Sharp Interpolation for Equidegree Hilbert–Burch Curves

Let H(t,a) be the Hilbert component of curves in projective three-space defined by the maximal minors of a (t+1)-by-t matrix of degree-a forms, over an algebraically closed field of characteristic zero. We prove that its interpolation number is t(t+1)(binom(a+3,3)-2)/2, exactly half its dimension. Smooth integral curves attain the bound, while the upper bound includes the whole component and its boundary. A bidiagonal specialization yields two triangular normal-evaluation systems indexed by intervals. Normal-section evaluation has maximal rank at every number of general markings on a general smooth member. The complete-intersection and linear cases, and a small quadratic example, are credited to prior work. This is a complete result for a specified infinite equidegree family, not a solution of arbitrary ACM degree matrices or the full AIM Hilbert-component question. Seven-page unrefereed AI-assisted preprint with exact supplementary checks; no independent review or absolute-priority claim.