# Verification and scope

The six-page manuscript proves a sharp bound for sampling every ordinary stratum of an essential real toric arrangement. For primitive normals and Smith period E_A > 1, every multiple m >= n E_A of E_A succeeds. Explicit families attain the multiplier n for every n,e >= 2. For arbitrary normals the exact successful set is E_A times the upper interval beginning at the maximum scaled-stratum codegree.

## Written proof audit

The argument identifies E_A with the exact common vertex denominator and the established lcm period. It proves codegree monotonicity and the characterization of full-dimensional lattice polytopes of codegree n+1. A primitive facet constant of E_A times a chamber closure is divisible by E_A; a unimodular simplex has primitive support constants summing to -1. This excludes the sole extremal case when E_A > 1. Lower-dimensional strata satisfy the required bound directly. The sharp family and the E_A=1 and nonprimitive exceptions are proved, not inferred from experiments.

## Exact regression checks

Normal and optimized Python runs agree on every case and these counts:

- 89 arrangements and 559 nonsingular full-rank bases.
- 2,555,030 grid points and 3,184 strata counted across cases.
- 70 independently reconstructed planar subdivisions, with 521 full-dimensional cells and 1,694 vertex incidences.
- 18 sharp-family cases, 564 nonmultiple obstructions, and 369 threshold comparisons.

The checker uses integers and fractions, not floating-point tolerances. Its reference grid uses the Hanlon-Painter sufficient theorem; the planar clipping comparison is a separate geometric implementation. These finite checks are regressions, not a computational proof of the universal theorem. The script source is pinned in source-provenance.json.

## Attribution and limitations

The interior-lattice-point framework is due to Hanlon and Painter. The Smith lcm period is established in Kamiya-Takemura-Terao. The elementary codegree facts are proved in the manuscript; no invention of codegree or Smith periods is claimed. The primitive bound was not located in the selected primary sources and bounded search, which does not establish absolute priority.

The ordinary-stratum necessity is not claimed for coarser Bondal labels. The toric-Frobenius application is sufficient and restricted to smooth varieties with no torus factor. The broad OWR-14299288-010 polytope question remains partially addressed. The subtorus question of Bauermeister et al. is not solved.

The work is AI-assisted, self-audited and unrefereed. There is no independent human review or proof-assistant certification. The package includes reproducible checks and source provenance, not third-party copyrighted PDFs.