Thermal heat-current cumulant ratios can be arbitrarily negative: three-state counterexamples to two sign conjectures

Unrefereed research candidate. Ptaszyński (Phys. Rev. E 106, 024119, 2022) conjectured from random-network calculations that the long-time heat-current cumulant ratios C3/C1 and C4/C2 are nonnegative in classical Markov systems driven only by temperature differences. This package gives exact three-state, two-bath counterexamples obeying local detailed balance. A fully connected family keeps energies, temperatures, stationary probabilities, mean heat current and entropy production fixed while both ratios tend to minus infinity like ε-2. In the standard state-jump model with one counted bath, two states never violate either sign. The mechanism is classical slow switching. The divergence tracks the variance: the variance-normalised ratios tend to -9/4 and -3/4, and variance-based bounds such as the thermodynamic uncertainty relation are untouched. With six states even the normalised ratios are unbounded below. A kinetic lower bound for positive mean and variance, in terms of the spectral separation of the stationary eigenvalue, captures the ε-2 rate. A negative ratio therefore does not certify quantum or non-thermal driving; slow switching must be excluded first. The archive contains the 13-page manuscript (PDF and LaTeX), exact rational and symbolic verifiers, a seeded random census, semantic negative controls, CI and SHA-256 sums. Assurance: written proofs, exact certificates, internal AI-assisted review only (not peer review, independent reproduction or formal verification). Creator: Anonymous. Text and data CC0-1.0; code MIT.