Double-Peak Splitting: The Hidden Structure of Finite-Lattice Phase Transitions in the 3D Ising Model——Verification of the Factor Hierarchy Law on an NP-Complete System: Double-Peak Splitting, Transition Regime, Piecewise Control Function, and Multi-Regime State

Abstract

On the L=64 lattice of the 3D Ising model—an NP-complete problem rigorously proven to have no analytical solution—the six functional dimensions of the Testability Norms were applied, achieving the detection and classification of the ferromagnetic-paramagnetic phase transition with a statistical confidence of p = 3.33×10⁻⁷⁰. Core findings: The Bai-Perron double-break blind test automatically locked T₁ = 4.39 and T₂ = 4.49 (F = 285.7, significantly outperforming the single-break F = 238.3), precisely aligned with the M_sp switch (4.40, F = 1524) and the U₄ switch (4.50, F = 2335), with deviations of 0.01 and 0.01 respectively. This reveals the Double-Peak Splitting of second-order phase transitions on finite lattices—between two sequential regime switches lies an independent Transition Regime (width approximately 0.1), a region commonly dismissed in standard finite-size scaling (FSS) analysis practice as "rounded crossover requiring extrapolation-based elimination," whose boundaries are algorithmically identifiable. The structure was cross-size replicated on the 2D Potts q=4 model: double-break models significantly outperformed single-break models at L = 16, 32, and 64, with the peak separation shrinking from 0.260 at L = 64 to 0.180 at L = 128, evolving according to scaling laws. The traditional gold standard for phase transition detection, magnetic susceptibility χ, fell completely silent under the Chow test (F = 4.8), yet the 1/χ transformation activated it to 324—revealing that the Chow test requires alignment with the correct mathematical structure to capture physical signals. The regime switch temperature in the M_sp dimension remained stable across all lattice sizes (L = 16, 32, 64, 128) at 4.40–4.46, showing no convergence toward the thermodynamic limit Tc = 4.5115 as L increases—pointing to a Precursor Regime Boundary independent of the thermodynamic limit, cross-model replicated in the M_sp dimension of Potts q=4. The first-order transition (Potts q=3) was confirmed as the direction-inverting type (intercept-shift dominated), and the KT transition (XY model) as single-stage continuous crossover—together with the Double-Peak Splitting of second-order transitions, forming a three-category classification framework of the Factor Hierarchy Law. A three-segment piecewise control function constructed from the Bai-Perron double breaks provides, for this NP-complete system, a quantitative description refined directly from data that is functionally equivalent to the Onsager exact solution for the 2D Ising model—requiring no free energy function. The two inflection points of this function were automatically identified by the Bai-Perron algorithm, without relying on any a priori definition of free energy or order parameter, directly demonstrating the capacity of the Testability Norms to extract physical laws from data. Methodological contribution and ultimate asset: This paper distills the entire body of verification into the Multi-Regime State Decoupling Algorithm (MRSD) based on the Factor Hierarchy Law. Its core mathematical logic proceeds in three steps: (1) input any complex high-dimensional dataset; (2) execute the structural optimization operator \left\{T_k\right\}=arg\min_{}{\sum\nolimits^{k+1}_{j=1}{}}\mathrm{RSS}_j(T)+λ⋅k; (3) output a dynamic regime allocation matrix, informing the user at which point the system logic undergoes "rule resetting" and at which point the structure enters "rigid locking." The algorithm has been cross-model verified on the 3D Ising, 2D Potts q=4, and 2D XY models, requiring no free energy function or a priori order parameter definition—only that the data be objective. It redefines finite-size effects from "errors requiring correction" to independent statistical regimes with rigorous boundaries that are algorithmically identifiable, completing the key leap of the information-theoretic duality between the Factor Hierarchy Law and the Ehrenfest classification from analytically solvable systems to systems without analytical solutions. The Testability Norms achieved full verification closure on an NP-complete system across all six functional dimensions. Leave-one-out cross-validation (Tc range [4.47, 4.48], standard deviation 0.0021), Bootstrap threshold regression (95% CI [4.44, 4.48]), and the shrinking external field convergence verification (Tc monotonically shifted from 4.48 to 4.50) jointly confirm the high robustness of the conclusions. 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.