Foundations of Physics: Geometry and Information

This book aims to establish a physical theoretical framework based on discrete information ontology.

Foreword

• Foreword: Reconstructing Reality via Algorithms and Geometry

Book Structure

Volume I: Discrete Ontology — Physical Foundations of Information

Part I: Finite Information Axioms and State Space Geometry

Chapter 1: Holographic Principle and Finiteness Axioms

• 1.1 Rigorous Derivation of Bekenstein Bound and Planck Information Density
• 1.2 Finite Information Axiom: Proof of Finite Hilbert Space Dimension for Physical Reality
• 1.3 Failure of Continuum Hypothesis: Physical Origin of UV Divergence and Natural Cutoff
• 1.4 Wheeler’s “It from Bit” and the Ontological Status of Quantum Information

Chapter 2: Parameterized Universe and Information Geometry

• 2.1 Riemannian Structure of State Space: Construction of Quantum Fisher Information Metric (QFIM)
• 2.2 Distance Defined by Distinguishability: Physical Meaning of Fubini-Study Metric
• 2.3 Berry Curvature and Geometric Phase: Geometric Origin of Gauge Fields
• 2.4 Dynamical Flow and Symplectic Structure on Projective Hilbert Space $\mathbb{C}{P}^{N-1}$

Part II: Discrete Dynamics and Causal Structure

Chapter 3: Quantum Cellular Automata (QCA) Axiomatic System

• 3.1 Six-Tuple Definition: Discrete Background $\Lambda$, Local Algebra $\mathcal{A}$, and Update Operator $U$
• 3.2 Causal Locality Theorem: Deriving Strict Light Cone Structure from Finite Propagation Radius
• 3.3 Discrete Noether Theorem: Symmetry, Conserved Currents, and Unitarity on Lattices
• 3.4 Computational Universality: Proof of Categorical Equivalence between Physical Dynamics and Universal Quantum Turing Machine

Chapter 4: Emergence of Field Theory from Discrete to Continuous

• 4.1 Path Integral Discretization: Lattice Sum Representation of Feynman Kernel and Continuous Limit
• 4.2 Derivation of Dirac Equation: Long-Wave Limit of Quantum Walk and Emergence of Spin
• 4.3 Gauge Fields as Connection Variables: Wilson Lines and Curvature on Lattices
• 4.4 Breaking and Restoration of Lorentz Symmetry: Fixed Point Analysis of Renormalization Group Flow (RG Flow)

Part III: Mathematical Foundations and Error Control

Chapter 5: Mathematical Tools for Discrete-Continuous Duality

• 5.1 Physical Sampling Theorem: Application of Poisson Summation Formula to Band-Limited Physical Fields
• 5.2 Discrete-Continuous Error Control (DCEC): Euler-Maclaurin Expansion and Precise Control of Boundary Terms
• 5.3 Prolate Spheroidal Wave Functions (PSWF): Optimal Basis and Information Truncation under Finite Observation Windows
• 5.4 Gödel Incompleteness Manifested in Physical Prediction Horizon: Undecidability Boundary

Volume II: Emergence of Time — Scattering, Thermodynamics, and Dynamics

Part IV: Microscopic Operator Definition of Time

Chapter 6: Scattering Time Delay Theory

• 6.1 The Time Problem in Quantum Mechanics: Pauli Theorem and Non-Existence of Self-Adjoint Time Operator
• 6.2 Complete Construction of Eisenbud-Wigner-Smith (EWS) Operator $\mathsf{Q}$
• 6.3 Proof of Physical Equivalence between Dwell Time and Group Delay
• 6.4 Time Delay Matrix in Multichannel Scattering and Application of Trace Formula

Chapter 7: Spectral-Scattering Duality and Thermodynamics

• 7.1 Kreĭn Spectral Shift Function $\xi (E)$: Rearrangement Mechanism of Perturbation on Density of States
• 7.2 Birman-Kreĭn Trace Formula: Exponential Mapping Relationship between Scattering Matrix Determinant and Spectral Shift Function
• 7.3 Generalization of Friedel Sum Rule: From Fermi Liquids to General Scattering Systems
• 7.4 Levinson Theorem: Relationship between Scattering Phase Topological Number and Number of Bound States

Part V: Unified Time Identity and Relativity

Chapter 8: Phase-Delay-Density Trinity

• 8.1 Derivation of Unified Time Identity: $\kappa (E)={\phi }^{\prime }/\pi ={\rho }_{\text{rel}}=\text{Tr}\mathsf{Q}/2\pi$
• 8.2 Time is Matter: Proportional Relationship between Time Flow Rate and Local Density of States
• 8.3 Scattering Mechanism of Gravitational Redshift: Rescaling Effect of Potential Field on Local Density of States
• 8.4 Relative Scattering Determinant: Effective Density of States Calculation and Metric Reconstruction in Curved Spacetime

Chapter 9: Cosmology and Thermodynamic Time

• 9.1 Non-Hermiticity of Hamiltonian: Information Dissipation and Hubble Expansion Caused by Cosmic Horizon
• 9.2 Redshift as Phase Drift: Adiabatic Scattering Interpretation of Cosmological Redshift
• 9.3 Thermodynamic Time Arrow: Locking Entropy Increase Direction from Density of States $\rho (E)$ to Partition Function
• 9.4 Geometric Nature of Dark Energy: Background Phase Drift Caused by Vacuum Density of States and Exponential Expansion

Part VI: Topological Structure of Time

Chapter 10: Breaking of Time Translation Symmetry

• 10.1 Definition of Discrete Time Crystal (DTC): Subharmonic Response of Floquet Systems
• 10.2 ${\mathbb{Z}}_2$ Holonomy: $\pi$-Modular Pairing in Parameter Space and Topological Protection
• 10.3 Null-Modular Double Cover Structure and Time Topology
• 10.4 Discrete Skeleton of Macroscopic Continuous Time: QCA Underlying Rhythm from Time Crystals

Volume III: Entropic Origin of Gravity and Geometry

Part VII: Foundations of Geometric Dynamics

Chapter 11: Causal Geometry and Generalized Entropy

• 11.1 Geometric Properties and Conformal Structure of Small Causal Diamonds
• 11.2 Generalized Entropy Functional: Definition of Geometric Area Term and Matter Entanglement Entropy Term
• 11.3 First Law of Entanglement: Modular Hamiltonian and Energy-Momentum Tensor
• 11.4 Raychaudhuri Equation: Precise Description of Geometric Focusing Effects

Chapter 12: Entropic Variational Principle (IGVP) and Field Equations

• 12.1 Maximum Entanglement Equilibrium Axiom: Stationary Condition for First-Order Variation of Vacuum Entanglement Entropy
• 12.2 Complete Derivation of Einstein Field Equations: From Scalar Entropy Balance to Tensor Dynamics
• 12.3 Emergence of Cosmological Constant $\Lambda$: As Integration Constant of Equation of State
• 12.4 Higher-Order Gravity Theories: Deriving Lovelock Gravity and $f(R)$ Gravity from Wald Entropy

Chapter 13: Variational Completeness and Boundary Terms

• 13.1 Boundary Term Problem in Variational Principle and Mathematical Completeness
• 13.2 Entropic Origin of Gibbons-Hawking-York (GHY) Term: Partition Function of Edge Modes
• 13.3 Hayward Corner Term: Entropy Additivity on Non-Smooth Geometry

Part VIII: Spacetime Stability and Singularities

Chapter 14: Information-Theoretic Origin of Energy Conditions

• 14.1 Failure of Classical Energy Conditions and Dilemma of Semi-Classical Gravity
• 14.2 Derivation of Quantum Null Energy Condition (QNEC): Non-Positivity of Second-Order Variation of Generalized Entropy
• 14.3 Information Flow and Geometric Focusing: Proving Generalized Horizon Monotonicity Theorem Using QNEC
• 14.4 Thermodynamic Interpretation of Singularity Theorems: Geometric Caustics Caused by Information Density Saturation

Chapter 15: Microscopic Statistics of Black Hole Thermodynamics

• 15.1 Horizon as Information Truncation Surface of QCA Network
• 15.2 Microscopic Counting Derivation of Bekenstein-Hawking Entropy $S=A/4G$
• 15.3 Locking of Coefficient $1/4$: Consistency Constraints of Unitarity and General Relativity
• 15.4 Black Hole Evaporation and Page Curve: Information Recovery Mechanism under Unitary QCA Evolution

Part IX: Geometric Unification of Interactions

Chapter 16: Total Space Geometry

• 16.1 Modern Fiber Bundle Formulation of Kaluza-Klein Idea
• 16.2 Unified Connection $\mathbb{A}$: Fusion of Gravitational Spin Connection and Yang-Mills Gauge Potential
• 16.3 Curvature Decomposition Theorem: Geometric Homology between Riemann Curvature and Gauge Field Strength
• 16.4 Geometricization of Forces: Lorentz Force as Geodesic Deviation in Total Space

Chapter 17: Topological Origin of Matter

• 17.1 Fermions as Topological Knots: ${\mathbb{Z}}_2$ Holonomy Classification of Self-Referential Scattering Loops
• 17.2 Geometric Proof of Spin-Statistics Theorem: Deriving Exchange Antisymmetry from Parameter Space Topology
• 17.3 Strong CP Problem and Axion: Dynamical Relaxation Mechanism of Topological Phase
• 17.4 Geometric Interpretation of Neutrino Oscillation: Flavor Mixing Mechanism on Discrete Spacetime Lattice

Volume IV: Physics of Agency — Observers and Complex Systems

Part X: Algebraic Structure of Observers

Chapter 18: Mathematical Definition of Observer

• 18.1 From Reference Frame Origin to Physical Entity: Structured Definition of Observer
• 18.2 Internal Algebra ${\mathcal{A}}_{\text{int}}$: Finite-Dimensional von Neumann Algebra and Information Processing Capacity
• 18.3 Memory Subsystem: Stability of Markov Substructures and Anti-Decoherence Conditions
• 18.4 Predictive Model: Homomorphic Mapping and Compression of External Environment in Internal Algebra

Chapter 19: Self-Referential Dynamics and Predictive Coding

• 19.1 Mechanical Causality and Teleological Causality: Dynamical Features Introduced by Self-Referential Loops
• 19.2 Self-Referential Update Operator: State Evolution Equation Including Predictive Feedback
• 19.3 Free Energy Principle: Predictive Error Minimization as Minimization of Physical Hamiltonian
• 19.4 Strange Attractors and Life Characteristics: Steady-State Behavior of Self-Referential Systems in Phase Space

Part XI: Topological Physics of Consciousness

Chapter 20: Topological Structure in Causal Networks

• 20.1 Fundamentals of Causal Graph Theory: Directed Acyclic Graphs (DAG) and Feedback Closed Loops
• 20.2 Minimal Strongly Connected Component (MSCC): As Irreducible “Atomic Self”
• 20.3 Topological Restatement of Integrated Information Theory (IIT): Strong Connectivity and $\Phi$ Value
• 20.4 Causal Emergence and Macroscopic Scale: Why Does Consciousness Exist at Coarse-Grained Levels?

Chapter 21: Holonomy Invariants and Consciousness Phase Transitions

• 21.1 Geometric Phase and ${\mathbb{Z}}_2$ Topological Index on Parameter Space Loops
• 21.2 Topological Soliton: Consciousness as Fermion-Like Structure Topologically Protected in Causal Networks
• 21.3 Criticality of Consciousness: Topological Phase Transition from Trivial Phase ($\nu =1$) to Non-Trivial Phase ($\nu =-1$)
• 21.4 New Solution to Quantum Measurement Problem: Observation-Induced Topological Fusion of System-Observer

Part XII: Multi-Agent Systems and Objectivity

Chapter 22: Consensus Geometry

• 22.1 Wigner’s Friend Paradox and Multi-Observer Consistency Problem
• 22.2 Bayesian Gravity: Observer Model Synchronization Driven by Relative Entropy Minimization
• 22.3 Objective Reality as Nash Equilibrium: Game-Theoretic Fixed Point of Inter-Subject Information Exchange
• 22.4 Physical Nature of Language and Knowledge Graphs: Tensor Network States in Observer Networks

Volume V: Metatheory — Logic, Computation, and Experimental Verification

Part XIII: Categorical Foundations of Physics

Chapter 23: Categorical Quantum Mechanics

• 23.1 Symmetric Monoidal Category as Axiomatic Language of Physics
• 23.2 Dagger Compact Category: Unification of Unitarity and Duality
• 23.3 String Diagrams: Topological Derivation and Computation of Quantum Processes
• 23.4 Completeness Theorem: Equivalence between Diagrammatic Language and Hilbert Space Operator Calculus

Chapter 24: Topos and Physical Logic

• 24.1 Physical Theory as Object in Topos
• 24.2 Intuitionistic Logic: Logical Duality between Quantum Superposition States and Failure of Law of Excluded Middle
• 24.3 Realization of Curry-Howard-Lambek Correspondence in Physics

Part XIV: Computational Foundations and Encoding

Chapter 25: Optimality of Physical Computation

• 25.1 Landauer’s Principle and Thermodynamic Cost of Irreversible Computation
• 25.2 Fibonacci Encoding and Zeckendorf Arithmetic: Optimal Encoding for Locally Reversible Updates
• 25.3 Physical Origin of Golden Ratio: As Optimal Information Transmission Rate in Causal Networks
• 25.4 Algorithmic Information Theory and Kolmogorov Complexity of Physical Laws

Part XV: Experimental Verification and Engineering Prospects

Chapter 26: Precision Measurement Experiments

• 26.1 Microwave Scattering Network: Verifying $\kappa (E)=\rho (E)$ Identity Using Chaotic Cavities
• 26.2 Time Delay Measurement in Mesoscopic Conductors: Electron Interference and Quantum Capacitance
• 26.3 Atomic Clock Gravitational Redshift Experiment: Testing Local Density of States Rescaling Effect

Chapter 27: Astronomical and Cosmological Observations

• 27.1 Gravitational Wave Dispersion: Search for High-Frequency Lorentz Violation in LIGO/Virgo Data
• 27.2 Spectral Scintillation of Fast Radio Bursts (FRB): Statistical Fingerprint of Microscopic Spacetime Structure
• 27.3 Discrete Features in Cosmic Microwave Background (CMB): Holographic Noise at Planck Scale

Chapter 28: Artificial Consciousness and Future Physics

• 28.1 Artificial Consciousness Engineering: Neuromorphic Chip Design Based on Self-Referential Dynamics
• 28.2 Consciousness Turing Test: Detecting Machine Agency Using ${\mathbb{Z}}_2$ Topological Index
• 28.3 Open Questions of Ultimate Theory: From “Discovering Physics” to “Constructing Geometry”

Appendices

• Appendix A: Mathematical Toolbox (Quick Reference for Operator Algebras, Differential Geometry, Category Theory)
• Appendix B: Geometric Correspondence Table for Standard Model Particles
• Appendix C: Discrete-Continuous Error Control Algorithm Code Examples
• Appendix D: Symbol Index and Axiom List for the Entire Book

Postscript

• Postscript: At the End of Geometry and Information