A Loewner divided-difference formula for the prime contribution in the localized Weil quadratic form, and a parity sign law

In the localized/truncated Weil quadratic form of Connes–Consani–Moscovici (arXiv:2511.22755) and Connes–van Suijlekom (arXiv:2511.23257), the prime contribution enters the Galerkin matrix as a divided difference (Loewner matrix) of an explicit oscillatory symbol. We record two consequences. First, for any test vector v the prime quadratic form admits an exact closed Loewner representation in terms of the Fourier symbol W_v and the arithmetic frequencies ω_q = 1 − log q / log c; the identity is exact and is validated numerically at c = 5, 13, 53 to working precision. Second, parity of the symbol fixes the sign structure: for the even vector C = cosh(x/2) the symbol W_C is real and positive, and we prove (with an explicit Galerkin-truncation bound) that the prime contribution in C is non-positive term by term, strictly negative for q < c — the primes cooperate. For the odd vector S = sinh(x/2) the symbol W_S is purely imaginary and sign-changing, so the per-prime contributions have mixed sign and the positivity of ⟨S, P S⟩ holds only as a sum in which the large primes outweigh the small ones — the primes compete. We make no claim about even-simplicity or the Riemann Hypothesis; rather, the odd sector is reduced to a single explicit term-by-term competition inequality, tied to the odd-sector positivity problem (RH-adjacent in the Yoshida sense).