The Dense Divisor Model over F_q[t]: Exact identities, the absence of faces, and the location of the selector problem

Abstract

This working note transfers the dense divisor model of the author's Parts I and II to the polynomial ring 𝔽_q[t], in order to separate the arithmetic features of the model from artefacts of the real place. Over 𝔽_q[t] the single-variable Selberg–Delange identity is exact, the lattice count has no Popoviciu defect in the volume range, and the model averaged over N is an exact product of single-variable identities, so the selector problem is purely pointwise. The main result is that the volume range, cut at its natural level, carries the Gamma factor 1/Γ(2z−1), with the singular series as coefficient at the selector, whereas the full amplitude carries 1/Γ(z)²; the same holds over ℤ, where the volume range carries 1/Γ(1+2w). Consequently the non-volume ranges must supply a linear term of the size of the binary Goldbach count, which quantifies Face Residue Cancellation. Exact computations support each statement. Nothing here bears directly on binary Goldbach over ℤ.