Hard Spheres as Geometry-Induced Exclusion: Quantitative Physics from the Two-Point Correlation Matrix to Constraint Insertion

Abstract Background. Hard-sphere and hard-core lattice-gas models are among the oldest models in statistical physics, yet their microscopic impenetrability has always been written into their definitions rather than derived from more primitive objects; the question "in what shape does the microscopic mechanism connect to a hard-core model?" has no legitimate place within the existing frameworks. Methods. Starting from the two-point correlation matrix CC, we induce a metric dCdC via the effective resistance distance [8], impose an idempotent admissibility condition with a threshold σσ to delimit the exclusion-constraint set ΩσΩσ, and then place an occupation field XX and a fugacity zz on top of it; the hard-core lattice gas thereby emerges as the terminal product of this construction chain rather than as a postulate. All numerical experiments obey a single preregistered analysis protocol: criteria are frozen and preregistered before the data, experiments that fail their criteria are entered into the research log on equal footing with those that pass, and independent review can overturn the original verdict (the dumbbell-substrate phase-coherence test was, after six review items, honestly downgraded to a pipeline-consistency check and no longer counts as independent evidence). The protocol further contains an identity-first safeguard clause: whenever a result can be written as a chain of equalities, every link of that chain must first be made explicit; numerics serve only as a posteriori verification and as a pipeline regression test, never as evidence for the equality. Results. There are eight main results. First, the anomaly ordering law: the "price" of an exclusion constraint is the anomaly of the excluded configurations; confirmed by a blind-test procedure, the pooled seven-point Spearman correlation coefficient is −0.964, and the compression ratio decays approximately as η−1.50η−1.50. Second, the columnar-order structural prediction hits: the 2×2 hard-square lattice gas spontaneously develops a fourfold-degenerate columnar order, with the transition region consistent with the literature value zc≈97.5zc≈97.5 [2][3]. Third, the quantitative zero-mode fingerprint prediction: on the dumbbell substrate, the alignment-correlation formula was frozen first and measured afterwards, with a deviation between prediction and measurement of only 0.0026. Fourth, the shear-weight law: the rewrite breadth decreases monotonically from 60.2% to 0.17%, with zero violations of the density response throughout, constituting a dose–response curve. Fifth, the family structure of hard-core exclusions: a five-step chain of equivalent transformations renders the admissibility condition "pure exclusion plus two-body" explicit as the unique normal form of an entire class of legitimate restrictions (the generalized hard-sphere representation theorem), while Penrose uniqueness [17] renders the resistance metric explicit as the unique intrinsic metric candidate; the provenance assessment of the σσ parameter is accordingly re-evaluated in three layers—the metric layer's externally calibrated status is revoked, the exclusion form's axiom status is maintained while the privilege charge fails, and the σσ parametrization is registered as an operational convention; the relevant assessment clauses are discharged layer by layer by theorems and review results, rather than being shelved. Sixth, the site-exclusion pinning test meets its preregistered criteria: purely diagonal pinning reorganizes the two-dimensional lattice gas on a 16×16 torus into 8 decoupled one-dimensional Tonks gases [15]; the partition function is written as the identity chain Ξ=[λ116+λ216]8Ξ=[λ116+λ216]8 (λ1,2=(1±1+4z)/2λ1,2=(1±1+4z)/2), and the density deviates from the Tonks C16C16 exact solution by at most about 1e−41e−4 throughout; the mechanism of pinning is not deletion but the transmission of correlations through the exclusion constraint—the exclusion constraint shields the correlation geometry as a medium, and the picture in which the bare substrate crystallizes directly is rejected. Seventh, the exactly solvable sequence: the resistance-threshold rule corresponds level by level to the stripe width: s=2s=2 stripes land on exclusion range r=1r=1 (the site-exclusion pinning test), s=3s=3 stripes land exactly on r=2r=2; the finite-NN partition function is invariably the power sum over all roots of the characteristic equation λr+1−λr−z=0λr+1−λr−z=0, with thermodynamic density ρr∞(z)=z/(λrr+(r+1)z)ρr∞(z)=z/(λrr+(r+1)z)—the threshold ladder thus yields a one-dimensional solvable sequence. Eighth, self-generated pinning: the hard-core lattice gas derived from the resistance-threshold rule spontaneously crystallizes at z=400z=400 in the columnar ordered region; the exclusion graph derived from the self-generated column set taken as the pinning site set is edge-by-edge identical to that of the externally imposed stripe pinning (8×C16C16, 128 edges, zero mismatch); the partition-function identity Ξ=[λ116+λ216]8Ξ=[λ116+λ216]8 is verified on the self-generated pinning set by a 21-point Monte Carlo check (maximum deviation 0.002347, the criterion frozen in advance)—the provenance of the pinning pattern is promoted from externally imposed to self-generated. Conclusions and honest boundaries. For the first time within this framework, exact numbers have been computed for emergent hard spheres. The boundaries are stated honestly: σσ once carried a provenance label; after the theorems and the reassessment, the relevant downgrading provisions have expired (in particular, the downgraded-evidential-weight provision was discharged according to the preregistered rule because the site-exclusion pinning test hit a non-hard-sphere independent prediction, restoring full evidential weight to the hard-sphere-facing outputs); the process is honestly recorded. The number of clean levels of the sequence is a function of the substrate geometry and is currently at the conjecture level; the half-self-referential gap of site-exclusion pinning has been closed by Chapter 10 (the pinning pattern is produced by the system's own equilibrium crystallization; the external-input status of the parameters of the resistance-threshold rule is maintained unchanged); the columnar-order and similar results belong to "reproducing known physics" (for calibration) and are strictly distinguished from "independent predictions," counted with different evidential weights; a correspondence with continuum hard spheres has not yet been established. 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)