Strict total positivity of a real Delannoy multiplication table: the sharp half-line threshold

We prove the sharp classification for the real-continuation Delannoy multiplication table M_x(i,j)=T(x,ij): it is strictly totally positive of order two exactly for x at least 1/2. The deposit contains the paper, a compact reproducibility archive, and deposit-level SHA-256 checksums. The archive derives the uniform two-cut expansion, rebuilds the 260-cell finite residue from the recurrence, checks the upper-strip Bernstein certificates, and reconstructs the universal dimension step using exact arithmetic.