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 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, 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 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 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 work constitutes the systematic and rigorous extension of the entire framework of Meta Operational Mathematics to the domain of quantum operations and their compositional inverses.Quantum mechanics, as the fundamental theory of nature at the microscopic scale, provides a uniquely rich testing ground for the meta-operational philosophy: operations that act upon quantum states—such as completely positive trace-preserving (CPTP) maps, quantum measurements, quantum instruments, and unitary evolutions—can themselves be operated upon, composed, iterated, added, multiplied, differentiated, exponentiated, logarithmized, and subjected to algebraic,analytic, and geometric transformations.We establish a complete four-level hierarchical framework tailored specifically to quantum theory: Level 0 consists of density matrices (quantum states) as elements of the base space F =Tsa(H); Level 1 consists of quantum operations as smooth mappings on F; Level 2 consists of quantum meta-operations as smooth mappings on quantum operations; and Level 3 consists of meta-meta-operations acting upon meta-operations. Within this framework, every fundamental operation of quantum mechanics is canonically lifted to a meta-operation, and the interactions among these meta-operations—through composition, pointwise addition, pointwise multiplication, differentiation, quantum randomization, exponentiation, and logarithm—generate the entire quantum operad QOp in its bornological closure.A fundamental distinction from the hyperbolic, elliptic, Gamma, Beta, Zeta, and probabilistic cases is rigorously established. In quantum theory, the base space F = Tsa(H) is not a subset of the complex plane but a non-commutative Banach space of trace-class self-adjoint operators.The compositional inverse of a quantum channel is not merely a formal inverse but the quantum error correction recovery map, which embodies the fundamental duality between CPTP maps and their Stinespring dilations. The complementary channel operation—corresponding to the environmental output of a Stinespring dilation—satisfies a twisted antipode axiom in the Hopf operad structure, reflecting the irreversible accumulation of information leaked to the environment. The completely positive constraint endows the quantum operad with a non-trivial boundary structure absent in all other cases.The eight fundamental meta-operations generating the entire quantum operad are: composition, pointwise addition, pointwise multiplication, differentiation, quantum randomization, the identity operation, the quantum channel evaluation operation, and the quantum measurement operation. Their irreducibility is proved through a careful analysis of the dependence structure of each generator on the remaining seven, using the Stinespring dilation theorem, the uniqueness of Kraus representations, and the Choi-Jamiołkowski isomorphism for complete positivity. The three essential features of quantum theory—positivity and complete positivity, complementarity and uncertainty relations, and entanglement and inseparability—are systematically elevated to the meta-operational level as algebraic axioms, analytic tools, and geometric objects, constructing a self-contained Quantum Meta-Operational Mathematics. All conjectures and open problems originally stated within this research program have either been resolved as theorems within the body of this work or are precisely formulated as remaining open problems with partial progress rigorously indicated.
Based on the two fundamental principles—Information Conservation (A1) andFinite-Step Computability (A2)—this paper derives the core structures of operatoralgebras, namely subfactors, the Jones index, and planar algebras. The researchdemonstrates that the factor structure of the observable algebra in quantum systems, the finite index of subfactor inclusions, and its discrete set of possible values(the Jones index) are not pure mathematical abstractions, but rather the inevitablealgebraic forms of organization for physical systems under information constraintsand realizability requirements. Information Conservation necessitates the existenceof a unique finite faithful trace on the algebra, forcing the observable algebra tomanifest as a type II factor. Computability requires the algebraic description tobe finitely generated, which locks the index of a subfactor inclusion to be finite andquantizes its spectrum into the classical Jones index set. This discrete spectrum directly corresponds to planar algebras (e.g., Temperley–Lieb algebras) which encodethe braiding statistics of anyons in topological quantum systems and the algebraicproperties of fault-tolerant quantum error-correcting codes. Furthermore, this paper proposes two operable experimental verification schemes: validating Jones polynomial evaluations by measuring the statistical phases of anyon braiding in quantumHall systems, and indirectly determining the effective dimension parameter relatedto the Jones index by analyzing the scaling behavior of logical error rates in surface codes implemented on superconducting quantum processors. These schemesestablish a falsifiable, unified bridge between abstract algebra, topological order,and quantum information.
This series presents a structured and logically sequential formulation of a discrete informational framework governed by global coherence constraints. Although it originates from the same research program developed in the earlier exploratory preprint series (P-series), the works presented here constitute an autonomous and self-contained series, organized in a progressively axiomatic form and focused on the explicit derivation of the framework’s effective structures. The goal of the construction is to investigate whether structures commonly associated with quantum mechanics, thermodynamics, and geometry may emerge from a discrete relational substrate without being introduced as fundamental postulates. Each article addresses a specific level of the theoretical derivation and forms part of a progressive logical structure based on results established in the preceding works of the series. This article develops the axiomatic foundation of the framework and introduces the discrete informational system underlying the construction. The paper defines the admissible configuration space, the local update structure, the global coherence constraint, and the dynamical selection principles leading to the emergence of a coherent sector. The analysis establishes the existence, invariance, and structural uniqueness of the coherent sector, together with the effective restrictions that prepare the minimal two-component structure developed in Paper 02 and the emergent quantum framework constructed in Paper 03. No quantum-mechanical, geometric, or probabilistic structure is assumed at the foundational level. The purpose of the work is to establish the minimal structural and dynamical layer from which the later effective constructions emerge.