This paper aims to provide a unified physical origin explanation for topological order that transcends the Landau paradigm, starting from principled axioms.Based on two fundamental axioms—Information Conservation (A1) and Finite-StepComputability (A2)—we argue that topological order is not an accidental productof specific models, but rather a macroscopic quantum structure that inevitablyemerges from quantum many-body systems under the joint action of A1 (globalinformation constraints) and A2 (local interaction rules) in order to achieve robustnon-local information storage and processing. This paper systematically demonstrates that core features such as topological degeneracy, anyonic fractional statistics, and topological entanglement entropy can be logically derived from these twoaxioms, and integrates topological order with many-body dynamics from a unifiedspatiotemporal perspective, further proposing numerical and experimental testingschemes. This framework provides a new principled foundation for understandingthe nature, classification, and role of topological order in quantum computing andquantum gravity
This paper aims to provide a unified physical origin explanation for topological order that transcends the Landau paradigm, starting from principled axioms.Based on two fundamental axioms—Information Conservation (A1) and Finite-StepComputability (A2)—we argue that topological order is not an accidental productof specific models, but rather a macroscopic quantum structure that inevitablyemerges from quantum many-body systems under the joint action of A1 (globalinformation constraints) and A2 (local interaction rules) in order to achieve robustnon-local information storage and processing. This paper systematically demonstrates that core features such as topological degeneracy, anyonic fractional statistics, and topological entanglement entropy can be logically derived from these twoaxioms, and integrates topological order with many-body dynamics from a unifiedspatiotemporal perspective, further proposing numerical and experimental testingschemes. This framework provides a new principled foundation for understandingthe nature, classification, and role of topological order in quantum computing andquantum gravity
This paper starts from two fundamental principles—Information Conservation(A1) and Finite-Step Computability (A2)—to argue the mechanism for the naturalemergence of inherent lower bounds of computational complexity in physically realizable computational processes. Research shows that core concepts of computationalcomplexity (such as polynomial-time upper bounds, circuit depth lower bounds) arenot mere mathematical abstractions but are quantitative laws that must be followedby physical systems when performing information transformations under fundamental constraints such as energy, time, and information conservation. Specifically, thequantum speed limit together with Axiom A2 jointly sets a polynomial-time upperbound for any finite-energy physical system; while the Information Conservationaxiom (A1) combined with the principle of minimal physical realization complexity(A3) ensures that the logical difficulty of a computational task is inevitably reflected in the topological complexity of the spatiotemporal structure of its physicalrealization. This sets a lower bound for the circuit depth determined by topological invariants, and in quantum many-body systems, it directly manifests asvolume-law growth of entanglement entropy, whose coefficient is determined by theunderlying hardware architecture. To test this theoretical framework, this paperdesigns two cross-scale experimental paths: at the microscopic level, using superconducting quantum processors to precisely measure the volume-law coefficient ofentanglement in random quantum circuits; at the macroscopic cosmological level,inferring the “computational complexity” growth information of the early-universeinflationary process by detecting the phase coherence of the primordial gravitationalwave background. These schemes aim to place foundational issues of computationalcomplexity within the domain of empirically testable physical science.
This paper proposes a theoretical framework for topological order emergencebased on three criteria: structural stability (S), emergent entropy (E), and decoherence resilience (D). Through axiomatic definitions of the mathematical formsand physical connotations of each criterion, a phenomenological criterion system fortopological order emergence is established. The article systematically explores thephysical origin and universality challenge of the threshold value 0.5, analyzes thecoupling relationships and independence conditions among the criteria, constructsa specific correspondence bridge with category theory invariants, and designs a setof jointly measurable experimental verification schemes. Furthermore, this framework is placed within the context of quantum information theory, proposing thatthe emergence of topological order may correspond to the fundamental limits ofquantum information processing, thereby providing new theoretical perspectivesand operational tools for the identification, classification, and detection of topological phases.
This paper, based on two fundamental physical axioms—Information Conservation (A1) and Computability (A2)—systematically argues for the intrinsic logical structure necessary for describing physical reality. Research shows that in aself-referential universe model constrained by local observations, the truth-valuejudgment logic of physical propositions necessarily deviates from classical Booleanlogic. The core root lies in the “local-global inequivalence” of physical systems: theinformation contained in the global physical state cannot be completely determinedby any finite observational procedure within a finite spacetime region. This leads tothe invalidity of the law of excluded middle within the system, and consequently,the logical structure mathematically manifests as a non-distributive orthomodularlattice, i.e., Birkhoff–von Neumann quantum logic. This structure is proven tocorrespond exactly to the subobject lattice in the ∞-topos TA constructed previously. This theoretical framework finally leads to a testable cosmological prediction:the central value of the non-Gaussianity parameter fNL of the cosmic microwavebackground (CMB) temperature fluctuations is about −3.2. Next-generation CMBexperiments (such as CMB-S4) will be able to test this prediction precisely, andthe results can be used to evaluate whether the quantum logical structure emergingfrom the basic axioms is suitable for describing the real universe.
This paper aims to establish a general theoretical framework that, starting fromtwo basic axioms about the realizability of the physical world, deduces the necessary form of algebraic structures required to describe quantum systems. These twoaxioms are: Generalized Information Flow Constraint (G1) and Finite-Step Computability (G2). We prove that for open quantum systems (non-Hermitian systems)incorporating dissipation and gain, their observable algebra must be organized intoa generalized factor with a specific “weight” structure. Under the requirement offinite generation (G2), the “relative size” index of subalgebra embeddings exhibitsdiscreteness, which constitutes the mathematical foundation for the emergent generalized braiding algebras and dissipative quantum error-correcting codes in a classof non-Hermitian topological orders. This work not only provides a unified algebraicperspective for understanding non-Hermitian topological matter but also proposesexperimental verification ideas based on dissipative topological superconductingsystems and noisy quantum processors.
This paper proposes and develops a research framework for deriving fundamental mathematical structures from underlying physical axioms. We prove that,starting from two basic principles that a self-referential computational system mustsatisfy Information Conservation (A1) and Computability (A2) the mathematicaldescription of its evolutionary process is necessarily organized into a specific category. The objects of this category correspond to discrete semantic levels of thesystem state, and the morphisms are completely positive trace-preserving (CPTP)maps, which directly embody the constraint of information conservation. The tensor product structure and dagger structure possessed by the category originate,respectively, from the parallel operations permitted by computability and the formal time-reversal symmetry implied by information conservation. Ultimately, wesystematically derive a complete dagger-compact category. Distinct from purelymathematical constructions, the credibility of this framework is rooted in an empirically testable physical prediction: in a quantum processor, the complexity growthdynamics of the system state should exhibit a periodic oscillation pattern related tothe logarithm of the system size (ln N). This explicit prediction provides a crucialfalsification condition for the theory.
This paper constructs a unified theoretical framework and classification system for the emergence of topological order. This framework, with structural stability (S), information efficiency (E), and dynamic accessibility (D) as its core criteria, systematically describes the emergence laws and universal classification from discrete symmetry to continuous Lie group topological phases. By establishing rigorous mappings between these criteria and topological invariants such as quantum dimension, chiral central charge, and modular matrices, this work connects the mathematical essence of topological order, its physical robustness, and its potential as a quantum information resource. The framework not only provides an integrated perspective for understanding the emergence mechanisms of topological order but also offers quantitative assessment tools and systematic guidance for theoretical screening, experimental detection, and implementation strategies of topological states on various physical platforms (such as frustrated magnets, topological superconductors, cold atoms, and superconducting quantum circuits).