Asymptotic Classification of Diophantine Equilibrium in the Base {2, 3, 5}: Lattice Geometry, Harmonic Proof, and Explicit Residues

We study nonnegative integer solutions (a,b,c) to 2a+3b+5c = n that minimize the dispersion of coefficients, equivalently the quadratic form Q(a,b,c) = 3(a^2+b^2+c^2) − (a+b+c)^2. We prove an asymptotic classification theorem: for sufficiently large n, the number m(n) of minimizers belongs to {1,2}, and the case m(n) = 2 occurs exactly on a finite set of congruence classes modulo a period T dividing 30. The geometric proof reduces the problem to a closest vector problem in a rank-2 lattice under a fixed quadratic metric, so ties correspond to Voronoi walls. A complementary harmonic proof uses a lattice theta series and the Poisson summation formula to show exact periodicity and a discrete Fourier spectrum in n. Computations up to n = 10^4 support the theory and indicate that the tie classes mod 30 stabilize. The framework clarifies why primes p ≥ 7 and prime powers q^k with q outside {2,3,5} exhibit uniqueness. We provide explicit residues observed empirically and exact, reproducible code.