Information-Theoretic Duality Between Regression Topology and Statistical Mechanics——A Data-Driven Equivalence of the Ehrenfest Classification Validated on the 2D Ising Model as the Physics Cornerstone for the Factor Hierarchy Law

Abstract

This paper establishes an exact information-theoretic duality between econometric regression topology and statistical mechanics. We demonstrate that an autonomous, data-driven testability norm—grounded in the Factor Hierarchy Law, with the Quadruple Test as its specific implementation in this paper—can blindly detect, precisely quantify, and correctly classify thermodynamic phase boundaries in perfect mathematical equivalence with the Ehrenfest paradigm, without any prior knowledge of Free Energy functions. The validation platform is the two-dimensional Ising model. Verification proceeds in two stages: pristine algebraic validation on the Onsager-Yang exact solution, and stochastic robustness testing on finite-lattice Monte Carlo simulations. We openly declare that because data derives from the known Onsager-Yang formula, this contribution is the rigorous proof of an exact informational duality between two independent frameworks—a mandatory metrological calibration, not an independent empirical discovery. Positive controls: Exhaustive search blindly locks onto Tc = 2.260 (deviation 0.009 at step 0.01). A grid-refinement study demonstrates monotonic convergence to ~10⁻⁸ under Golden Section Search—the limit of 64-bit machine precision. An analytical proof confirms the Chow F-statistic achieves a unique global maximum exactly at T₀ = Tc; in the analytical limit, localization error is strictly zero. Chow F = 118,074 against a null control of 3.12 (266-fold difference, permutation p = 0.000). Interaction effect: p = 0.000, ΔR² = 0.991. A symmetry-breaking switch at h = 0 yields Chow F = 989.41 (p = 0.000). Interaction R² peaks at Tc (deviation 0.03). Multi-response-function and anisotropic validations all lock onto theoretical Tc values (deviations < 0.007). Negative controls: A 3,000-point scan over 4 variables finds no false positive of comparable magnitude (maximum SNR 164,000:1). Monte Carlo simulations (L=16-128, 8 observables) detect Tc in all sizes; all cross-size candidate peaks are excluded by F-value decay criterion. Core discoveries: (1) Two distinct regime-switching topologies—"Rule-Reset" (interaction-dominated) and "Direction-Reversal" (intercept-jump-dominated)—map in exact informational duality onto Ehrenfest's second-order and first-order transitions. (2) Chow F(h=0) is an informational proxy for the order parameter, decaying from 691 million to 55 across 7 orders of magnitude and precisely mirroring latent heat vanishing. The F-statistic's ~38-fold amplification originates from its quadratic M² structure (theoretical lower bound β_F/β_M ≥ 2 confirmed). (3) Interaction R² is a precise proxy for second-order transition intensity (peaks at 0.9992, deviation 0.03). Methodological contribution: A Severe Test (sensu Mayo) is completed—22 independent verification checkpoints spanning five dimensions, all passed. Cross-disciplinary integration with the Tang Break (Tang, 2026h-j) and financial regime switches (Tang, 2026g) establishes the physics cornerstone for the Factor Hierarchy Law, proving it is an informational dual of thermodynamic symmetry-breaking structures. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.