{
  "canonical_path": "review/independent-audit-2026-11-11.txt",
  "text": "Public copy of the independent audit report. The manuscript path below\nis package-relative; the private absolute path has been removed.\nThe auditor's programs and numerical results are preserved without\nchanges in checks/independent-audit/.\n\nMATHEMATICAL AUDIT OF THE FIFTEEN-VALUE EVEN-BETA PAPER\n11 November 2026\n\nAudited PDF\noutput/pdf/fifteen-beta-values.pdf\nSHA-256 2635118a4d725bfe9b1667c4df6873fa1f13f692e8ea42be73e76939b3572724\nFourteen pages. Source and PDF hashes agree with paper/build-record.json.\nThe individual source and certificate hashes are in audited-files.json.\n\nVerdict\n\nI found no mathematical error or gap affecting Theorem 1.1, which asserts\nthat at least one of beta(4), beta(6), ..., beta(32) is irrational.\nI checked the complete argument in Sections 1--5 and Appendix A and\nindependently reproduced the finite arithmetic, phase and nonzero-seed\ninputs. The proof provides both decay after clearing coefficients and\nnonvanishing along infinitely many prime-dilated indices under the joint\nrationality hypothesis.\n\nThe conservative cleared-form upper exponent is\n\n  41/5 + 15 log(2) - 18.83\n    = -0.2327922916008203587415181781273514788...\n\nThus the quantitative comparison has a substantial strict margin.\n\n1. General mathematical arguments checked\n\nConstruction and support\n\nThe nested blocks have gamma*n+31 denominator factors, with\ngamma=341/16. The numerator has 15*n/2+6 zero factors including\nmultiplicity, and one additional linear factor. Consequently\nG=221/16 and deg R_n=-G*n-24. Reflection gives R_n(-n-1-t)=R_n(t)\nfor every positive multiple of 96. It induces a_(i,n-a)=(-1)^i a_(i,a).\nThe alternating sum therefore retains only even i. Twice differentiating\nleaves exactly the candidate set beta(4), ..., beta(32). I checked the\nfactors of two, i(i+1), finite harmonic corrections and absolute\nconvergence of the beta expansion.\n\nThe isolated-pole estimate\n\nI checked the local-series estimate v_p(a_(i,a)) >= i-F_p(a), including\nthe zero constant term of the central prefactor. For p^2>7n every\nnonzero divisible integer factor involved has valuation exactly one.\nAn isolated address has p-unit differences from every other pole, so\nits entire principal part is p-integral for every odd p.\n\nRegrouping the rational constant at the integer zeros gives the two\ncomplementary sums in (3.2). For each evaluation only one address class\nhas p-divisible denominators. A singleton can be avoided by taking the\nopposite side, and every remaining coefficient has the required bound.\nFor a repeated class, the valuation loss i+2 in evaluation cancels\nthe i in the local coefficient bound. This proves the stated exponent\nmax(0,M_p+2). The same argument gives integrality for p>n, including\nevaluation denominators of higher valuation.\n\nSmall primes and dyadic cost\n\nI checked the definitions of nu_p, the common Taylor-degree losses and\nthe exponents in Appendix A. The negative and positive odd numerator\nproducts each start at 1, permitting their expression as (2A)!/(2^A A!).\nThe Legendre bounds give the displayed 128(lambda_p+1) small odd-prime\ncost and 15n+36+107(lambda_2+1) dyadic cost after subtracting G v_p(n!).\nThese contribute o(n) and at most 15n log(2)+o(n), respectively.\n\nArithmetic envelope and passage from primes\n\nI checked the closed-interval count N_j, the half-shift in C, and the\nfact that C omits the two endpoint zeros and the prefactor. Those\nomissions only increase the envelope. Repeated classes have y<=x-1\nwhen 1<x<2. Every class is repeated for x>=2.\n\nOn each open interval between affine-offset collisions and integer\nevents, the residue-cell ordering and all floor counts are constant.\nAt a non-event x only one offset jumps at an interior cell boundary.\nIts exact value agrees with a neighboring cell. At 0 and x-1 the\nouter pole block has the indicated one-sided value. Thus the cell\nmaxima include the closed-boundary maxima. Event values of n/p\ncontribute O(log n) for the fixed finite partition.\n\nThe bound e(x)<=36+G*{x} follows from the interval lengths. The\npositive-part tail integral is at most 36/X+G/(2(X-1)). PNT on each\nfinite partition, followed by the Chebyshev bound beyond a fixed M\nand then M tending to infinity, proves the claimed asymptotic prime\nsum. Negative body contributions are retained with their correct sign.\n\nReal estimate\n\nI checked the gamma product and the uniform Stirling estimate down to\nt-rn=1. Its O(log(n+t)) error becomes only a fixed polynomial factor.\nThe logarithmic derivative and its derivative give the uniform\nO((1+log n)^2) cost for R_n''. The critical polynomial has exactly\none positive root by its coefficient signs. The tail inequality\nx Phi'(x)<=-383/64 for x>=3 suffices to sum the infinite tail while\nabsorbing the polynomial factor. These estimates give the printed\nuniform bound for L_n, rather than a finite-index extrapolation.\n\nPrime dilation and final contradiction\n\nI checked R_n(t)=2^(Gn+25) F_n(2t+n+1) and the two further powers\nof two from differentiation. Selecting p-divisible zero factors gives\nthe trimmed seed with odd roots through 239, rather than through 241.\nThere are 3L selected zero factors, W+31 selected denominator factors\nand one selected prefactor. Thus A_0=1356 and tau=1358.\n\nWilson's theorem gives the stated combined unit -1. Coefficient\nextraction after pole removal is legitimate in the local integral\npower series. A nonzero address class contains W weighted poles and\nat least L numerator roots before cubing. Its resulting bound is\nv_p(c_(Np;a,i))>=-GN+i. After scaling, the beta coefficients vanish\nmodulo p. Nonselected rational-constant terms have valuation at least\n30. The selected harmonic denominators give the character chi_4(p),\nyielding p^tau q_(Np)=-chi_4(p)*q_tilde modulo p.\n\nThe seed residues below prove q_tilde is a nonzero rational number.\nUnder joint rationality of the fifteen beta values, excluding the\nfinite set of denominator and seed-numerator primes supplies infinitely\nmany nonzero L_(Np). Combining this with integrality and decay of\nT D_(Np) L_(Np) gives the stated contradiction.\n\n2. Independently reproduced finite inputs\n\nFresh programs in this scratch directory import none of the supplied\ncheckers. Supplied results are compared only after the calculations.\n\nArithmetic certificate\n\nindependent_arithmetic.py enumerates affine crossings by reducing their\nnumerators to arithmetic progressions. independent_floor.cpp evaluates\nthe closed boundary points and all open residue cells directly using\ninteger floor and ceiling formulas. This gives 38,372 event points and\n38,371 intervals, agreeing with every supplied midpoint and maximum.\nThere are 23 cells with negative envelope value. Upward rounding at\n96 bits gives the total bound\n\n  3203914250644786841689526785574147 /\n  395150460539893383747800452300800\n    = 8.108086844356158... < 41/5.\n\nThe independently calculated bound agrees with the manuscript's\n64-bit upward-rounded bound to the displayed precision.\n\nPhase certificate\n\nindependent_phase.py constructs the derivative numerator directly from\nthe rational parameters. It checks degree 33, positive coefficients\nin degrees 0,1,2 and negative coefficients in degrees 3 through 33.\nIt verifies the printed root bracket and bisects it further. 512-bit\nArb arithmetic gives\n\n  max Phi = -18.8351123668081936003718677400690 +/- 7.31e-32,\n\nwhich is strictly below the printed rational upper bound\n-18835112315041/10^12 and the simpler bound -18.83.\nThe full independent arithmetic and phase enclosures together yield\na cleared exponent at most -0.3298178140528555490509126132735.\nThe manuscript uses the weaker conservative exponent above.\n\nSeed certificate\n\nindependent_seed.py constructs the complete seed numerator and\ndenominator over each finite field. For each pole it uses a polynomial\nBezout inverse modulo the power of its pole factor, followed by\nbinomial translation of the polynomial residue. This provides a third\nmethod alongside the supplied product and logarithmic-derivative\ncalculations. It reproduces all 2,077 coefficients at both primes,\nall odd-index cancellations and the constant residues\n\n  q_tilde = 3738 mod 10007,\n  q_tilde = 5333 mod 10009.\n\nBoth moduli are checked prime by trial division.\n\nLocal arithmetic spot checks\n\nindependent_local_bounds.py also verifies the exact small-prime cost\nbounds in 56 cases and the large-prime residue/envelope correspondence\nin 222 cases at n=96,192,384,768. It checks 32,722 repeated residue\nclasses, including the half-shift and omitted endpoints in C.\nThese finite checks supplement the general proofs above.\n\nPackaged checks\n\nThe complete supplied runner, including --with-product, passed in a\nscratch copy. Its report is replay/checks/check-report.json. This replay\nis additional to the independently written calculations.\n\n3. Mathematical errors and exposition\n\nNo mathematical errors or gaps were found affecting the conclusion.\n\nTwo optional exposition improvements would make the prime-dilation and\nsmall-prime counts quicker for a human reader to verify.\n\nAt Section 5.2, explicitly state that a block [p a,p b] contains b-a\nrepresentatives of every nonzero residue and b-a+1 representatives of\nzero. Summing gives W and W+31. This exposes the source of the\nselected/nonselected valuation separation.\n\nIn Appendix A, give the two odd-product lengths explicitly as\nA_minus=(1+r)n-a+1 and A_plus=a+rn+1. This makes the double-factorial\nestimate and its six logarithmic error terms immediately visible.\n\n4. Accepted inputs and scope\n\nAccepted classical inputs are partial fractions, elementary local\npower-series arithmetic, Legendre's formula, Wilson's and Fermat's\ntheorems, Descartes' rule of signs, uniform real Stirling estimates,\nthe prime number theorem, the Chebyshev bound and ordinary convergence\nestimates. Computational results use exact integer/finite-field\narithmetic and outward-rounded Arb interval arithmetic.\n\nThe quantitative lemmas needed for the theorem were checked as proved\nin this manuscript. Krisman's cached v2 PDF was checked for the claimed\nseventeen-value comparison and construction context. Full literature\ndependency audits, priority claims, discovery-session independence and\nexternal Lean formalizations are outside this audit's scope. The\nverification method is mathematical reasoning with independently\nimplemented finite computations.\n",
  "path_note": "Paths and hashes in the report refer to the canonical repository layout."
}