Generalized field mixing for endpoint criticality with marginal flow: resolving the four-state Potts endpoint in the square-lattice $J_1$--$J_2$ Ising model

Abstract: Identifying the asymptotic criticality of a critical endpoint is challenging, as pseudo-first-order signatures persist over accessible system sizes and mask its underlying critical nature. This ambiguity is amplified at endpoints controlled by a marginally irrelevant scaling field, where logarithmic flow delays the onset of asymptotic scaling. Here we develop a generalized field-mixing framework for endpoint criticality governed by one relevant scaling field together with a marginally irrelevant one, a setting that lies outside the conventional two-relevant-field formulation. By constructing a finite-size pseudocritical manifold, the framework removes the normal relevant detuning and exposes the residual marginal drift, enabling controlled histogram- and Binder-based finite-size analyses. We apply this approach to the frustrated square-lattice $J_1$--$J_2$ Ising model, where the location and even the nature of the stripe-ordering endpoint have remained controversial for decades. The endpoint is isolated directly as a distinct singular point, rather than inferred from where the phase boundary appears most Potts-like, and its asymptotic criticality is shown to follow four-state Potts universality with logarithmic corrections. This identification is independently supported by direct comparison with the Potts point of the Ashkin--Teller model and by consistent Binder scaling in both the magnetic and nematic sectors. Our results resolve a longstanding numerical ambiguity in a paradigmatic frustrated Ising system and establish a general framework for extracting asymptotic endpoint criticality in the presence of marginal flow.