Canonical criterion for third-order transitions identified by microcanonical inflection-point analysis

Microcanonical inflection-point analysis (MIPA) identifies third-order transitions from derivatives of the microcanonical entropy, yet this classification has lacked a direct canonical counterpart accessible from measurable fluctuations without reconstructing the density of states. Here we establish such a fluctuation-based canonical framework for MIPA-type third-order transitions using a cumulant-ratio diagnostic. Its signed extrema identify the corresponding canonical signatures and, in the single-saddle regime, are asymptotically linked to microcanonical classification. Because the diagnostic depends only on energy cumulants, it avoids explicit density-of-states reconstruction and is accessible from measurable fluctuations. Physically, it is a canonical fluctuation-asymmetry observable whose extrema mark high- and low-energy tail reorganizations of the canonical energy distribution. Benchmarks on Ising, Potts, and nonreciprocal Ising models support the framework.