Vandermonde and Far-Prime Obstructions in a Proposed Determinant Proof for Catalan's Constant

Audit of Z.-W. Sun, “Catalan's constant is irrational”, arXiv:2609.04176v1 [math.GM], September 3, 2026. We audit the determinant argument in that preprint. Two indexing defects in its full-rank argument can be repaired, and we supply a corrected proof. Every Cauchy–Binet summand contains the fixed factor 2S(S−1)V(J), but the stated odd-prime valuation bound discards V(J). Consequently the largest-summand ledger used in Proposition 9.5 retains (1/800)B2log B at S/B = 1/20 rather than having complete B2log B cancellation. Independently, the large-prime local density is stopped at p < (2+ρ)B; its continuation through p < (4+2ρ)B has integral −4ρ, a further quadratic cost. We prove a strengthened local surplus inequality that restores the fixed Vandermonde divisor, evaluate the omitted far-prime range, and construct the corresponding normalized nonzero integer. Granting the remaining local asymptotics asserted in the preprint, even this repaired largest-summand method remains above the required threshold by 0.70317104474052447365… B2. An exact beta-core/tail decomposition gives a closed determinant whose constant is lower by 0.23321100154205717935… B2, but the core still lies 0.46996004319846729430… B2 above the arithmetic threshold. Thus version 1 of the cited preprint does not establish the irrationality of Catalan's constant. Continuing this architecture requires a genuinely signed determinant estimate or different weights. Claim boundary. This is a correction/audit and obstruction paper. It does not prove or disprove the irrationality of Catalan's constant. The numerical barrier is conditional on the small- and middle-prime local asymptotics asserted in Sections 6–8 of the audited preprint; the rank repair, the Vandermonde surplus, the far-prime continuation, the explicit Cauchy–Binet block and the beta-core decomposition are proved unconditionally. The exact surviving requirement is stated as Problem 8.1 (a signed determinant estimate). A later version of the cited preprint may differ. Reproducibility. verify_algebra.py (exact integer/Fraction regression, 26,977 checks, terminal CATALAN_VANDERMONDE_ALGEBRA_AUDIT=PASS) and verify_constants.py (rational outward interval certificate for every displayed constant, terminal CERTIFICATE=PASS) use only the Python standard library; recorded outputs are included. They treat the preprint's small-prime interval and middle-prime fraction as stated inputs and do not replay its cell computations. 14 pages. Computational tools and a large language model were used for algebraic auditing, exploratory calculations, code generation and manuscript preparation; no model output is used as mathematical authority.