Paper038_PNT_Exact_Matrix_Arithmetic

The Prime Number Theorem states that the number of primes up to N is asymptotic to N/ln N. We prove this without complex analysis, contour integration, Tauberian theorems, or the Riemann zeta function as an analytic object. The proof uses only the coprimality matrix CN ∈ MN(ℤ), its Möbius decomposition, the spectral projectors satisfying P2 = P, and the trace identity tr(CN) = 1. The prime distribution is forced by the spectral structure of exact integer matrices over ℚ. The word "analytic" appears only in the Not Used column.