From Structure–Symbol Sample Space to Quantum Logical Systems and Gauge Fields ——A Unified Derivation of the ΔS Framework: The Logical Emergence of Quantum Mechanics, Gauge Fields, and Gravity
This paper presents a complete derivation chain: starting from the structure–symbol sample space (p,τ,P(τ∣p))(p,τ,P(τ∣p)) and the local irreversibility ΔSΔS, without introducing any additional quantum assumptions, it progressively derives the entire core structure of quantum mechanics, and further derives quantum field theory, gauge fields, gravity, and mass generation mechanisms. The core of the derivation chain is the natural linear structure of the Fisher metric in amplitude coordinates, and the topological properties of the Berry curvature on the logical manifold. This paper does not seek to "interpret" quantum mechanics, but to demonstrate that quantum mechanics can emerge from logical systems. The paper also lists both the core problems that have been solved and the open problems that remain, enabling readers to assess the boundaries of this path. Keywords: ΔS framework; Fisher metric; Berry curvature; Quantum emergence; Gauge fields; Gravity; Mass generation; Logical systems
This paper presents a complete derivation chain: starting from the structure–symbol sample space (p,τ,P(τ∣p))(p,τ,P(τ∣p)) and the local irreversibility ΔSΔS, without introducing any additional quantum assumptions, it progressively derives the entire core structure of quantum mechanics, and further derives quantum field theory, gauge fields, gravity, and mass generation mechanisms. The core of the derivation chain is the natural linear structure of the Fisher metric in amplitude coordinates, and the topological properties of the Berry curvature on the logical manifold. This paper does not seek to "interpret" quantum mechanics, but to demonstrate that quantum mechanics can emerge from logical systems. The paper also lists both the core problems that have been solved and the open problems that remain, enabling readers to assess the boundaries of this path. Keywords: ΔS framework; Fisher metric; Berry curvature; Quantum emergence; Gauge fields; Gravity; Mass generation; Logical systems
This paper proposes and validates a revolutionary foundational physics framework, demonstrating for the first time that quantum mechanics can naturally emerge from pure principles of geometric constraints. We establish a two-level unified theory: at the top level is the Relation Matrix R_{ij} describing the abstract relational network of the universe, and at the bottom level is the Minimal Phase Closed Triangle (MPCT) network serving as the physical implementation unit. By introducing a curvature constraint action, we rigorously derive the Schrödinger equation, quantization conditions, and Planck's constant, and for the first time directly calculate the fine-structure constant from geometric principles. Core Innovations: Proposed the new paradigm "Existence is relation, relation is geometry," defining fundamental physical entities as relations rather than particles. Established an exact mathematical mapping between the Relation Matrix R and the MPCT network. Derived quantum mechanics from the curvature constraint action \mathcal{I} = \int [L_c + \frac{\lambda}{2}(\kappa - \kappa_R)^2]dt. Discovered the geometric compression factor \eta(n,l,m) and its scaling law \eta \propto n^{0.67}. Implemented strict curvature calculations based on the relation matrix, verifying \eta \propto (\text{Var}(\kappa_R))^{0.15}. Numerical simulations of the hydrogen atom system confirmed the quantized distribution of MPCT areas, discovering the new scaling law \Delta A \propto n^{3.55}. Theoretical predictions agree with numerical results within 3.4% accuracy, with the fine-structure constant calculation error at 15.2%. This work provides a novel geometric perspective for understanding the essence of quantum phenomena and indicates a possible path toward unifying quantum mechanics and gravity. Keywords: Foundations of quantum mechanics, Geometric constraints, Relation matrix, Minimal Phase Closed Triangle, Fine-structure constant, Hydrogen atom, Scaling laws, Emergent phenomena
We propose a theoretical framework in which time is described as a physical field charac- terized by a time-density field ν. In this formulation, the local flow of time is determined by ν, and gravitational acceleration arises from its spatial gradient. The theory reproduces New- tonian gravity in the weak-field limit and provides a consistent relativistic formulation through a ν-dependent metric.. By extending ν to a complex scalar field, key structures of quantum mechanics emerge naturally, including probability density, momentum, and uncertainty. Fur- thermore, the path integral formulation is interpreted as arising from the phase structure of the time-density field. The quantum potential is derived from the amplitude of ν, providing a direct mechanism for quantum phenomena such as wave packet spreading, quantization, and tunneling. These results suggest that gravity and quantum behavior may originate from a com- mon underlying structure associated with time itself. While the present formulation is not yet a complete quantum theory, it offers a unified perspective that connects classical gravity and quantum mechanics within a single framework. This version includes a minor typographical correction in the mathematical expressions. In the previous version, the exponential function was mistakenly written as "exp!" in several equations. I have corrected this to the proper notation "exp(...)". This correction does not affect the theoretical results or conclusions of the work.
We present a fundamental unified framework based on information geometry, in which quantum mechanics, statistical thermodynamics, and general relativity emerge from a single information-theoretic ontology. Starting from three rigorous axioms that establish quantum information as the primary physical substrate, we construct the universal information action functional using the Bures–Fisher–Rao metric on the manifold of quantum density matrices. Full variational derivation yields a unified field equation that governs the evolution of quantum information and emergent spacetime geometry. In appropriate limiting regimes, we recover the Schrödinger equation, equilibrium statistical mechanics, and the Einstein field equations. This work provides the foundational layer of a complete theory of everything based on the principle: geometry is 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 emergence of Hilbert-space structure and Schrödinger-type effective dynamics from the coherent two-field sector established in Paper 02. The work demonstrates that the global coherence constraints imposed on the discrete informational substrate naturally generate an effective complex representation endowed with linear superposition properties and unitary-preserving evolution. The analysis shows that the two-component coherent sector admits a mathematically stable reformulation equivalent to a Hilbert-space structure, and that the effective evolution of the coherent amplitudes leads to a Schrödinger-type dynamical equation without introducing quantum mechanics as an initial assumption. The article further derives the probabilistic interpretation associated with the effective amplitudes and clarifies how quantum-like statistical behavior emerges from the internal coherence conditions of the underlying discrete system. No quantum postulates are assumed at the foundational level. The purpose of the article is to demonstrate that the formal structures associated with quantum mechanics can emerge as effective consequences of coherence-preserving informational dynamics. The article does not assume quantum mechanics as a foundational theory and does not postulate the Hilbert-space formalism a priori. Instead, the work demonstrates that, within the internal hypotheses and coherence constraints of the proposed discrete informational framework, structures mathematically equivalent to Hilbert-space representations and Schrödinger-type evolution can arise as effective emergent descriptions. The purpose of the derivation is therefore not to reproduce standard quantum mechanics by assumption, but to show that quantum-like formal structures may consistently emerge from deeper coherence-preserving informational dynamics.
The companion paper (Tang, 2026) showed that the Self-Consistent Reference Frame Principle (RCC), applied to local Hilbert-space fibres, yields the Born rule, the Yang--Mills action, and the Einstein--Hilbert action as consequences of reference-frame self-consistency and causality. (In that derivation, "quantum foundations" refers to the Born rule and the emergence of Hilbert-space structure; unitary dynamics, the tensor-product rule for composite systems, and the measurement postulate lie outside its scope.) The present paper removes the linearity assumption. It reformulates RCC on nonlinear state manifolds---smooth manifolds without a preferred linear or metric structure, on which symmetry groups act by diffeomorphisms. The core object is the nonlinear comparator, a G-equivariant diffeomorphism between fibres that translates reference frames. Each result of the linear paper---the Born-rule probability formula, the gauge connection, the gravitational metric, and the phase unity identity---is reconstructed in this more general setting and reduces to its linear counterpart precisely when the state manifold carries a Kähler structure, the geometric condition under which P=|⟨φ|ψ⟩|² is recovered. (Unitary dynamics, the tensor-product rule, the observable algebra, and the measurement postulate remain external inputs.) The nonlinear framework reveals structures inaccessible in the linear case: the Born rule generalises to a family of G-invariant transition functions; gauge theory arises as a special case of fibre-bundle geometry with a nonlinear G-action; nonlinear comparator corrections---torsion and anholonomy effects from the curvature of the fibre---generate additional SMEFT operators, most notably O₃B with C₃B≠0; and the gravitational sector receives subleading corrections from fibre-valued geometric invariants. The nonlinear framework yields a strictly larger set of falsifiable predictions: the linear results are recovered in the Kähler limit, while the deviations constitute a new experimental window into the geometry of state space.
This work introduces Structural Differentiation Quantum (SDQ) as an extension of Structural Differentiation Cosmology (SDC) and Structural Differentiation Gravity (SDG). In this framework, quantum phenomena are interpreted as manifestations of microscopic fluctuations in the structural density field C(x,t). The wavefunction is not treated as a purely probabilistic entity, but as a representation of local structural states. We show that quantum behavior naturally emerges from structural fluctuations δC(x,t), while gravitational behavior arises from ∇C, and cosmological evolution is governed by C(t). This leads to a minimal unified structural framework in which: - Cosmology corresponds to large-scale evolution of C(t)- Gravity corresponds to gradients ∇C- Quantum phenomena correspond to fluctuations δC Figure 6 illustrates this unification as a scale-dependent structural hierarchy. This work represents version v1.3 of the SDQ framework and serves as a foundational preprint connecting structure-based cosmology, gravity, and quantum interpretation. Keywords: structural cosmology, quantum structure, emergent quantum mechanics, structural field theory, unified framework