Information-Theoretic Duality Between Regression Topology and Statistical Mechanics

This paper establishes an exact information-theoretic duality between econometric regression topology and statistical mechanics. We demonstrate that an autonomous, data-driven Quadruple Test, grounded in the Factor Hierarchy Law, 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, one of the few systems in statistical physics possessing a mathematically rigorous exact solution (Onsager, 1944; Yang, 1952). Verification proceeds in two logical stages: Stage A (pristine algebraic validation) on Onsager's exact solution, and Stage B (stochastic robustness testing) on finite-lattice Monte Carlo simulations. We openly declare that because the data derives from the known Onsager-Yang formula, the contribution is not an independent empirical discovery of new physics, but rather the rigorous proof of an exact informational duality between two independent frameworks. In the language of metrology, this is not an endogeneity flaw but a mandatory calibration requirement—before a telescope is deployed to observe unknown deep space, it must first be calibrated against a known, invariant light source in a controlled laboratory. Positive Controls and Asymptotic Convergence (9 items): Exhaustive search blindly locks onto the critical temperature at Tc = 2. 260 (deviation 0. 009 at a coarse step size of 0. 01). A grid-refinement study demonstrates monotonic convergence: the deviation shrinks to zero within 6-decimal precision at a step size of 0. 001, and further converges to ~10⁻⁸ under a Golden Section Search—the absolute limit of 64-bit double-precision machine arithmetic. An analytical proof formally demonstrates that the Chow F-statistic achieves a unique global maximum exactly at T₀ = Tc; therefore, in the analytical limit, the localization error is strictly zero. The Chow test at Tc yields F = 118, 074 against a null control of F = 3. 12 (a 266-fold difference), with permutation test p = 0. 000. The interaction term is overwhelmingly significant (p = 0. 000, ΔR² = 0. 991). A symmetry-breaking regime switch at the external field boundary h = 0 is detected with Chow F = 989. 41 (p = 0. 000). Interaction R² peaks sharply at Tc (deviation 0. 03). Multi-response-function validation (specific heat C, nearest-neighbor spin correlation) and anisotropic validation (three Jx/Jy ratios) all lock onto their respective theoretical Tc values with deviations under 0. 007. A synthetic double-break dataset is tested with both breaks successfully detected. Negative Controls (4 items): A 3, 000-temperature-point exhaustive scan over 4 response variables finds no false positive of comparable magnitude to the true peak (maximum artifact F = 481 vs. Tc peak F = 78, 754, 162; a signal-to-noise ratio of 164, 000: 1). Monte Carlo simulations (L = 16, 32, 64, 128; 8 observables including the Binder cumulant U₄ and multi-body correlation functions) successfully detect the Tc break in all sizes; all cross-size candidate peaks are excluded by the criterion of F-value decay with increasing lattice size. Curvature artifact tests confirm that Chow F for a smooth sigmoidal curve does not diverge with sample size, maintaining a stable ~11-fold gap from the true break. Robustness (4 items): Under 10% Gaussian noise, Chow F remains at 22. 3. F-values grow strictly monotonically with sample size (100 → 1, 000: 11, 436 → 118, 074), confirming genuine physical signal characteristics. F-values grow overall with lattice size L (L = 16 → 128: 170. 9 → 289. 2, Kendall τ = 0. 33), confirming qualitative consistency with Fisher Finite-Size Scaling theory. Detection accuracy remains invariant under anisotropic conditions. Physical Scaling (3 items): Chow F (h = 0) establishes a strictly monotonic mapping with the order parameter Mₛp—F-values decay monotonically from 691 million at T → 0 to 55 at T → Tc, spanning 7 orders of magnitude and tracing the full lifecycle of the order parameter. This decay curve precisely mirrors the physical vanishing process of latent heat. The F-statistic's ~38-fold amplification effect is proven to originate from the quadratic structure of the F-statistic based on the sum of squared residuals (M²) —the theoretical lower bound βF / βM ≥ 2 is empirically confirmed (ratio 1. 97 ≈ 2), with the actual 38-fold amplification representing the composite contribution of the quadratic structure and residual difference structure. This algebraic guarantee proves that the amplifier property of F is an intrinsic feature of its mathematical structure, not a sampling accident. Interaction R² peaks at Tc at 0. 9992 (deviation 0. 03). Core Theoretical Contributions: Contribution 1: Informational duality between the Factor Hierarchy Law and the Ehrenfest classification. This paper rigorously proves two distinct regime-switching topologies with fundamentally different statistical signatures—"Rule-Reset" (interaction-dominated, p = 0. 000, ΔR² = 0. 991) and "Direction-Reversal" (intercept-jump-dominated, interaction p = 0. 978). Rule-Reset maps precisely onto Ehrenfest's second-order phase transition, and Direction-Reversal maps precisely onto Ehrenfest's first-order phase transition. This correspondence is not an empirical coincidence, but a functional duality—a bijective informational mapping exists between the calculus operations on the thermodynamic potential (∂G/∂h, ∂²G/∂T²) and the statistical operations of regression geometry (Δ Intercept, Δ Interaction Slope). The Factor Hierarchy Law independently arrives at all conclusions of the Ehrenfest classification purely through regression analysis of observational data, without any knowledge of the Free Energy function. Contribution 2: Chow F-statistic as an informational proxy for the order parameter and an early-warning signal. This paper discovers and proves that Chow F (h = 0) is a statistical proxy variable for the thermodynamic order parameter Mₛp—their relationship is not a linear mapping, but a nonlinear high-gain amplification guaranteed by the quadratic structure (M²) of the F-statistic. The 38-fold amplification effect has been confirmed through algebraic root analysis. This enables Chow F to serve as a more sensitive early-warning signal than the order parameter itself in complex systems where the order parameter is difficult to measure directly. The complete decay curve of F (h = 0), which monotonically attenuates to zero at Tc with rising temperature, provides a definitive statistical proxy for the vanishing of latent heat. Contribution 3: Interaction R² as a precise proxy for second-order transition intensity. Interaction effect incremental R² peaks at Tc at 0. 9992, with a deviation of only 0. 03. This provides a precise quantitative metric for the "Rule-Reset" switching topology within the Factor Hierarchy Law. Methodological Contribution: This paper completes a "Severe Test" (sensu Deborah Mayo) of the Quadruple Test, establishing both the sensitivity (all positive controls passed) and specificity (all negative controls passed) of the methodology. A total of 22 independent verification checkpoints—spanning five dimensions (9 positive controls, 4 negative controls, 4 robustness checks, 2 statistical rigor checks, and 3 physical scaling checks) —are all passed. The analytical proof further confirms that the localization error of the method is strictly zero in the analytical limit. Cross-Disciplinary Integration: Together with the interest-rate-spread regime switch discovered by Tang (2026a–2026f) across five major financial markets (institutional systems), the Tang Break (a five-dimensional stellar regime boundary at 4762 K) discovered by Tang (2026h, 2026i, 2026j) across five independent astronomical dimensions (physical observation systems), and the informational duality proven in this paper on a first-principles physics model, the Factor Hierarchy Law has now received evidential support from three completely independent disciplines. This paper provides the physics cornerstone for the Law—proving that the hierarchical structure of Rule Factors and Execution Factors, and the critical behavior of regime switches, are not accidental products of data noise, but an informational dual of thermodynamic symmetry-breaking structures, a universal principle by which complex systems self-organize. Much like the historical realization that information-theoretic entropy reflects thermodynamic states, this paper demonstrates that regression variance partitioning serves as a direct informational proxy for physical symmetry structures.