A three-state thermal counterexample to a heat-current kurtosis conjecture

An explicit three-state classical thermal rate model with two positive-temperature baths has a negative fourth-to-second spectral heat-cumulant ratio. The total off-diagonal rates are positive integers. Every supported bath transition satisfies local detailed balance. The exact ratio is −3470947982622836/176627429595846691. This contradicts Eq. (19) of Ptaszyński, Physical Review E 106, 024119 (2022), in the finite tilted-generator formulation used in that paper. The proof constructs a local four-times continuously differentiable dominant eigenvalue branch and computes its actual derivatives. Lean 4.19 verifies the finite matrices, thermal conditions, complex spectrum, strict dominance, and derivative ratio. Exact rational calculations reproduce the determinant and all four cumulants. The formalization does not construct a trajectory probability space or prove the standard long-time stochastic identification theorem. The result does not refute the separate skewness conjecture or claim a minimal-state or optimal-bound theorem.