolute_real_zero_gap",
    "page": "OAI/NumberTheory/SiegelZeros/Conclusions/Theorem.lean.html#L11"
   },
   "hypotheses": [
    "(q : ℕ) [NeZero q] (hq : 3 ≤ q)",
    "(χ : DirichletCharacter ℂ q)",
    "(β : ℝ) (hβ : β < 1)",
    "(hzero : χ.LFunction (β : ℂ) = 0)"
   ],
   "conclusion": "Real.log 3 / 8 ≤ (1 - β) * Real.log (q : ℝ)",
   "commit": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
   "status": "proved",
   "verification": {
    "state": "kernel-checked",
    "detail": "Lean's kernel checked seven_eighths_gap against this contract on 2026-10-10T22:56:22Z (Lean (version 4.34.1, x86_64-w64-windows-gnu, commit 5045d0056413266e57c625dcd7c365b10e377c52, Release), Mathlib d13f23b723b8, corpus fd4aeeb2ee4f; axioms: propext, Classical.choice, Quot.sound). This build did not run Lean: it shows that record, after comparing the contract, the bridge file and the commit with it, and the corpus files it read.",
    "provenance": "recorded",
    "record": {
     "status": "checked",
     "checked_at": "2026-10-10T22:56:22Z",
     "environment": "362897e7c89fc6ed2d1c",
     "toolchain": "Lean (version 4.34.1, x86_64-w64-windows-gnu, commit 5045d0056413266e57c625dcd7c365b10e377c52, Release)",
     "mathlib": "d13f23b723b8a846827a245b89c10fc7d3f11612",
     "corpus_revision": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
     "fingerprint": "80fb8a612103a10af380902a",
     "bridge_sha256": "8b56c323c7e21deee5b3fef458c094fa52adb8996f18d0ee24f59dffaee6ffa9",
     "inputs": {
      "ComparatorChallenges/DirichletSevenEighths.lean": "8fb13ad977bca104591fcbc6ef01a58496e7385053c9b1445f95749afc0eff54",
      "ComparatorChallenges/SiegelZeros.lean": "bedaac0248a641910f79a380d9702d82c6bfbbfd3760a69d27c72c5d8fb136ce"
     },
     "axioms": [
      "propext",
      "Classical.choice",
      "Quot.sound"
     ],
     "expected": "(∀ {q : ℕ} [inst : NeZero q] (χ : DirichletCharacter ℂ q) {s : ℂ},\n    7 / 8 < s.re → ¬(χ = 1 ∧ s = 1) → DirichletCharacter.LFunction χ s ≠ 0) →\n  ∀ (q : ℕ) [inst : NeZero q],\n    3 ≤ q →\n      ∀ (χ : DirichletCharacter ℂ q),\n        ∀ β < 1, DirichletCharacter.LFunction χ ↑β = 0 → Real.log 3 / 8 ≤ (1 - β) * Real.log ↑q"
    }
   },
   "stale": false,
   "origin": "Suggested by ChatGPT as “7/8 nonvanishing supplies an explicit real-zero constant”, relayed by the lean4-explainer maintainer, 2026-10-10.",
   "argument": "Suppose 7/8 < β. Take s = β: its real part is β > 7/8, and s ≠ 1 because β < 1, so the pole exception ¬(χ = 1 ∧ s = 1) holds whatever χ is. The source theorem then gives χ.LFunction β ≠ 0, against hzero. So β ≤ 7/8, and 1 − β ≥ 1/8 > 0. Since q ≥ 3, log q ≥ log 3 > 0. Multiplying, (1 − β) · log q ≥ (1/8) · log q ≥ (log 3)/8. This is the target's statement with c = (log 3)/8 ≈ 0.137 > 0, and it uses none of the target's hypotheses that χ is primitive, that χ ≠ 1, that χ is real-valued, or that 0 < β. Argued by Claude (Opus 5.5), 2026-10-10; the bridge in bridges/SevenEighthsGap.lean is the proof Lean checked.",
   "bridge": {
    "name": "seven_eighths_gap",
    "file": "bridges/SevenEighthsGap.lean",
    "page": "implications/bridges/SevenEighthsGap.lean.html"
   },
   "relation": "seven_eighths_gap_gives_absolute_gap",
   "improvement": "An explicit constant, c = log 3 / 8, where the target says only that some c > 0 exists; and fewer hypotheses: the contract needs none of the target's χ primitive, χ ≠ 1, χ real-valued or 0 < β.",
   "construction": null,
   "evidence": "",
   "note": "The target gives no value of c: “No explicit value of $c$ is given.” (lean/docs/003.md, line 16). Its released proof applies SiegelZerosAwei.W50.uniform_exclusion_of_local_isolated_bezout. The bridge is checked against the statements in ComparatorChallenges/DirichletSevenEighths.lean and ComparatorChallenges/SiegelZeros.lean, the copies openai/math's comparator checks the released proofs against. The released modules were not built here (the source's import closure is 2,924 files), so that each copy states the same proposition as its released theorem is the comparator's claim, not checked here."
  }
 ],
 "candidates": [
  {
   "id": "fluid-halting-with-single-fold",
   "kind": "composes-with",
   "from": [
    {
     "name": "OAI.BalancedTransport.balanced_three_stack_realization",
     "page": "OAI/MathematicalPhysics/NavierStokes/BalancedTransport/MainResult.lean.html#L12"
    },
    {
     "name": "OAI.SingleFold.main",
     "page": "OAI/NumberTheory/SingleFold/Main.lean.html#L21"
    }
   ],
   "to": null,
   "phrase": "Fluid halting detection plus single-fold representation",
   "origin": "Suggested by ChatGPT, relayed by the lean4-explainer maintainer, 2026-10-10.",
   "missing": "A stated conclusion, and the link between two models of computation. The fluid theorem says the particle that starts at fixedLabel reaches the observer exactly when Halts M w, for a FiniteMachine M and an input w. SingleFold.main gives every set of natural-number tuples that is REPred (Mathlib's recursively enumerable) a polynomial with exactly one witness on each member and none off it. Composing them needs an encoding of (M, w) as a tuple of natural numbers and a proof that Halts is REPred under it; neither file states one (the fluid proof uses an inputCode internally). With both, the composite would read: the particle reaches the observer exactly when one integer polynomial in the encoded (M, w) has a natural-number solution, and that solution is unique.",
   "note": ""
  }
 ]
}