{
  "report": "PUBLIC COPY OF THE INDEPENDENT AUDIT\n\nLocal filesystem references have been replaced by document descriptions\nor portable commands. The outcome, mathematical findings, input hashes\nand reported check counts are preserved. Path changes to the public\nimplementations are recorded in assembly-review.json.\n\nMathematical audit of the powered-kernel result\n11 November 2026\n\nI found no substantive mathematical error or gap in the current proof.\nBoth assertions of Theorem 1 pass this audit:\n\n  - At least one of beta(6), beta(8), ..., beta(74) is irrational.\n  - At least one of beta(8), beta(10), ..., beta(110) is irrational.\n\nThe proof's general power criterion is also valid with its stated\nnonzero-seed hypothesis. The manuscript verifies that hypothesis for\nq = 1, 2, 3; it does not establish it for every q. The two asserted\nirrational members need not be distinct.\n\nThe audited manuscript is the 13-page draft dated 10 November 2026:\n\n  Original powered_beta_proof.tex (audited snapshot)\n  SHA-256:\n  0db642938a7a6d3fae930b50d104dc2e4a7554a4046acf87da4bbdf60279ebeb\n\n  Original powered_beta_proof.pdf (audited snapshot)\n  SHA-256:\n  21aca68e2f5987dada1ab7af644020feb371fd13a307decb58746bb3494f8ff9\n\nThis audit combines a check of the general arguments with independently\nwritten exact calculations. In the preceding preliminary review I had\ninspected and replayed the supplied checkers. The calculations for this\naudit were written afresh and did not run or import those checkers.\nThis is therefore a fresh mathematical audit and computational\nreconstruction, but not a blind review conducted without prior exposure\nto the authors' implementations. Standard arithmetic libraries were\nused as described below.\n\n1. Construction and target beta values\n\nThe block endpoints are integral when n is a multiple of 480. The\nnumerator and denominator counts give\n\n  3 #Lambda_n = 10n + 6,\n  sum_j B_j(n) = (279/10)n + 35,\n  deg R_n = -(179/10)n - 28.\n\nThe reflection identity is correct, including the sign contributed by\nthe central linear factor. Keeping the nominal central pole order and\nallowing its top coefficient to vanish is legitimate.\n\nRaising R_n to the qth power gives zeros of order 3q. Differentiating\n2q times therefore preserves the integer zeros needed both to remove\nthe initial summands and to control the rational constant. The\npartial-fraction expansion and alternating summation are valid; all\nthe differentiated series converge absolutely. Reflection cancels the\nodd beta indices. The remaining indices run from 2q+2 to the largest\neven integer not exceeding 37q. Thus q = 2 and q = 3 give precisely\nthe two claimed intervals, containing 35 and 52 values respectively.\n\n2. Denominator transfer\n\nThe essential convolution argument is valid for the full local series:\nif its degree-h coefficient has valuation at least nu-lambda*h, its\nqth power has valuation at least q*nu-lambda*h. The slope does not\nacquire an extra factor q. Combining this with the harmonic denominator\nof exponent i+2q produces exactly q times the original valuation cost.\n\nI checked all four prime ranges. For small odd primes, the required\nwhole-series estimate follows from the local-factor argument in\nKrisman's Lemma 4.3, underlying Corollary 4.4. At 2, the harmonic\ndenominators are odd and the displayed power of 2 introduces no loss.\nFor odd primes greater than n, the grouped cancellation at the negative\ninteger zeros supplies the integrality of the rational constant: a\nprime cannot divide denominators in both complementary groups because\ntheir pole addresses differ by less than that prime.\n\nSome E_p(n) can be negative. This causes no problem: the multiplier\nD_n is a positive rational number, and the proof establishes the\nintegrality of the resulting coefficients, which is the property the\ncontradiction needs. It does not require D_n itself to be an integer.\n\n3. Inherited arithmetic cost and real decay\n\nI read and checked the relevant arguments in Sections 2--6 of the\npinned Krisman paper, including the passage from the prime-counting\ncertificate to the bound on log D_n. The source files used were:\n\n  Krisman, Hexagon 2610.00285v2 (PDF)\n  Text extracted from that PDF\n\nThe finite arithmetic partition accounts for the discontinuities of\nthe defining counts. The treatment of its endpoints and small primes\nhas sublinear logarithmic cost; the tail estimate and the\nprime-number-theorem passage give the stated constant A.\n\nThe real product estimate is uniform over t >= rn+1, including the\nunbounded tail. In particular, the residual logarithmic terms in\nKrisman's Proposition 6.4 admit the stated uniform bound. The phase\nand tail bounds used by the manuscript are valid. The logarithmic derivative\nestimate introduces only a polynomial factor for each fixed q, and\nthe separate tail sum converges with the stated bound. All parameters\nother than n are fixed when n tends to infinity.\n\nFresh exact arithmetic gives\n\n  A = 12507029707866288048422972493354331\n      / 790300921079786767495600904601600,\n\n  Q = -744/25 + A + 20 log 2\n    = -0.07140140819199213756... < -0.0714.\n\nConsequently |D_n^q L_{q,n}| <= exp(qnQ + o(n)) for every fixed q.\nTaking a power preserves the strict margin between denominator cost\nand real decay.\n\n4. Prime dilation and nonvanishing\n\nThe change of variables and its normalization are correct:\n\n  R_n(t) = 2^(Gn+29) F_n(2t+n+1),\n  L_{q,n} = 2^[q(Gn+29)+2q] ell_{q,n}.\n\nAt n = 480p, the selected numerator zeros reduce to the trimmed seed\nwith endpoints +/-1599, not the untrimmed endpoints +/-1601. The\nselected-factor counts give A_0 = 8626. The unselected numerator and\ndenominator units give the sign -1 in the rational-function congruence.\n\nThe coefficient extraction has the necessary local-series\njustification. After removing the selected pole and applying the\nnormalization, the series lies in Z_(p)[[z]]; its remaining denominator\nfactors have unit constant terms. Reduction therefore commutes with\ncoefficient extraction, giving the stated scaling p^(qA_0-i).\n\nThe bound for nonselected coefficients is also correct. Combined with\nthe harmonic denominators and tau_q = 8628q, it leaves positive\nvaluation q(S-1) in the nonselected constant contributions. At selected\npoles, the surviving odd harmonic denominators supply precisely\nchi_4(p). Thus the constant congruence has the stated factor\n(-1)^q chi_4(p), while every scaled beta coefficient vanishes modulo p.\n\nUnder joint rationality, exclude the finitely many primes dividing the\nfixed common denominator or the numerator or denominator of the\nnonzero seed constant. The congruence then makes the forms nonzero for\nevery sufficiently large remaining prime. Together with integrality\nand decay, this yields infinitely many nonzero integers tending to\nzero. The argument does not require an irrationality claim for\nCatalan's constant or Krisman's own nonvanishing argument.\n\n5. Fresh finite calculations\n\nFor the actual n = 480 seed, I constructed its full numerator and\ndenominator polynomials over each finite field. Their degrees are\n4801 and 13427. I Taylor-shifted these global polynomials at every\npole and recovered the local coefficients by series inversion. This\nuses neither the reflection shortcut nor the supplied logarithmic-\nderivative/factorial algorithm. Reflection was checked afterwards.\n\nThe independently reconstructed seed constants are:\n\n  q                 modulo 10007       modulo 10009\n  1 (control)              6332               9058\n  2                        3182               5359\n  3                        3335               6971\n\nBoth moduli were checked prime by trial division. All five residues\nreported in the manuscript agree; q = 1 modulo 10009 is an additional\ncontrol calculation. All 191 even-beta coefficients in the five\nsupplied records also agree. There were 161,124 coefficient reflection\nchecks and 18 full rational-function reconstruction checks over these\nfields. These residues prove the required rational seed constants\nnonzero, because every inverted denominator is a unit at the chosen\nprime.\n\nFor the arithmetic budget, a separate evaluator directly counted the\ndefining closed intervals and odd integers at every discontinuity in\nthe auxiliary variable and in every intervening open cell. This\ndiffers from the supplied accumulated-jump algorithm. It independently\nreconstructed all 62,132 event points and 62,131 interval maxima,\nincluding the exact upward-rounded sum\n\n  1243345293752876677540040193981\n\nat scale 2^96. The resulting constant A agrees exactly. Rational\nlogarithm enclosures, using 70 terms of the elementary atanh series,\nverified the curvature, tangent, monotonicity and tail inequalities\nand the strict bound on Q.\n\nFor rational dilation tests, I used full integer polynomials, Hasse\nderivatives, explicit rational long division and Cauchy products.\nThese tests covered q = 1, 2, 3 in four dilation geometries, including\nuniform and nonuniform blocks and both odd residue classes modulo 4.\nThe additional seed n = 6, rn = 7 and prime 29 gives n = 174 and uses\nthe manuscript's ratio r = 7/6. Every test passed:\n\n  Selected coefficient congruences                         1,644\n  Other-pole coefficient valuation bounds                 24,552\n  Constant congruences                                        12\n  Scaled beta coefficient slots checked                      468\n  Cleared coefficient/constant slots checked                 360\n  Grouped zero identities                                     36\n  Exact rational-function reconstructions                     90\n  Exact rational coefficient reflection checks             28,872\n\nThe slot counts include zero or unused beta entries. The smaller\ngeometries test the formulas; they do not substitute for the general\nprime-dilation and denominator arguments checked above.\n\n6. Scope, preservation and reproduction\n\nNo mathematical correction is required by this audit. It is a\nmathematical and computational review, not formal verification or a\ncertification of priority. I accepted standard classical inputs,\nincluding the prime number theorem, Wilson's and Fermat's theorems,\nand elementary formal-series and asymptotic facts. I did not rebuild\nthe cited Lean project. The separate determinant pilot and other\nother research projects are outside this audit.\n\nI did not edit the research workspace. All 71 pinned input files retained their original\nSHA-256 hashes. The public audit implementations and numerical records are in:\n\n  checks/audit/\n\nThe machine-readable summary is review/independent-audit-2026-11-11.json. The detailed\noutputs are fresh-seeds.json, fresh-budget.json and fresh-rational.json.\nThe complete local input inventory and preservation comparison are retained\nprivately; the audited manuscript hashes are recorded above.\n\nPortable reproduction commands (run from the paper directory with\npython-flint installed; no manuscript checker is imported):\n\n  python3 -B checks/audit/fresh_seeds.py\n\n  g++ -O2 -std=c++17 checks/audit/direct_estars.cpp -o checks/audit/direct_estars\n\n  python3 -B checks/audit/fresh_budget.py\n\n  python3 -B checks/audit/fresh_rational.py\n",
  "scope": "Public audit report with normalized file references; original audit scope and input hashes retained."
}