# 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.