# Verification of the cyclic vertex criterion

The six-page manuscript *Cyclic Vertex Groups and Integral Toric Restriction*
contains a complete self-audited proof of the stated coordinate restriction
criteria, connected-kernel Kirwan corollary, fixed-sector extension and
primitive disconnected counterexample. This report is not independent peer
review or proof-assistant certification.

## Analytic proof checks

The proof distinguishes coordinate restriction from the actual Borel Kirwan
map. It uses the natural additive Koszul model of Luo-Matsumura-Moore and Franz;
it does not assume that the ordinary Koszul product computes cup products.
Over the integers, the local quotient dimension is
`max(1, n - min_vertex rank_mod_p(B_vertex))`. Over the prime field it is
`n - min_vertex rank_mod_p(B_vertex)`. Cohen-Macaulay regularity and Smith normal
form yield the cyclic-cokernel and prime-to-p determinant criteria.

The local dimension proof uses a finite union of closed spectra, not an
invalid distribution of ideals over intersections. Homogeneous localization
detects global regularity. A bounded-below graded argument handles Koszul
rigidity. Connectedness is exactly lattice surjectivity, not merely primitive
facet normals. The fixed-sector proof retains ghost vertices and maximal-face
boundary cases.

The disconnected example uses the matrix with rows `(1,1,1)`, `(1,-1,1)` and
`(1,1,-1)`, together with its signed columns. Its Smith invariants are `(1,2,2)`.
The four-element even-sign kernel acts faithfully on a product of three
projective lines, with no generic divisor isotropy. Successive sphere-bundle
Gysin sequences apply because each Euler class is monic in its own variable.
The coefficient ring retains the odd torsion in the finite group's integral
cohomology. Thus the actual integral Kirwan map is onto even though coordinate
restriction is not.

## Exact finite regression

The portable standard-library checks retain all enumeration domains and raw
outputs. They check 2,112 nonsingular two-by-two matrices with entries in
`[-3,3]` and 236 nonsingular matrices among 240 seeded three-by-three trials.
Determinantal divisors and ranks at determinant primes agree. Thirteen Koszul
fixtures include a non-Cohen-Macaulay negative control; tests check integral
differential squares, rational and finite-field ranks, and UCT torsion-summand
counts in the recorded finite degree ranges, not full torsion exponents.
There are also 511 monic Euler-class smoke tests over a finite truncated ring.

A second checker verifies all eight signed vertex matrices and all four
kernel elements of the primitive example. The adjugate identity bounds the
kernel enumeration to fourth roots of unity. Both scripts pass under normal
and optimized Python, with identical output bytes in each pair. These tests
are regression evidence; the general statements follow from the written proof.

## Sources and limits

Known weighted-projective computations, product obstructions and explicit
cyclic-necessity discussion in Goldin-Holm-Knutson are credited. The additive
model and regular-sequence criterion are established prior work. Holm's 2006
talk is correctly identified as *Integral cohomology of symplectic quotients*,
based on joint work with Tolman; the separately cited in-preparation work has
a different title. The bounded primary-source review through 10 October 2026
does not certify absolute priority.

No classification of the general disconnected Kirwan map, coarse-space
cohomology or integral Chen-Ruan products is claimed. The frozen source record
is not closed in its entirety. Publication adds no source-closure count.

## Package checks

The desktop LaTeX compiler succeeds. Tectonic exports a six-page PDF with no
undefined references, missing citations, overfull boxes or LaTeX errors. All
six rendered pages were inspected. One nonfatal underfull-box warning occurs
in the Pomerleano-Teleman bibliography item; that reference is complete and
legible. The final page has all nine references, with no orphaned extra page.
Source archive membership and bytes are verified against its manifest.

The author's affiliation and contact details are in a footnote; AI assistance
is disclosed in ordinary manuscript prose. CC BY 4.0 applies to the authored
package. Independent review, formal verification and certified priority remain
false; a DOI or website release would not change those facts.