# Verification report

The complete theorem in the manuscript was accepted after the
originating researcher's detailed self-audit. Its argument is
self-contained: negligible-neighborhood volume estimates, an empty
tail ray, covered lacunary FCC octahedra, nonempty rooms, spherical-cap
path rerouting, and annular attainment of upper-density optima.

The critical endpoint uses closed occupied balls and open vacancy.
The two center sets are fixed independently of outer radius. Every
large-radius room is nonempty; the nonpercolating example is not an
empty-vacancy convention. Pairwise distances remain at least two.

The exact checker passes 57,716 conditions in both normal and optimized
Python. The reports agree after deleting only the execution timestamp.
It uses integer/rational arithmetic and symbolic polynomial identities,
and does not depend on removable assertions.

Finite domains include shell sizes 2, 4 and 8; rational triangular
grids with denominators at most 24; room-to-shell checks through index
25; and 732 rational radii for the corridor and attainment estimates.
The boundary topology and infinite estimates rely on the written
proof, not these finite tests.

The local package verification and manifest bind the actual compiled
PDF, source ZIP, final page inspection and portable reruns. A package
receipt is not a DOI or evidence of external publication.

No independent mathematical review or proof-assistant verification
occurred. The broad source remains unresolved beyond the explicitly
stated three-dimensional unrestricted-density theorem. Absolute
novelty is not certified.