A candidate scalar determinant bound on ten moment-lattice pairs

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The tested expression

A candidate closed-form lower bound for the normalized covolume of the sublattices spanned by a degree pattern was tested on exactly ten prescribed pairs $(d,\ell)$ and all nonempty degree unions. For $A\subseteq\{0,\ldots,\ell\}$, set $h_j=\binom dj-\binom d{j-1}$, $A_j=A\cap[j,d-j]$, $N_j=|A_j|$, and

The computed expression was

This expression is not used as a theorem; its proposed ballot-basis proof is unnecessary for the independent compression argument, which supersedes the diagnostic.

There are 16,694 patterns in total and zero violations of the tested comparison. Full prefixes attain each displayed minimum; uniqueness of the minimizer was not tested or asserted. The diagnostic ends at its finite bound.

What is checked

Run, from the repository root,

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

The checker rescans every pattern of every pair with exact rational arithmetic in the standard library and the shared root tools checker; it performs no search and finishes in a few seconds. Its named checks are that no pattern falls below the full normalized covolume $(d-\ell)\binom d\ell/(2\sum_{s\le\ell}\binom ds)$, that the full prefix attains the recorded minimum for each pair, and that 16,694 patterns are scanned. A failed clause exits nonzero.