Smith-form covolumes of Walsh degree-band lattices

../

Purpose

Before the compression proof of semistability was obtained, a finite diagnostic tested a necessary spectral-subspace condition: that no degree band of Walsh characters spans a lattice of smaller normalized covolume than the full band. The compression theorem now proves the relevant inequality for all sublattices; this diagnostic is retained as an exact finite record and supports no infinite claim.

The 53 degree-union matrices

The finite range is $(d,\ell)=(4,2),(6,3),(6,4)$ and every nonempty subset $J$ of the degrees $0,\ldots,\ell$: exactly $7+15+31=53$ matrices. For each, the rows are all Walsh characters whose degree belongs to $J$, in ascending mask order, and the columns are all $2^d$ points in ascending mask order; the entry is $(-1)^{\mathrm{popcount}(s\wedge x)}$. The product of the nonzero Smith invariants is the index of the row lattice in the saturated spectral lattice. The lattice generated by the Walsh rows has covolume $2^{d\cdot\mathrm{rank}/2}$, so subtracting the binary valuation of the index gives the exact saturated logarithmic covolume. All invariants encountered are powers of two.

Run, from the repository root,

uv run --no-sync python wiki/research/erdos_774/evidence/moment_band_determinants/main.py

The checker recomputes every Smith form with an exact integer reduction in the standard library and the shared root tools checker; it needs no symbolic algebra package and finishes in a few seconds. Its named checks are that every invariant is a power of two, that the retained record of degree sets, Smith multiplicities and covolumes is reproduced case by case, that the three full-band values in the table hold, and that no degree union has smaller normalized covolume than its full band. A failed clause exits nonzero. The record is the output of the original run, retained as the exact input the checker compares against.

Nine even-parity moment matrices

An earlier hand-run diagnostic, whose script is not retained, fixed the range $3\le d\le7$ and $1\le k<d/2$ in advance, exactly nine pairs. For each pair the rows were the masks $s$ with $\mathrm{bit\_count}(s)\le k$ in ascending order, including $s=0$; the columns were the masks $x$ with even $\mathrm{bit\_count}(x)$, also ascending; and the entry was $1$ precisely when $(x\wedge s)=s$. These are the squarefree monomial rows restricted to even-parity columns, not Walsh rows. The absolute values of the first (number of rows) diagonal entries of the Smith form were counted. Here $a^b$ means that the value $a$ occurs $b$ times.

These nine observations prove no general Smith-form rule, uniform extraction, covering-radius estimate, or chromatic growth. No theorem in the main argument depends on either table.