The parity-carpet stack spectrum

• 2026-08-28 [Proved] The flat odd-scale parity-carpet stack's spectrum is the divisor field of the frequency gcd and nothing more: the sine coefficient of the L-layer stack G_L at odd (a,b) is (1/(pi^2 ab))[1 - sigma_1^S(a)/L - sigma_1^S(b)/L + sigma_2^S(gcd(a,b))/L] and vanishes at any even index, the interaction part carrying exactly sigma_2(gcd(a,b))/(ab); Parseval splits the variance blockwise into the two terms of the carpet law, re-proving the moire variance formula; every spectral statistic is an Estermann-Ramanujan zeta quotient, sum sigma_2(gcd)(ab)^(-w) = lambda(w)^2 lambda(2w-2) and sum sigma_2(gcd)^2 (ab)^(-w) = lambda(w)^2 lambda(2w-2)^2 lambda(2w-4)/lambda(4w-4) with lambda the odd zeta; a stack weighted n^(-s) renders sigma_(2-s) as its spectrum; divisor information only, no new L-function; coefficients checked cell-exactly at L = 14 to 47 digits and Parseval against the exact rational variance; the object is the flat odd-scale stack, not the all-scales Farey stack. Witness: moire-correlation-laws.