# Verification of the Cayley coloring counterexample

The result is proved for the entire infinite group in `main.tex`:
`Gamma = (C5 * (C2 x C2)) x C2`, with generating set
`{a, a^-1, at, a^-1 t, bt, ct, bct}` and central involution t.
The graph has ordinary chromatic number3 but no invariant random
proper three-coloring; an invariant random proper four-coloring is
constructed from a finite quotient.

The proof has four separately checked components: the seven-element
generating set and two finite subgroup graphs; an explicit proper
three-coloring for every reduced group word; incompatible constraints
`5q >= 4` and `4q <= 1` on any invariant law; and a finite-quotient
proper four-coloring. The five-cycle lower bound makes ordinary
chromatic number exactly3. No assumption of FIID, mixing, independence
or color symmetry enters the nonexistence proof.

The included `reproducibility/verify_counterexample.py` uses only the
Python standard library and exact integer arithmetic. It checks every
assignment in the two finite pieces (59049 and6561 assignments),42
boundary extensions,818104 directed Cayley edges from58436 reduced
words,116872 translated local constraints,100 quotient edges and a
twenty-coloring invariant orbit. Outgoing edges leaving the enumerated
set are checked by the same formula, not discarded. Normal and optimized
executions agree. These finite checks support but do not replace the
all-word mathematical proof.

The exact source question is answered negatively under its standard
invariant-random-law meaning. The complete proof was accepted by the
originating researcher after self-audit. No independent review or
proof-assistant verification is claimed. The bounded prior-art check
found no earlier identical counterexample, but absolute priority is
not certified. The preprint was developed with AI assistance.