# 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`.