{
  "source_repository_path": "shifted-even-beta/review/mathematical-review.txt",
  "text": "MATHEMATICAL REVIEW OF THE V2 LATEX ASSEMBLY\n6 November 2026\n\nScope\n  Constructing-assistant review of the typeset exposition against the\n  independently audited plain-text proof identified in\n  independent-audit-2026-11-06.json. This is not a new independent\n  audit. The report's scope and the post-audit example are separate.\n\nTheorem statements and parameters\n  Theorem 1.1 gives one irrational value from beta(m+1) through\n  beta(19m-1). It is the B=1 case of the general theorem, with\n  D_1=1 and a_1=18.\n  Theorem 1.2 uses odd d<=2B-1, D_B=sum phi(d), and\n  C_B=1+2 ceil[D_B(9+(40/9)log(2B-1))].\n  The endpoint is C_B m-1 for every positive odd m and B>=1.\n  Section 8 sets a_B=C_B-1 and a=m a_B, preserving the pole order\n  and all bounds from the previous notation.\n  Proposition B.1 retains s=m-1+2 ceil((879m-50)/100)<18.58m.\n  The criterion uses even a>mD(2r+1), with endpoint a+m-1.\n  Every displayed individual parameter row matches endpoints.json.\n  The uniform comparison columns were recomputed from the stated\n  formulas. The source comparison statements retain their recorded\n  reading scopes; no new literature audit was undertaken.\n\nConstruction and Gram argument\n  Checked reduced odd fraction classes, D odd, simple numerator zeros,\n  reflection, properness and absolute convergence. The weighted pole\n  sums annihilate odd beta coefficients for odd n.\n  The m=1 conditionally convergent component sums are explicitly treated.\n  Checked the residue sign after even differentiation, exact kernel\n  pole orders, integral h_b coefficients, degree J=m(D-1)/2, homogeneous\n  multiplier degree E=m(D-1), and clockwise-contour sign.\n  The central cancellation uses 2c=m+1. The weight is positive on all\n  of the real line, the integrals converge, and kernel independence\n  gives rank for every B.\n  The worked B=2,m=1,r=2,a=70 example has a separate exact replay:\n  its moment matrix equals minus the seed matrix, the determinant\n  is 5184(mu_0 mu_2-mu_1^2), and the cleared integer multiplier has\n  precisely the prescribed nonzero last seed coordinate.\n\nDenominators and decay\n  Checked the progression lemma at primes dividing 2d and other primes.\n  The normalization cancels the other-pole constants exactly.\n  The common clearing exponent is a+k, where k=m-1.\n  Grouped cancellation uses surviving grid zeros and the fact that\n  differences 2(l-l') divide delta_n; it includes primes dividing b.\n  Checked the gamma-product profile, cancellation of n log n and\n  linear terms, uniformity as t/n tends to zero, fixed Cauchy radius,\n  polynomial cost of differentiation and a separate infinite-tail bound.\n  The unique maximum and f(u)<1 give x_0<u. The decreasing ratio in G\n  then gives (e^(a+k)G)^2<U<1.\n\nPrime dilation and contradiction\n  Checked the odd fraction-class permutation, factorial unit,\n  cancellation of the two Wilson signs and degree E in the scaling.\n  The local power-series argument precedes coefficient extraction.\n  It shows explicitly that the order-i coefficient scales by\n  p^(a-E-i), and that a-E-i=tau-nu after differentiation.\n  The weaker nonselected-pole estimate v_p(A)>=nu-k is sufficient\n  because tau=a-E+k>k. The surviving constant has factor chi_4(p).\n  The last generalized Vandermonde minor is nonzero. Fixed integer\n  weights therefore cancel all possible irrational coefficients while\n  preserving a nonzero seed constant.\n  The prime exclusions name the numerator and denominator of q_w(1).\n  The cleared sequence is integral, tends to zero, and is nonzero at\n  infinitely many prime indices, yielding the contradiction.\n\nEndpoints and appendices\n  Checked the uniform K bound, rational comparisons, ceiling formula\n  and the treatment of m=1 and m=3 in the sharper one-value bound.\n  Replayed all 21 endpoint certificates, the six uniform parameter\n  choices, all uniform rational comparisons and seven integer-r\n  comparisons. The numerical limiting profile is labeled numerical.\n  The beta-integral moment formula in Appendix A has denominator\n  (1+y^2)^a and the correct normalization. Its required polynomial\n  degree is less than a.\n  All 8,909 structural self-checks and the additional worked example\n  pass. These finite checks support identities; the general proof\n  supplies the all-parameter statements.\n\nSubsequent exposition revision\n  The added single-grid example follows by substituting\n  j=(p-1)/2+ell p; the product of its selected factors is p^L P_1(t).\n  Exact polynomial and residue checks passed in 20 cases.\n  Substitution of D=1, K=2^L, k=m-1 and x=2y gives the exact\n  identity a+k+Phi(2y)=m Psi_r(a/m,y)-1. Exact rational logarithm\n  coefficient comparisons passed in 48 cases. Taking the maximum\n  gives the stated relation to log(e^(a+k)G).\n  The integer-r certificates give Psi_r(18,0)>1, so the finite\n  correction cannot yield decay for these r at a<=18m.\n  These checks concern new explanatory identities, not a new audit.\n  The existing checker sources and results were verified unchanged;\n  the full replays recorded above were not repeated for this revision.\n\nHeadline and abstract revision\n  Restored the displayed list in the abstract, with endpoint 19m-1.\n  Moved the sharper one-value statement and proof to Appendix B;\n  its formulas and rational comparisons are unchanged.\n  Recomputed all seven opening-table uniform endpoints from a_B.\n  All twelve specific parameter rows and the abstract's three\n  examples are unchanged. The general theorem and the seven labeled\n  equations affected by relocation were checked against the prior\n  sources. No new proof or full certificate replay was needed.\n\nDirect comparison of the two general theorems\n  Restated v1 with C_B^(1)=s_B+1, preserving its original totient sum,\n  ceiling formula, quantifiers and count. The current theorem uses\n  C_B=a_B+1; its constant and endpoint are algebraically unchanged.\n  Checked the multipliers for B=1,2,3: 67,197,463 in v1 and\n  19,85,229 in v2. The source formulas match the corresponding\n  versions; numerical values were recomputed with 80-digit logarithms.\n  The abstract, all endpoint tables, proof sections outside the\n  endpoint-notation changes, and checker sources/results are unchanged.\n  No full certificate replay or new independent audit was needed.\n\nResult\n  No substantive discrepancy between the v2 manuscript and the audited\n  argument was found. The additional worked example and notation\n  changes pass the stated exact self-checks. Audit recommendations are\n  implemented in the proof and retained as recommendations only in the\n  review records, not in the manuscript's audit note.\n"
}