The layers as a dilation system

• 2026-09-09 [Proved] The stack's layers are a dilation system: the square wave s(x) = (-1)^floor(x) is the odd 2-periodic extension of the constant 1, so the layers s(nx) are the dilates of one function in the sense of Hedenmalm, Lindqvist and Seip, with sine coefficients a_n = 2 sqrt 2/(pi n) on odd n and 0 on even, and the symbol S(s) = (2 sqrt 2/pi)(1 - 2^(-1-s)) zeta(1 + s); the moire correlation law is exactly that system's Gram matrix sum_j a_(j n') a_(j m'), its both-odd hypothesis being the symbol's support on the odd integers, the lcm-grid integral and the symbol sum agreeing at all 78 pairs m <= n <= 12 (1/15 at (3,5), 1/3 at (3,9), 0 at (2,3)); the Gram vanishes unless v_2(m) = v_2(n) and every block is the odd Gram, so the full system's Gram operator is a direct sum of copies of the odd one and the half-period form is one block over 4. Witness: lab/py/stack-dilations check_dilation_shift, check_gram_two_ways, check_blocks.
• 2026-09-09 [Proved] The parity layers are complete and minimal in L^2(0,1) but neither a Riesz basis nor a frame: the Riesz criterion needs the symbol bounded on Re s > 0 and S is unbounded at the pole of zeta(1 + s) (Hedenmalm-Lindqvist-Seip Theorems 5.2 and 3.1), the normalised coefficients a_n/a_1 being totally multiplicative with divergent prime sum (Corollary 5.3), completeness and minimality following from Corollary 5.8 with the biorthogonal system built from the Dirichlet inverse mu(n)/n on odd n and not itself a dilation system; the sharp frame bounds are the supremum and infimum of |S| on the half-plane, infinity and zero; the parity stack is the boundary case tau = 1 of the source's example zeta(tau + s), Riesz iff tau > 1. Witness: lab/py/stack-dilations check_inverse, REFS.md.
• 2026-09-09 [Proved] No weighting of the scales moves the stack's symbol off the line Re s = 1: a weight multiplies the symbol by its own Dirichlet series at the same s, and the Mobius weight gives a_1^2/S, the reciprocal up to a_1^2 = 8/pi^2; over the first K odd scales lambda_max rises 2.01467 to 2.47224 and lambda_min falls 0.4393 to 0.3570 for K = 25 to 200, condition number 4.586 to 6.926, the determinant exact against the Smith product at K = 1..13. Witness: lab/py/stack-dilations check_inverse, spectrum, check_determinant.
• 2026-09-09 [Refuted] The pole heuristic that the largest Gram eigenvalue grows like (log N)^2: the spectral norm of a gcd matrix at exponent one over k distinct integers is of order (log log k)^2 by Lewko and Radziwill 2014 Theorem 2, which settles the exponent-one case that Gal 1949 bounded for the gcd sum, and the window agrees, the (log N)^2 ratio falling 1.93 across K = 25..200 against 1.41 for (log log N)^2. Witness: REFS.md, lab/py/stack-dilations spectrum.