Chu-Vandermonde convolution, closed form at symbolic parameters

The base case is DECIDED, not sampled. At p = 0 two Gamma factors of the summand become parameter-free -- Gamma(k+1)^-1 forces k >= 0 and Gamma(-k+1)^-1 forces k <= 0 -- so the support of F(0, .) is the single point k = 0, and every other point of the scanned window [-2, 12] is CHECKED to vanish rather than assumed. At k = 0 the surviving Gamma powers cancel pairwise (Gamma(m+1) against Gamma(m+1)^-1, Gamma(n+1) against Gamma(n+1)^-1), leaving the rational 1 for every value of m and n. The claimed closed form C(m+n,p) evaluates the same way to 1. The leading recurrence coefficient -(p+1) has no integer zero at or above 0, decided by the rational-root theorem, so the recurrence runs forward. Tamper controls in the same file: 2*C(m+n,p) (right ratio, wrong symbolic base) is rejected by the base case; C(m+n,p+1) (right base value, wrong ratio) by the annihilation check; C(m+n,p)*Gamma(m+1) is refused as not comparable to a rational rather than silently compared; and a window that does not STRICTLY contain the forced support is rejected.

This row exists so the fact rests on an ARTIFACT rather than on re-running the producer. The sweep reads every committed certificate, parses it, and hands it to the independent checker; nothing in that path calls the search. Tamper controls in the same file: a certificate numerator edited by one, a recurrence coefficient edited by one, and a certificate re-pointed at a neighbouring file's summand are each rejected, and a truncated / foreign-format / decimal-bearing file is refused by the reader before the checker sees it.