A Markov representation of the Weil quadratic form

We construct an explicit closed Markov form from the positive theta kernel whose Fouriertransform is the completed Riemann zeta function. On a natural core of real even functions,multiplication by this kernel transforms Weil’s quadratic form into the Markov energy minustwice the variance for an explicit probability measure. We identify the closure with the maximalfinite-energy domain and obtain an equivalent formulation of the Riemann hypothesis as aspectral gap of size two in the even mean-zero space. The threshold is an eigenvalue of infinitemultiplicity. The essential spectrum is exactly [2,∞), and the number of eigenvalues belowtwo in that sector equals the number of distinct conjugation-and-reflection orbits of nonrealzero ordinates. Compactly supported tests give exact finite-prime forms and a rank-one inertiacriterion, including the singular case. We also prove that decreasing any positive prime-powerweight, with all other weights no larger, creates infinitely many negative directions for thethreshold form. These results concern a specific realization of the classical Weil criterion; theydo not establish the threshold inequality.