stack-dilations

• The parity stack's layers read as a system of dilated functions in L^2(0,1), and the Dirichlet series that is its symbol.
• The square wave s(x) = (-1)^floor(x) is the odd 2-periodic extension of the constant 1 on (0,1), so the layers s(nx) are the dilates phi(nx) of one function and the Hedenmalm-Lindqvist-Seip theory applies verbatim.
• Its sine coefficients against e_n(x) = sqrt 2 sin(n pi x) are a_n = 2 sqrt 2/(pi n) at odd n and 0 at even n, so the symbol is S(s) = sum_n a_n n^-s = (2 sqrt 2/pi) (1 - 2^(-1-s)) zeta(1 + s).
• check_dilation_shift verifies the one identity the whole reading rests on, <s(nx), e_k> = a_(k/n) when n | k and 0 otherwise, by exact piecewise integration, max deviation 1.16e-15 over n <= 8 and k <= 60: the dilate carries the Dirichlet series n^-s S(s).
• check_gram_two_ways prints the Gram matrix of the layers two ways at all 78 pairs m <= n <= 12: grid_integral integrates s(mx) s(nx) exactly in rationals on the lcm(m,n) grid, symbol_entry reads the symbol sum sum_j a_(j n') a_(j m'), and they agree entry for entry, reading 1/15 at (3,5), 1/3 at (3,9) and 0 at (2,3).
• symbol_series sums the symbol numerically to 20000 terms and lands within its own tail bound of the closed form at every pair, largest gap 2.03e-05.
• check_blocks shows the Gram entry vanishes whenever v_2(m) != v_2(n), and that the block at v_2 = a is the odd Gram gcd(m,n)^2/(mn) itself, at all 2730 ordered pairs to 64: the full dilation system's Gram is a countable direct sum of copies of the odd one, so the odd stack carries the whole spectrum.
• check_determinant re-derives the Smith determinant prod over odd k of prod over p | k of (1 - p^-2) over the first K odd scales at K = 1..13, the value over the 13 odd scales n <= 25 being 11399736556781568/21994507608198125, the number lab/py/stack-levels prints.
• check_inverse verifies that the Dirichlet inverse of a_n/a_1 is mu(n)/n at odd n and 0 at even n, at every n <= 400, so 1/S(s) = 1/(a_1 (1 - 2^(-1-s)) zeta(1 + s)) and the Mobius square wave, coefficients mu(n) a_n, has symbol a_1^2/S(s).
• a_n itself is not totally multiplicative, a_1 = 2 sqrt 2/pi; a_n/a_1 and its inverse are, which is what HLS Corollaries 5.3 and 5.8 need, and both verdicts are invariant under scaling phi by a_1. The prime sum those corollaries test already reads 1.9854 at p <= 100000 and diverges by Mertens.
• Not-a-Riesz-basis needs no multiplicativity at all: HLS Theorem 5.2 with Theorem 3.1 demands S bounded on Re s > 0, and S(s) = a_1 (1 - 2^(-1-s)) zeta(1 + s) blows up as s -> 0+.
• spectrum computes the extreme eigenvalues of the odd Gram over the first K odd scales with a symmetric eigensolver at K = 25, 50, 100, 150, 200, printing lambda_max, lambda_min, the condition number and two normalisations.
• The window reads lambda_max from 2.01467 to 2.47224 and lambda_min from 4.393e-01 to 3.570e-01, condition number 4.586 to 6.926, with lambda_max/(log N)^2 falling from 0.1330 to 0.0689 and lambda_max/(log log N)^2 falling from 1.0910 to 0.7717.
• The ceiling those numbers sit under is not printed here and belongs to the literature: the spectral norm of a gcd matrix at exponent one over any k distinct integers is of order (log log k)^2, Lewko-Radziwill 2014 Theorem 2, which settles the exponent-one spectral norm that Gal's 1949 bound on the gcd sum itself had left open.

RUN

uv run python research/lab/py/stack-dilations/stack_dilations.py

Under a second, prints only, writes nothing.

WITNESSES

• The dilation reading of stack.md, "The layers as a dilation system": the symbol S(s), the Gram identity, the block splitting, the failed Riesz condition and the completeness.
• The eigenvalue window it prints is the finite half of the layer Gram spectrum named as the open object in "What a breakthrough would look like".
• It does not re-derive the covariance law of lab/py/stack-levels; it matches it, and extends it to the even scales the odd convention drops.