{
 "what": "The audit of P. Anand, 'Zeta 7 is Irrational' (Zenodo 22920911 v1, 2026-09-23), and of issue #1 on gmDevi/zeta-7-21-lean (opened 2026-10-03 by GitHub user huntrontrakkr, a third party): the outer integral I_out of (73)/(122) recomputed exactly from the paper's own formulas under each reading, the margin (118) that Theorem 1.1 rests on recomputed for every correction, the calibration on Fauzan's ζ(5) paper, the paper's finite statements checked at small K, its own family measured, and the Lean repository read.",
 "generatedBy": "instruments/zeta7audit/decide.py",
 "generatedOn": "2026-10-05",
 "env": {
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 "rule": {
  "exact": "every integral is an exact rational: pwaff.integrate certifies each branch of the integrand constant on each cell (an affine difference evaluated at both closed endpoints) and cuts a cell at the exact point where one is not",
  "balls": "every logarithm is an arb ball; [lo, hi] endpoints rounded outward to doubles",
  "independence": "no code shared with the claimant or with the issue's reproduction script; the calibration on Fauzan is the control that the same evaluator reproduces a printed value that is right"
 },
 "scope": "Decided: the arithmetic of the paper's proof route — I_out, (121), A0, (85), the U ingredients, (117)–(119) — as exact rationals or balls, from the formulas as printed. Not decided: whether ζ(7) is irrational (open); Theorem 1.2 as an asymptotic statement (measured at n ≤ 3 only); the inner estimate Proposition 4.4 and its limit (70); Lemma 6.1's sup (94) (grid-measured); the Lean repository's own theorem (read, not built).",
 "claim": {
  "title": "Zeta 7 is Irrational",
  "author": "Prince Anand",
  "doi": "10.5281/zenodo.22920911",
  "version": "v1 (the record's only version)",
  "date": "2026-09-23",
  "license": "CC-BY-4.0",
  "pages": 42,
  "statement": "Theorem 1.1: ζ(7) is irrational. Theorem 1.2: Q_{40n,200} ∈ ℤ[X], deg 37n, 0 < Q_{40n,200}(ζ(7)) < exp(−19000 n²) for large n. Corollary 1.3: |ζ(7) − a/b| > b^−270.",
  "route": "lim sup K^−2 log Q_{K,M}(ζ(7)) ≤ A_M + U (117), with A_M from the prime sum (89) and U = 44/25 from the real bound; (118) A200 + U < −11 < 0 is what makes the integer b^{37n} Q(a/b) tend to zero.",
  "authorNote": "The Zenodo record's own note: \"an optimality mismatch error in 1.3 theorem proof particularly the A200 which uses −11 and gets exp(−17600 n²) … would be fixed in v2\" — about the decay constant's rounding, not about I_out."
 },
 "issue": {
  "url": "https://github.com/gmDevi/zeta-7-21-lean/issues/1",
  "number": 1,
  "title": "Anand's ζ(7) preprint: I_out in Table 5 / (122) doesn't follow from (59)",
  "author": "huntrontrakkr",
  "authorIsThisLab": false,
  "opened": "2026-10-03T19:55:13Z",
  "updated": "2026-10-03T20:00:42Z",
  "state": "open",
  "comments": 0,
  "fetched": "2026-10-05",
  "points": [
   "on (1, 37/20] the outer integrand is 37/20 − y ≥ 0, not −239/20 − 2y; this alone moves A200 + U from −11.99 to +0.95",
   "integrating (54), (58), (59), (62) over (1/3, 37/20] gives I_out = 453803/288000 ≈ +1.576, not −11002997/720000; the printed value is the integral of Table 5; the same steps on Fauzan reproduce his 127751/96000",
   "A0 equals Fauzan's whole ζ(5) constant A*, used as a bound for ∫₂₀^∞ R/x³, which (85) bounds by about −0.057",
   "with I_out from (59), A200 + U ≈ +4.86, or about +3.49 with A0 also replaced; (118) fails"
  ]
 },
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  "what": "L_p(K) = v_p(S_K) + gamma_p^out, the third branch of (60), as exact integers from (54), (58), (59), (62) at K = 40n. The integer sum over every integer p in (K/3, 2h] is a Riemann sum of the limit integrand (error O(1/K)); the prime sum with log p in arb is the outer-range part of log m_{K,M}/K^2 itself at that K, which tends to I_out by the prime number theorem, as the paper's Proposition 5.4 says."
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  "what": "R(x) = -Gamma(x) - N(x) of (64)-(69), proved affine on each of its cells and integrated against x^-3 exactly (inner.py). This decides the printed (121) as a consequence of (64)-(69); whether (64)-(69) is the right limit of the inner exponents (Proposition 4.4, (70)) is not decided here."
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   "statement": "A_M > B0 for every M > 0 (the M-terms of (89) form (1/M)(9λ − 11549/(720M) + 44/M²), whose quadratic has negative discriminant), so A_M + U < 0 for some M needs B0 + U < 0; the B0 + U column is the best any cutoff can do."
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  ],
  "scope": "Per n: exact (Delta_K, P_K and the content are exact rationals) and ball-certified (log P_K(zeta(7))). The paper's Theorem 1.2 is a statement for all sufficiently large n; these three n are a MEASUREMENT of the family, not a decision of that limit.",
  "family": "K = 40n, N = 3n, W = D_N^8/D_K, h = 37n — the paper's own (7), through instruments/zetahankel/hankel.py"
 },
 "lean": {
  "commit": "4e48690ab5f9d04cbf3cede2b657f8db91ef6878",
  "files": 38,
  "lines": 15635,
  "counts": {
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   "admit": 0,
   "axiom declaration": 0,
   "native_decide": 0,
   "implemented_by": 0,
   "@[extern": 0,
   "unsafe": 0,
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   "ofReduceBool": 0,
   "decide +kernel": 20
  },
  "where": {
   "sorry": [
    "Challenge.lean:32"
   ]
  },
  "mainTheorem": "theorem ZetaWindow.zeta_7_to_21_not_all_rational : ¬ ∀ k ∈ ({7, 9, 11, 13, 15, 17, 19, 21} : Finset ℕ), ∃ q : ℚ, riemannZeta (k : ℂ) = (q : ℂ)",
  "mainTheoremHasHypotheses": false,
  "sorryIsTheChallengeStub": true,
  "mentionsAnand": [
   "README.md",
   "formalization.yaml"
  ],
  "readmeOnAnand": "* **Claims about single values.** An unrefereed Zenodo preprint (P. Anand, *Zeta 7 is Irrational*, 23 September 2026, doi:10.5281/zenodo.22920911) claims that ζ(7) is irrational, which would imply the theorem. Our own check could not confirm two constants on which its final inequality rests, so we do not regard ζ(7) as settled. It adapts the method of another unrefereed preprint (A. Fauzan, *ζ(5) is irrational*, 17 September 2026, doi:10.5281/zenodo.22826419), for which Lean formalisations have been announced; ζ(5) does not occur in the theorem here."
 },
 "verdicts": [
  {
   "id": "table5-is-the-integral",
   "claim": "The printed (122), I_out = −11002997/720000, is the integral of the printed Table 5",
   "verdict": "CERTIFIED",
   "by": "exact rational integration of the nine printed rows",
   "values": [
    "-11002997/720000"
   ]
  },
  {
   "id": "issue-point-2",
   "claim": "I_out from the paper's (54), (58), (59), (62) is 453803/288000 (huntrontrakkr, issue #1, point 2)",
   "verdict": "CERTIFIED",
   "by": "pwaff: the integrand proved piecewise affine, integrated exactly; the finite-K integer and prime sums converge to it",
   "values": [
    "453803/288000",
    "445163/288000"
   ],
   "note": "453803/288000 under reading R2 (the faithful one); the literal reading R1 gives 445163/288000. Either is positive."
  },
  {
   "id": "paper-122",
   "claim": "(122): I_out = −11002997/720000 follows from the paper's own formulas",
   "verdict": "REFUTED",
   "kind": "arithmetic-slip",
   "by": "the same integral, both readings",
   "values": [
    "453803/288000",
    "445163/288000"
   ],
   "note": "Table 5 does not follow from (54), (58), (59), (62) on (1/3, 1] or on (1, 37/20]; the printed value is Table 5's integral."
  },
  {
   "id": "table5-last-row",
   "claim": "Table 5, last row: T(y) = −239/20 − 2y on [1, 37/20]",
   "verdict": "REFUTED",
   "kind": "arithmetic-slip",
   "by": "for p > K the paper sets γ_p^out = 0 and d = 0, so T(y) = 37/20 − y ≥ 0 under every reading; measured v_p(cont Δ_K) = 0 at every prime in (K, 2h] for K = 40, 80, 120",
   "values": [
    "289/800",
    "-629/50"
   ]
  },
  {
   "id": "issue-point-1",
   "claim": "Correcting (1, 37/20] alone moves A200 + U from −11.99 to +0.95 (issue #1, point 1)",
   "verdict": "CERTIFIED",
   "by": "exact",
   "values": [
    "-16336874545130483832646567/1362217364503734816000000",
    "1291920923253474354913433/1362217364503734816000000"
   ],
   "note": "True with the printed A0 and U. With A0 and U also corrected in the claimant's favour (below), the last row alone would leave A200 + U = -1.6302 < 0: the failure needs the (1/3, 1] part as well, which the formulas also decide."
  },
  {
   "id": "issue-point-3",
   "claim": "A0 equals Fauzan's I_out + (5.18) + (5.16), his whole ζ(5) constant A* (5.19); the paper's own (85) bounds ∫₂₀^∞ R/x³ by about −0.057 (issue #1, point 3)",
   "verdict": "CERTIFIED",
   "by": "exact rational identities; (85) evaluated at T = 20 with (83) and |C| < 22 as printed",
   "values": [
    "9928298118277006344769/7535670527041937280000",
    "-2069/36000"
   ]
  },
  {
   "id": "issue-final",
   "claim": "With I_out from (59), A200 + U ≈ +4.86, or about +3.49 with A0 also replaced (issue #1)",
   "verdict": "CERTIFIED",
   "by": "exact",
   "values": [
    "6626902387705935050696033/1362217364503734816000000",
    "61800464894085719767009279/17708825738548552608000000"
   ]
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  {
   "id": "paper-121",
   "claim": "(121): ∫₃²⁰ R(x)/x³ dx = 120667538150827615739401/708353029541942104320000 from (64)–(69)",
   "verdict": "CERTIFIED",
   "by": "inner.py: R proved affine on 131 cells, integrated exactly",
   "values": [
    "120667538150827615739401/708353029541942104320000"
   ]
  },
  {
   "id": "paper-96",
   "claim": "(96): λM0 − I(ρ) + C* ≤ U = 44/25, with I(ρ) = −3.143 by (110)",
   "verdict": "CERTIFIED",
   "kind": "arithmetic-slip",
   "by": "arb: the inequality holds (the sum is 0.556); the printed I(ρ) mis-sums the nested energies (−1.948 correctly) — in the claimant's disfavour",
   "values": [
    "278147/500000"
   ]
  },
  {
   "id": "paper-118",
   "claim": "(118): A200 + U < −11 < 0, the inequality Theorem 1.1 rests on",
   "verdict": "REFUTED",
   "kind": "arithmetic-slip",
   "by": "exact: with I_out from the formulas A200 + U > 0 under both readings, with the printed A0 and U and with every claimant-favourable correction decided here; no cutoff M rescues it",
   "values": [
    "6626902387705935050696033/1362217364503734816000000",
    "39952980327483939123204031/17708825738548552608000000",
    "39222854419674408091150777/17708825738548552608000000"
   ]
  },
  {
   "id": "paper-119",
   "claim": "(119): A100000 + U < −12, the input to Corollary 1.3",
   "verdict": "REFUTED",
   "kind": "arithmetic-slip",
   "by": "exact, as (118)",
   "values": [
    "102669566542915492290714348509561/21284646320370856500000000000000"
   ]
  },
  {
   "id": "paper-prop46",
   "claim": "Proposition 4.6: v_p(Δ_K) ≥ γ_p^out under (52), and ≥ 0 for p > K",
   "verdict": "PARTIAL",
   "by": "exact content of Δ_K at K = 40, 80, 120: holds at every prime tested, with equality at most of them",
   "note": "a finite statement checked at three K; not a proof for all K"
  },
  {
   "id": "paper-prop63",
   "claim": "Proposition 6.3 (for every K ∈ 40ℤ): log F_K(ζ(7)) ≤ U K² + 24K log K + 220K",
   "verdict": "PARTIAL",
   "by": "arb at K = 40, 80, 120: holds",
   "note": "checked at three K; the real bound is not where the proof fails"
  },
  {
   "id": "paper-thm12",
   "claim": "Theorem 1.2: 0 < Q_{40n,200}(ζ(7)) < exp(−19000 n²) for all sufficiently large n",
   "verdict": "REFUTED",
   "kind": "arithmetic-slip",
   "by": "the proof's own bound (117) gives a positive exponent once I_out is computed from the formulas",
   "note": "The PROOF is refuted. The statement is a limit in n and is not decided by any finite computation; measured, the best integer normalisation of this determinant has log P_K(ζ(7))/K² = +0.80, +0.87, +0.88 at n = 1, 2, 3, where the statement needs < −11.87.",
   "scope": "the derivation, not the asymptotic statement"
  },
  {
   "id": "paper-thm11",
   "claim": "Theorem 1.1: ζ(7) is irrational",
   "verdict": "REFUSED",
   "by": "its only route, Theorem 1.2 with M = 200, rests on (118), which is refuted",
   "note": "Not proved by this paper; not disproved either. Whether ζ(7) is irrational is open, and nothing here bears on it beyond this route."
  },
  {
   "id": "paper-cor13",
   "claim": "Corollary 1.3: |ζ(7) − a/b| > b^−270 for large b",
   "verdict": "REFUSED",
   "by": "rests on (119), refuted"
  },
  {
   "id": "lean-statement",
   "claim": "What gmDevi/zeta-7-21-lean proves: its README's theorem — ζ(7), ζ(9), …, ζ(21) are not all rational — with no hypotheses, one intended sorry (Challenge.lean, the Comparator statement stub), no axiom declarations, no native_decide",
   "verdict": "PARTIAL",
   "by": "grep-level reading at commit 4e48690 (not built here): the statement and the census match the README",
   "note": "It is not a formalisation of Anand's paper and does not say it is: it cites the paper as background and states that its own check could not confirm two constants of the final inequality. \"ζ(7) is irrational\" appears nowhere as a formal statement; the formal theorem does not imply it."
  }
 ],
 "notDecided": [
  "Whether ζ(7) is irrational. It is open; this audit refutes one proof route and says nothing else about it.",
  "Theorem 1.2 as a statement about large n: no finite computation decides it. Measured at n = 1, 2, 3 only.",
  "Proposition 4.4 (the inner range) and the limit (70): they need K ≥ 200M² and are not tested here; (121) is decided only as a consequence of (64)–(69).",
  "Lemma 6.1's potential bound (94) as a sup: measured on a grid, not certified; the paper's A.3 partition is not re-run.",
  "The Lean repository's own theorem (one of ζ(7), …, ζ(21) is irrational): read at the pinned commit, not built."
 ],
 "verdict": "REFUTED",
 "verdictLine": "The proof route is refuted, not the theorem: I_out computed from the paper's own formulas is +1.576 (or +1.546), not −15.28, and A200 + U, which must be negative, is positive under every reading and every correction decided here. ζ(7)'s irrationality is untouched.",
 "kind": "arithmetic-slip",
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