The Ladder of Isomorphism Experimental Paper: Forgetful Tower Operations from Ising Model, Surface Code to Euler Equation and the Far-Deep Isomorphism Derivation of Kitaev Gap
This paper establishes a directly operable forgetful tower protocol, taking theadjoint functor pair of hierarchical categories as the mathematical skeleton, andunifies the microscopic, mesoscopic and macroscopic levels of physical systems as abidirectional limit structure of forward expansion and backward convergence. Selecting three highly mature systems—the two-dimensional Ising model, the surfacecode topological quantum error correction, and the incompressible Euler equation—as demonstration platforms, it shows how the universality class of statistical mechanics, the modular tensor category of quantum information, and the vorticityhomotopy class of fluid mechanics each condense into the generative source pointof their respective forgetful tower. As a core application, the paper re-derives theanyonic excitation gap of the Kitaev toric code using the far-deep isomorphismframework, proving that the strict positivity of the gap directly follows from thedeep rigidity of the modular tensor category, without requiring explicit diagonalisation of the microscopic Hamiltonian. This derivation compresses the computationalobject from an exponential-dimensional Hilbert space to a fixed-size fusion rule, significantly simplifying the conceptual path and generalisability, and applies directlyto non-Abelian topological codes. The paper strictly distinguishes proven theorems, constructive propositions and conditional conjectures, and honestly delimitsthe applicability boundaries of the framework.
This paper establishes a directly operable forgetful tower protocol, taking theadjoint functor pair of hierarchical categories as the mathematical skeleton, andunifies the microscopic, mesoscopic and macroscopic levels of physical systems as abidirectional limit structure of forward expansion and backward convergence. Selecting three highly mature systems—the two-dimensional Ising model, the surfacecode topological quantum error correction, and the incompressible Euler equation—as demonstration platforms, it shows how the universality class of statistical mechanics, the modular tensor category of quantum information, and the vorticityhomotopy class of fluid mechanics each condense into the generative source pointof their respective forgetful tower. As a core application, the paper re-derives theanyonic excitation gap of the Kitaev toric code using the far-deep isomorphismframework, proving that the strict positivity of the gap directly follows from thedeep rigidity of the modular tensor category, without requiring explicit diagonalisation of the microscopic Hamiltonian. This derivation compresses the computationalobject from an exponential-dimensional Hilbert space to a fixed-size fusion rule, significantly simplifying the conceptual path and generalisability, and applies directlyto non-Abelian topological codes. The paper strictly distinguishes proven theorems, constructive propositions and conditional conjectures, and honestly delimitsthe applicability boundaries of the framework.
This paper constructs a categorical framework of the “Ladder of Isomorphism”based on the unified formal foundation of hierarchical generative systems and towers of forgetful functors, situating always-isomorphism and far-deep isomorphismwithin a common theoretical basis. Always-isomorphism addresses the deep equivalence between the initial and final states of the same system under temporalevolution, while far-deep isomorphism deals with the synchronic deep equivalencebetween two simultaneously existing systems that have no direct superficial causalconnection. The two form a precise hierarchical relation of conditions: far-deepisomorphism depends on the triple conditions of formalizability, nonempty deeptopology, and hierarchical connectivity; always-isomorphism adds the condition ofinformation conservation on top of these, becoming a strengthening of far-deepisomorphism for dynamically closed systems. This paper establishes their strictphysical realization in topologically ordered quantum many-body systems. Theanyon category, as a physical carrier of the inverse limit of the forgetful tower,has its modular tensor structure completely locked by time evolution under thedual constraints of a spectral gap and information conservation; the automorphisminduced by the evolution operator at the inverse limit level is the identity map.Based on this, we derive dual experimental predictions: the baseline prediction —time-translation invariance of braiding statistics under adiabatic evolution; and theincremental prediction unique to always-isomorphism — identity recovery of theanyon category under non-adiabatic unitary cycles and its recovery time bound.Cross-system comparisons of quantum dimension, topological spin, and modularmatrix provide direct experimental criteria for far-deep isomorphism. This paperestablishes clear layer-by-layer falsifiability conditions and a breakdown diagnosisframework for both types of isomorphism, transforming the Ladder of Isomorphismfrom an abstract formal statement into a strictly falsifiable physical proposition.
The far-deep isomorphism proposition asserts that systems with infinite surfaceseparation must converge in deep topology, but previously lacked quantitative predictions across different systems. This paper rigorously applies this proposition tothe cross-system comparison of topological quantum error-correcting codes and derives a new conditional conjecture: if two topological codes are drastically differentin surface structure but converge to the same equivalence class at the deep anyoncategory level, and if the functoriality axiom together with the Lipschitz continuitycondition holds, then their optimal fault-tolerance thresholds necessarily approachthe same value in the thermodynamic limit, and the deviation of thresholds atfinite sizes is bounded above by the deep topological distance. This paper takesthe square-lattice toric code and the hexagonal-lattice color code as a test pairfor the deep-convergence group, and the Z2 toric code with the Z3 toric code asa control group for the deep-divergence case. A complete numerical test schemeand a statistical operationalization procedure are provided, together with explicitcriteria for strong and weak falsification as well as a systematic diagnosis protocol after falsification. This framework advances the far-deep isomorphism from anepistemological proposition in category theory to an operational general programof physical universality.
This paper demonstrates that topological order constitutes a rigorous physicalrealization of the always-isomorphism proposition in quantum many-body systems.The always-isomorphism proposition, introduced as a cross-scale structural convergence hypothesis, asserts that for systems satisfying hierarchical generative structure and information conservation conditions, although the generative origin andthe evolutionary terminus are maximally separated in the superficial causal chain,they must converge in deep topology onto the same generative kernel. We firstestablish a formal framework and physical interpretation of this hypothesis, clearlydistinguishing three processes: renormalization group flow, dynamical time evolution, and adiabatic deformation, and a priori delineate three failure boundaries:gap closing, non-unitary environment coupling, and short-range entanglement. Onthis basis, we prove that the ground-state degeneracy stability, rigidity of longrange entanglement, and modular invariance of the anyon modular tensor categoryof topological order exactly correspond to the rigorous physical versions of the threenecessary conditions for always-isomorphism. By decomposing the theorem into anRGconvergence theorem and a dynamical preservation theorem, we eliminate conceptual confusion and provide a complete categorical equivalence proof. The toriccode model and the fractional quantum Hall effect serve as concrete examples showing the precise correspondence among the generative origin, evolutionary terminus,and deep generative kernel. Failure analysis reveals gradual failure behavior in thecritical region, offering feasible schemes for experimental testing. We conclude thatthe relationship between topological order and always-isomorphism is not an accidental analogy but a necessary coincidence of deep structures — topological orderprovides a mapping channel from category theory to condensed matter physics foralways-isomorphism, while always-isomorphism provides a genetic interpretationin the temporal dimension and hierarchical structure for topological order.
The energy gaps appearing in different physical systems– the anyonic excitation gap of Kitaev’s toric code, the first eigenvalue of the Laplacian on a compactRiemannian manifold, and the mass gap of Yang–Mills theory– traditionally belong to three isolated domains: topological order, spectral geometry, and quantumfield theory. In this paper we construct a covariant functor, called the spectralfunctor, between the category of weighted constraint networks and the category ofself-adjoint operators, and prove that the three types of gaps are not independentphenomena but manifestations of a common spectral structure on different physicalsource objects. The codomain of the spectral functor is the SA-Hilb category whichcarries self-adjoint operator information, thereby avoiding the categorical defect ofattaching spectral information externally to Hilbert spaces. Rigorous verification ofthe functor axioms ensures covariance between coarse-graining and spectral preservation. For the toric code, the gap ∆ = 4J is strictly derived from the stabilizeralgebra; under coarse-graining the effective coupling constant is strictly given bythe compression of matrix elements of the Galerkin projection, and the dimensionless gap remains invariant. For the Laplacian, the finite element discretisationresolves the conflict between negative stiffness matrix entries and the positivityaxiom of weights by distinguishing geometric weights from coupling operators; theclosed-form solution of the first eigenvalue (4/h2)sin2(h/2) converges exactly tothe continuum spectral gap, and the deviation from asymptotic commutativity ofcoarse-graining is controlled by O(H2). For Yang–Mills theory, we draw an honestboundary between classical constraint networks and quantum operators, formulatethe mass gap problem as a uniform positivity condition for the spectral gap of lattice constraint networks, and establish a one-way logical closure, clearly indicatingthe mathematical obstacles for the reverse reconstruction. The Bures informationgeometric metric based on thermal states further unifies the spectral gap and theinformation-geometric distance through two-way inequalities, making the physicalenergy gap, the operator spectral gap, and the information-geometric rigidity radius equivalent. The commutativity verification of the three examples– strictlycommutative, asymptotically commutative, and conditionally commutative– fullyconfirms the spectral functor as a categorical foundation for the transfer of stabilityacross scales.
The low-energy excitations of non-Abelian topological order are described byanyon fusion categories. In the thermodynamic limit, the algebra of local observables may form a tower of von Neumann subfactor inclusions, and the correspondence between the Jones index and the total quantum dimension of anyons hasbeen supported by Longo’s theorem in algebraic quantum field theory and by explicit calculations in Kitaev’s quantum double model. Taking this as a startingpoint, this paper does not presuppose the universal validity of subfactor structures in generic strongly correlated lattice systems; instead, it sets them as explicittheoretical hypotheses and systematically constructs a derivation chain from thediscrete sequence of Jones indices to constraints on physical excitation gap ratios.The central link is the definition of a subfactorizable Hamiltonian– expressing thelow-energy effective Hamiltonian, within an energy cutoff, as a quadratic form inTemperley-Lieb projections– thereby using the representation theory of the TLalgebra to establish a mapping between anyon excitations and Bratteli diagramnodes, leading to the conclusion that the gap ratio depends only on the Jones parameter τ. The discrete valueing of the Jones index when it is less than 4 forcesτ to be locked to a countable set, yielding the rigid prediction that within thesame topological phase region the ratio of different excitation gaps cannot driftcontinuously with external parameters. This paper designs an experimental testscheme based on scanning tunneling spectroscopy and transport measurements,and systematically provides falsification conditions, so that each hypothesis in thetheoretical framework can be independently tested. The contribution of this paper lies in extending subfactor algebra theory from exactly solvable models to atheoretical framework with operational physical predictions, through a cautioushypothesis–deduction–test structure.
Within the theoretical framework of the Ladder of Stability series, this paperadvances the rigidity classification of Jones subfactor indices and Coxeter–Dynkindiagrams into a testable quantitative prediction on topological quantum platforms.Based on the standard correspondence between the quantum dimensions of fusion categories in non-abelian topological order and subfactor inclusion indices, wepropose an effective bridging hypothesis B-J that couples the low-energy excitation gap ∆ to the Jones index [M : N] through a monotonic inverse relationship.This leads to the derivation that, within regions where topological invariants remain constant, the energy gap is locked to a set of discrete steps determined by{4cos2(π/n) | n ≥ 3}, with inter-step ratios that are universal across materialplatforms. The paper systematically analyses the spatial fluctuations of the scaling constant C, derives an upper bound nmax for the number of distinguishablelevels, and thereby strictly confines the experimental test to the low-level regionn ≤ 6. A bivariate test protocol based on scanning tunnelling spectroscopy andtransport measurements is presented, together with precise falsification conditionsthat include Kolmogorov–Smirnov statistical tests and false-positive control provisions. This work transforms the structural rigidity of operator algebras into anoperationally falsifiable empirical prediction, completing a crucial step from mathematical structure to physical observation.