# Verification of the connected path counterexample

The exact integer certificate proves t(P4,17)=8<t(P4,18)=9 at threshold 1/e for the uniform-site, uniform-all-colors proposal chain. Full-state enumeration and an independently implemented color-orbit quotient agree on all 100 rational distances. Each passes normal and optimized execution. Both verify the finite state spaces of 69,632 and 88,434 proper colorings, the five initial equality patterns, and rational Taylor bounds for 1/e.

The full-state implementation uses int64 only while (4q)^9<2^63-1 and converts values to Python integers before total-variation products. The portable quotient uses arbitrary-precision arithmetic throughout. Its 261 edgeless comparisons are consistency tests, not a general proof by testing.

The paper gives the recurrence, the complete finite domain, certified interval table and exact decisive fractions. The analytic coupling inequality C_n(t)-n/q <= d(G,q,t) <= C_n(t)+(t*Delta+m)/q proves mixing time 9 for all q>=512 on P4. No enumeration over infinitely many q is asserted.

The upstream edgeless example and classical coupling methods are credited. No known gap remains in this stated result after the originating researcher's detailed self-audit. This is not independent expert review or proof-assistant formalization. The bounded primary-source search is not an absolute priority certificate.

Question 5's color-number monotonicity target is answered negatively, including on connected graphs. The separate lattice-rate discussion and Questions 6-16 in the frozen record are not claimed solved. The frozen combined record remains partial.

The final release package is compiled, checked for diagnostics, visually inspected on every page and replayed from its portable source archive; the machine-readable build and readiness receipts bind the final artifacts.