Near-Minimal Ehrhart Data on the Primitive-Triangle Boundary of Denominator-Two Polygons

Let P be a convex polygon of denominator two with no boundary lattice points and with b(2P) = 3. Writing N = i(P) and I = i(2P), we prove the necessary congruence 2I+1 congruent to 4N plus or minus 1 modulo 8 and determine the sharp near-minimal boundary. We classify all values through I ≤ 3N-3, prove that I = 3N-2 occurs exactly at (N,I) = (10,28), and construct an infinite family with N = 4k+5 and I = 3N-1. The proof combines primitive-triangle parity, internal- and outer-hull geometry, lattice width, and a bounded exhaustive check after theoretical reduction. Status: Public Beta v0.1; internally verified candidate proof; external mathematical review pending.