Capped floating-point basis reduction for projected moment lattices

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Question and scope

The phase-rounding criterion would give uniform extraction from a rational basis of the projected lattice whose Gram–Schmidt coordinates are at most $1/4$ in magnitude and whose total squared length is $O(2^d)$. This exploration asked whether basis exchanges could satisfy that criterion on small cubes. It is float64, not an exact certificate, and supplies no uniform-energy theorem. Neither extraction nor coloring conclusions depend on it.

The seven pairs were $d=4,\ldots,7$ and $k\in\{1,2\}$ with $2k<d$. Each run stopped after a complete cursor scan, 20,000 cursor steps, or 1,000 accepted swaps, whichever came first. There was no randomness. Initial representatives were the monomials $x_A$ with $|A|>k$, in increasing degree and combination order; points were ascending integer masks with bit $i$ as coordinate $i$. Orthogonal projection removed the Walsh rows of degree at most $k$. Integer size reductions and Lovász-parameter $0.75$ adjacent swaps were allowed only if the explicitly updated Gram–Schmidt coordinate cap stayed at most $0.25+10^{-9}$; the inequality test used tolerance $10^{-10}$, and the QR factorization was recomputed after each accepted swap.

No swap was blocked by the floating cap test. These values ceased to discriminate and no further computation was undertaken.

Script

The exploration script records the complete algorithm and writes its summary to the ignored output/ folder beside it. It needs NumPy, which the repository does not declare, and the original run set OPENBLAS_NUM_THREADS=1. There is no checker: floating-point observations are not evidence, and this folder contributes nothing to the aggregate evidence entry point.