Exact finite witness

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The entry point computes the finite witness displayed on L217. At $N=63$, $k=9$, block length $21$ and occupancy $3$, it enumerates every subset of sizes two, three and four and every one of the 21 distinct block rotations. Falling-factorial inclusion probabilities and all accumulations are exact. An explicit bound excludes integer overflow.

A separate control enumerates every outcome and rotation of the law with three blocks of length four and occupancy two, comparing the resulting short-pair mean, variance and repeated-difference mass with the moment calculation. The unmodified variance is also checked to be positive, while both the zeroed and renormalized centered-square expectations are negative at the retained witness.

The calculation also treats the mixture with weight $33/82$ on the uniform law on all nine-element sets. It satisfies every difference and sum capacity, with equality at difference one, yet its zeroed and renormalized short-pair square is negative. The uniform component's mean and variance are checked against independent graph intersection counts, and the mixture against the law of total variance.

For the signed cyclic polynomial $F$ in the final part of the proof, the same small mixture instead gives the positive value

$$\mathcal L((F-\mathbb E F)^2) =\frac{26206418542771452345267619483}{89042295412084245797931582420}.$$

This is a finite limitation: that part of the theorem asserts negativity only for sufficiently large parameters. Signed genuine moments have a separate direct-enumeration control on the small block law. Accumulations use bounded integer arrays or exact Python integers where needed.

From the repository root, run uv run --no-sync python wiki/theory/additive_bases/L217_sidon_block_quartic_obstruction/evidence/main.py. The full run and --quick have the same finite coverage, take under ten seconds and use under 300 MB with one thread. Dependencies are NumPy and the root tools package. No external input or saved output is required. This witness does not prove the asymptotic statement; that is the scope of the written argument.