Inheritance of Exact Ring-Gas Solutions by Spontaneous Columnar Pinning in a Solvable Lattice Model

Abstract In the hard-sphere paradigm, the exclusion scale (the range of the exclusion neighborhood on a lattice) has remained an external input parameter ever since Tonks's exact one-dimensional solution. In a fully solvable lattice model, this Letter studies whether one component of it can be endogenous: can the pinning site set be produced by the system's own equilibrium crystallization? Methodologically, we convert the geometry of a host medium into an exclusion graph via the resistance distance and a resistance-threshold rule, constructing a hard-core lattice gas with eight-neighbor (king adjacency) exclusion, equivalent to the 2×2 hard-square plaquette model. There are three results. First (numerical evidence): in the high-fugacity columnar-ordered regime (operational definition: fugacity far above the ordering crossover, with the order parameter already on its plateau; we take z=400z=400), the system spontaneously forms a columnar-ordered configuration without any manual symmetry breaking; the domain-symmetrized order parameter MM has plateau values falling in 1.02–1.07 across the three sizes L=16,32,48L=16,32,48 (statistical spread over independent runs about 0.05); the density deviates from the close-packed value 1/4 by no more than 2.4% (a defect-free snapshot attains exactly 0.250000). The pinning site set thereby becomes an output of the system's own equilibrium order (the columnar phase itself is due to Ramola-Damle-Dhar [1]; we do not claim its discovery). Second (structural lemma): the exclusion graph derived by applying the resistance-threshold rule to the self-generated column set is edge-by-edge identical to that of the externally imposed stripe pinning—8 disjoint 16-rings, 128 undirected edges, zero cross-column edges, zero mismatches—and is invariant under the intra-column sliding soft modes. Third (exact identity with numerical verification): the pinned partition-function identity Ξpinned(z)=[λ116+λ216]8Ξpinned(z)=[λ116+λ216]8 is, for the first time, verified by direct Monte Carlo sampling on a spontaneously crystallized pinning set; all 21 test points of the isotherm (7 fugacity points × 3 isomorphic realizations, coinciding point by point) fall within the pre-frozen criterion, with maximum deviation 0.002347; statistical independence is confirmed by a recheck of 9 points with 3 additional fresh seeds. We explicitly delimit the boundary: the exclusion-rule parameters of the host layer (σσ, the target exclusion number, and the shell-gap midpoint convention) remain external inputs; what is endogenous here is the pinning site set and its induced exclusion graph. This provides a strictly testable candidate mechanism for whether the exclusion scale can be produced by the geometry of the medium. The simulation code and raw data are available from the author upon request. ORCID: 0009-0007-1999-0293 Email: [email protected]/[email protected] phone:+86 18278275520(china)