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.
We discuss the entanglement spectrum of the ground state of a (1+1)-dimensional system in a gapped phase near a quantum phase transition. In particular, in proximity to a quantum phase transition described by a conformal field theory (CFT), the system is represented by a gapped Lorentz invariant field theory in the ``scaling limit'' (correlation length $\ensuremath{\xi}$ much larger than microscopic ``lattice'' scale ``
a''), and can be thought of as a CFT perturbed by a relevant perturbation. We show that for such (1+1) gapped Lorentz invariant field theories in infinite space, the low-lying entanglement spectrum obtained by tracing out, say, left half-infinite space, is precisely equal to the physical spectrum of the unperturbed gapless, i.e., conformal field theory defined on a finite interval of length ${L}_{\ensuremath{\xi}}=\phantom{\rule{0.16em}{0ex}}ln(\ensuremath{\xi}/a)$ with certain boundary conditions. In particular, the low-lying entanglement spectrum of the gapped theory is the finite-size spectrum of a boundary conformal field theory, and is always discrete and universal. Each relevant perturbation, and thus each gapped phase in proximity to the quantum phase transition, maps into a particular boundary condition. A similar property has been known to hold for Baxter's corner transfer matrices in a very special class of fine-tuned, namely, integrable off-critical lattice models, for the entire entanglement spectrum and independent of the scaling limit. In contrast, our result applies to completely general gapped Lorentz invariant theories in the scaling limit, without the requirement of integrability, for the low-lying entanglement spectrum. While the entanglement spectrum of the ground state of a gapped theory on a finite interval of length
2R with suitable boundary conditions, bipartitioned into two equal pieces, turns out to exhibit a crossover between the finite-size spectra of the same CFT with in general different boundary conditions as the system size
R crosses the correlation length from the ``critical regime'' $R\ensuremath{\ll}\ensuremath{\xi}$ to the ``gapped regime'' $R\ensuremath{\gg}\ensuremath{\xi}$, the physical spectrum on a finite interval of length
R with the same boundary conditions, on the other hand, is known to undergo a dramatic reorganization during the same crossover from being discrete to being continuous.
A point where several critical lines intersect is called a multicritical point.Near such a point, multiple critical phenomena interfere and a finite-size correction becomes large.Therefore, conventional methods cannot be applied near the multicritical point.We propose a new method to numerically calculate a transition point for a quantum spin chain near the multicritical point.We treat a bond-alternating (BA) XXZ model for S = 1/2, 1, 3/2.In a ground-state phase diagram of this model, a 2D Gaussian universality transition line bifurcates into two 2D Ising universality transition lines, which make a multicritical point.At this point, Berezinskii-Kosterlitz-Thouless transition occurs, where the correlation length diverges singularly.To deal with a 2D Ising universality, we review a transverse field 1D quantum Ising (TFI) model, corresponding to the classical 2D Ising model with transfer matrix.A relation between a disorder phase and an order phase is known as the Kramers-Wannier duality transformation.We find a proper duality transformation that is exact in the finite-size TFI model with a periodic and anti-periodic boundary condition.This shows that the energies for the two boundary conditions are crossing at the transition point.A free fermion field theory is derived from the critical Ising model by taking a continuum limit.From a conformal field theory, we verify that the energy-crossing is realized in the 2D Ising universality class, not only in the TFI model.We apply our method to the BA XXZ model.The energy-crossing happens in boundary conditions that are twisted around a z-axis and y-axis.In an anisotropic limit, two energies in a finite-size are crossing at the transition point since the BA XXZ model is identical to the TFI model.on the other hand, at the multicritical point, the finite-size correction vanishes by an isotropy and a twist translation symmetry.Therefore, near the multicritical point, our method has a smaller finite-size correction.In this paper, we also review the 2D Gaussian universality.By a bosonization, the BA XXZ model can be transformed to a phase Hamiltonian composed of boson operators.The degenerate energies for a twisted boundary condition are split by a perturbation around a Gaussian fixed point.By our method, we numerically calculate a transition point of the BA XXZ model for S = 1/2, 1, 3/2.As expected, the finite-size correction of numerical results becomes very small near the multicritical point.i