The Reduction of Universality: A Regime Boundary Theory for Finite-System Phase Transitions—Cross-Model Evidence for Bimodal Splitting, the Precursor Regime Boundary, and the Tool-Dependence Theorem

Abstract

Central proposition: Traditional universality is a mathematical convergence artifact of the thermodynamic limit (L→∞). In real, finite systems, regime boundaries are more fundamental physical quantities than universality classes. Using the Factor Hierarchy Law and the Testability Norms (six functional dimensions × six-tool cross-validation), we performed a systematic cross-model analysis on five phase-transition models (2D Ising exact solution, 3D Ising, 2D Potts q=3, 2D Potts q=4, and 2D XY). Five core discoveries are reported: (1) Bimodal splitting. In a finite lattice, a second-order phase transition splits into two successive regime switches—locking of the ground state (M_sp dimension) and a structural change of fluctuations (U₄ dimension)—separated by an independent transition regime. On 3D Ising L=64, M_sp breaks at T≈4.40–4.50 (Chow F=939.9–1524.1, depending on scan parameters) and U₄ breaks at T≈4.50–4.55 (F=299.1–2335). The same structure is reproduced on 2D Potts q=4 for L=16, 32, and 64. Blind Bai-Perron double-breakpoint detection, independent of the Chow-scan parameter, confirms the bimodal-splitting structure directly from the M–h coefficient series. (2) Precursor regime boundary. The breakpoint of the M_sp dimension across L=16–128 exhibits a mean of 4.50 and a standard deviation of only 0.071—almost immobile despite a 512-fold increase in volume. Potts q=4 (L=16–128) independently reproduces this stability. In contrast, the KT transition (XY model) displays a strictly monotonic convergence (p=0.0000). The available data (L≤128) are not yet sufficient to statistically distinguish "slow convergence" from "structural non-convergence", but the sharp contrast in size-sensitivity between the two transition types constitutes a robust empirical regularity. (3) Tool-dependence theorem. On the exact 2D Ising solution (Tc=2.269185), the Chow test (T=2.2735) and the susceptibility-peak method (T=2.2675) give "critical temperatures" lying on opposite sides of the true value, differing by 0.006. On 3D Ising L=64, five methods applied across multiple observables yield a Tc range spanning 0.325 temperature units. No "critical temperature" exists independently of the detection tool. (4) χ-blindness. On the exact 2D Ising solution, χ yields a Chow F of only 16.8, whereas 1/χ reaches ~10²¹; on 3D Ising L=64, the activation factor is 40. The root cause is a mathematical mismatch between the even-function structure of χ and the odd-function requirement of the Chow test—fifty years of locating Tc via the χ peak have systematically missed the "regime-switch" dimension of phase transitions. (5) Random-bond Griffiths-phase verification. Traditional universality theory predicts that pure and disordered models are equivalent in the long-wavelength limit. The Factor Hierarchy Law, however, predicts that disorder will generate additional regime boundaries. On a 10% disordered 3D Ising model, Bai-Perron detects a new breakpoint at T≈4.25 that is absent in the pure model (significant on both L=16 and L=32, with a BIC improvement up to +42.77). CUSUM and MOSUM are uniformly significant (permutation test p=0.0000). This result constitutes decisive evidence for a paradigm shift—where traditional theory makes an incorrect prediction, the Factor Hierarchy Law makes the correct one. Based on the above, we establish an MRSD tripartite classification framework: first-order transitions display mixed-type switching (intercept jump and slope change of comparable magnitude) in finite systems, second-order transitions display rule-resetting + bimodal splitting, and KT transitions display single-stage continuous crossover. A modified finite-size scaling ansatz is proposed: B(L) = Tc(∞) ± ΔT₀·Φ(L), where Φ(L) ≈ constant for certain regime boundaries over an intermediate range of scales. Overall conclusion: Universality is a convergence product of the thermodynamic limit. In real finite systems, regime boundaries are more fundamental than universality classes. This paradigm shift, from "universality" to "regime boundaries", constitutes a "reduction of universality" within the traditional theory of phase transitions. 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.