# Verification report

The complete theorem is established by the written primitive-direction,
chain-population and periodic-cap arguments in the manuscript. The originating
researcher checked uniqueness of counted directions, chain boundaries, integer
population convexity, floor losses, modular parameter repetitions, averaging
over translates, prime selection and the excluded low-density endpoint.
No mathematical gap was identified in the stated dense-regime theorem.

The supplementary checker performed 68,705 exact checks in normal Python and
another identical 68,705 in optimized Python. Checks use integers and exact
rational arithmetic; failure conditions remain active under optimization.
The seed for sampled subsets is fixed at 15427019. Domains and individual
counts appear in the code and retained outputs. They include:

- Primitive counts for r=1,...,35 and signed chain partitions for n=2,...,6.
- Population and floor bounds, with thirteen explicit dense sets up to n=36.
- Direct cross-product triple counts versus independent pair-line counts.
- Every affine field line for p=3,5,7,11 and selected periodic lifts.
- Exact translation averages and the prime-selection constant estimates.
- The already known 25-vertex knight graph: 48 edges and exact matrix rank 47.

These are finite regressions, not formal verification of the theorem and not
independent mathematical review. The known upper construction and rigidity
result are attributed. The source's exact-minimum and threshold questions
remain outside the accepted statement. The preprint has not been refereed.