Classical information theory and quantum information theory are respectively based on Boolean algebra and Hilbert space projection lattices, lacking a unified logical framework for cross-scale physical propositions covering classical, mesoscopic and quantum domains. Traditional physics attributes the classical-quantum transition to continuous spatial scale variation, which is essentially external parameter determinism and cannot explain the peculiar intermediate characteristics of mesoscopic systems. This paper constructs the philosophical and mathematical foundation of mesoscopic information theory based on the coupling of S4 modal logic and the non-distributive modular lattice M₃. A composite axiom system is proposed, and Axiom 3 (a–g) realizes the consistent embedding of S4 modal operators on non-distributive lattices. The cross-scale evolution of physical systems is reinterpreted as discrete phase transition of propositional lattice types, with an endogenous CHSH-type correlation threshold τ≈2.414 serving as the operational criterion for mesoscopic phase. A Möbius-weighted lattice entropy independent of Shannon and von Neumann entropy is established to characterize structural information of mesoscopic logical systems. Noise and decoherence are redefined as intrinsic signals of lattice structural transition rather than simple information dissipation. This study establishes the independent ontological status of mesoscopic propositional systems, breaks the scale-phenomenological paradigm in traditional cross-scale physics, and provides a new foundational framework for the unification of cross-scale information theory.
Classical information theory and quantum information theory are respectively based on Boolean algebra and Hilbert space projection lattices, lacking a unified logical framework for cross-scale physical propositions covering classical, mesoscopic and quantum domains. Traditional physics attributes the classical-quantum transition to continuous spatial scale variation, which is essentially external parameter determinism and cannot explain the peculiar intermediate characteristics of mesoscopic systems. This paper constructs the philosophical and mathematical foundation of mesoscopic information theory based on the coupling of S4 modal logic and the non-distributive modular lattice M₃. A composite axiom system is proposed, and Axiom 3 (a–g) realizes the consistent embedding of S4 modal operators on non-distributive lattices. The cross-scale evolution of physical systems is reinterpreted as discrete phase transition of propositional lattice types, with an endogenous CHSH-type correlation threshold τ≈2.414 serving as the operational criterion for mesoscopic phase. A Möbius-weighted lattice entropy independent of Shannon and von Neumann entropy is established to characterize structural information of mesoscopic logical systems. Noise and decoherence are redefined as intrinsic signals of lattice structural transition rather than simple information dissipation. This study establishes the independent ontological status of mesoscopic propositional systems, breaks the scale-phenomenological paradigm in traditional cross-scale physics, and provides a new foundational framework for the unification of cross-scale information theory.
This paper lays out the philosophical foundation of mesoscopic information theory. Taking S4 modal logic and the M₃ diamond modular lattice as its mathematical skeleton, this work puts forward four modal‑probability bridging axioms which form a minimally‑sufficient axiom set: the modal necessity axiom, the modal‑probability consistency axiom, the modal‑probability regularity axiom, and the distributive‑law breaking axiom. This constraint system realizes a controlled transition from the Boolean base of S4 to the non‑Boolean M₃ lattice, embedding modal operators and probability measures within one unified formal framework. Global completeness is not pursued; only internal self‑consistency and traceable reasoning are required. This axiomatic apparatus yields a unified philosophical account of mesoscopic‑level features including modal tension, quasi‑quantum‑like behaviors and informational phase transitions. Empirical adequacy and real‑world case studies are reserved for future versions.
This paper reconstructs its philosophical foundation of Modal Information Theory (MIT). A complete deductive chain is constructed through Stone duality of Boolean lattices, Alexandroff topology and S4 modal logic correspondence, four modal-probability bridging axioms confining the structure to M3 diamond lattice, two-dimensional Hilbert space quantization, Gleason-Busch theorem and Tsirelson bound. The midpoint of the classical-quantum interval is defined as the mesoscopic order parameter η=1+√2≈2.4142, namely the silver critical value. Serving as the theoretical basis between classical and quantum regimes, it provides fundamental mathematical criterion for Liu Entropy and validates its dedicated mesoscopic information unit mobit. The paper demonstrates the minimality of the axiom system and extremal origin of the critical value, establishes partitioning criteria for classical, mesoscopic and quantum domains from both philosophy of science and quantum logic perspectives, and solidifies the axiomatic system of Modal Information Theory.
Addressing cross-disciplinary fundamental issues including paradigm separation between classical and quantum observation, logical drift in general artificial intelligence probabilistic frameworks, and ambiguous uncertainty tracing in information systems, this study proposes a universal axiomatic framework of the binary excluded middle law and establishes a dual theoretical architecture consisting of ontology entropy and cognitive entropy. Strict logical demonstration verifies that all objective physical systems, regardless of microscopic or macroscopic scale, obey binary exclusive truth-value constraints without inherent superposition states or intrinsic probabilistic uncertainty. Phenomena such as quantum superposition, observational fuzziness, and probabilistic ambiguity are cognitive representations arising from information isolation and insufficient observational coupling, rather than inherent ontological properties of physical systems. This study redefines the essential nature of observation as a threshold-induced phase transition process, in which continuous information aggregation reduces cognitive entropy and eventually drives the system toward a deterministic binary steady state. Under this dual-entropy paradigm, the Schrödinger’s cat superposition paradox is fully and self-consistently resolved, eliminating long-standing logical ambiguities in quantum observation interpretation. This paper further conducts cross-domain adaptability analysis and constructs a unified fundamental logical framework for physical observation, intelligent cognition, information evaluation, and numerical computation, while outlining a phased systematic research roadmap. The results indicate that the binary excluded middle law serves as a universal underlying logical meta-rule for natural and intelligent information systems. The proposed dual ontology–cognitive entropy system fundamentally unifies the paradigms of classical and quantum physics, natural science, and artificial intelligence, possessing significant fundamental theoretical innovative value and broad engineering application prospects.
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, authored by Aidan McGill in October 2024, proposes a novel reinterpretation of quantum decoherence by framing it as a physical phase transition in probability distributions. It challenges conventional interpretations, such as the Copenhagen and many-worlds theories, which rely on the concept of wave function "collapse" driven by an external observer or measurement. Key aspects of this framework include: Observer-Independent Decoherence: The model removes the necessity of an observer, proposing that classical behavior is an emergent property of quantum complexity rather than a separate reality. The Phase Transition Mechanism: Decoherence is defined as a transition in the probability distribution of a system's wave function, triggered when interactions with external systems cross a specific threshold of complexity. Information and Energy Dynamics: The shift from quantum to classical behavior is driven by informational exchanges and energy transfers between a system and its environment. Empirical Alignment: The framework is supported by real-world examples, including buckyball double-slit experiments and superconducting qubits, which demonstrate quantum effects in macroscopic systems. Consistency with the Born Rule: This reinterpretation maintains the probabilistic outcomes of standard quantum mechanics and remains fully consistent with the Born rule. By focusing on the intrinsic properties of system interactions, this work offers a unified view where quantum mechanics governs all scales, and classical reality is a byproduct of informational complexity.
Classical information theory, built on Shannon entropy, and quantum information theory, built on von Neumann entropy, quantify uncertainty and compression properties of probability distributions and density operators. Both are powerful, but both treat information as a property of states of individual systems or ensembles. In this manuscript we argue that entanglement—the pattern of non-classical correlations between systems—is more fundamental than either Shannon or von Neumann information. We motivate this claim with a condensate thought experiment in which many bosons occupy the same single-particle state: Shannon and von Neumann entropies assign them identical information content, yet they can be distinguished by their distinct entanglement histories with external systems. Formally, states with identical marginal density operators but different purifications represent different global information, accessible only through their entanglement structure. Entropic quantities are then seen as coarse-grained summaries of this underlying relational information. We connect this perspective to an entanglement-geometric view of physics, including Entanglement Manifold Cosmology (EMC), in which spacetime and physical “truth” emerge from the geometry of correlations. The goal is an information-theoretic narrative that is rigorous enough for physics and information theory while readable to a non-specialist who is comfortable with basic linear algebra and quantum mechanics.