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Lectures on Algorithmic Geometrodynamics
Vol. 1 · Fragile Mechanics
Overview
Part I - Foundations
The Control Loop: Representation and Control
Part II - The Sieve (Runtime Safety)
Limits: Barriers (The Limits of Control)
Failure Modes (Observed Pathologies)
Infeasible Implementation Replacements
Part III - Implementation Architecture
Computational Considerations
TopoEncoder Architecture: Attentive Atlas with Typed Latents
Optimization: Thermodynamic Governor and Stable Updates
Part IV - Control & Belief
Belief Dynamics: Prediction, Update, Projection
Implementation Note: Entropy-Regularized Optimal Transport Bridge
Part V - Geometric Dynamics
Wasserstein-Fisher-Rao Geometry: Unified Transport on Hybrid State Spaces
Radial Generation: Entropic Drift and Policy Control
The Equations of Motion: Geodesic Jump-Diffusion
Part VI - Holography & Field Theory
The Reward Field: Value Forms and Hodge Geometry
The Causal Information Bound
Part VII - Cognitive Extensions
Theory of Meta-Stability: The Universal Governor as Homeostatic Controller
Non-Local Memory as Self-Interaction Functional
Ontological Expansion: Topological Fission and the Semantic Vacuum
Computational Metabolism: The Landauer Bound and Deliberation Dynamics
Causal Discovery: Interventional Geometry and the Singularity of Action
The Metabolic Transducer: Autopoiesis and the Szilard Engine
The Inter-Subjective Metric: Gauge Locking and the Emergence of Objective Reality
Covariant Memory Attention: Self-Attention and Lorentzian Cross-Attention for Causal Retrieval
Part VIII - Multi-Agent Gauge Theory
The Standard Model of Cognition: Gauge-Theoretic Formulation
The Parameter Space Sieve: Deriving Fundamental Constants
The Geometric Micro-Architecture
Covariant Cross-Attention: The World Model as Geodesic Integrator
End-to-End Architecture: The Universal Geometric Network
Part IX - Economics
Appendices
Appendix B
: Units, Parameters, and Coefficients (Audit Table)
Appendix C
: WFR Stress-Energy Tensor (Full Derivation)
Appendix D
: Frequently Asked Questions
Appendix E
: Rigorous Proof Sketches for Ontological and Metabolic Laws
Appendix F
: Loss Terms Reference
Appendix G: Architecture Modules Reference
Part XI - Implementation
The Covariant World Model: Geodesic Dynamics on the Poincare Ball
Geometric Dreamer — Model-Based RL on the Poincare Ball
Vol. 2 · Fractal Gas
Overview
Part I — Algorithms and Foundations
The Fractal Gas Algorithm Family
Variants of the Fractal Gas
Mathematical Foundations of Fragile
Euclidean Gas: Canonical Transition and Operator Estimates
Single Walker Observables and Probability Fields
Latent Fractal Gas: Algorithm and Analytic Conditions
Architecture
Part II — Finite-Particle Convergence
The Keystone Principle and the Contractive Nature of Cloning
Wasserstein-2 Control via Keystone-Based Variance Proxy
Hypocoercivity and Convergence of the Euclidean Gas
Convergence, Survival, and Parameter Dependence
Part III — Mean-Field Limits and Equilibrium
The Mean-Field Law of the Euclidean Gas
The Discrete Population Limit and Propagation of Chaos
Equilibrium Profiles and Their Analytical Characterization
Exchangeability, Empirical Laws, and Functional Inequalities
Part IV — Entropy, Regularity, and Bounds
Hypocoercive Entropy: From Kinetic Mixing to the Full Swarm
Mass, Hellinger, and Transport Convergence
C³ Regularity of the Fractal Gas Fitness
Smooth and Analytic Regularity of the Fractal Gas Fitness
Logarithmic Sobolev inequalities and entropy convergence
The Geometric Gas: Regularity, Stability, and Entropy
Quantitative Error Bounds
Parameter Constraints and Tuning
Part V — Fractal Set and Continuum Limits
The Fractal Set
Emergent Geometry from Adaptive Diffusion
Scutoid Spacetime: Moving Cells and Neighbor Changes
Continuum Limits: Hypotheses and Consistency
Fractal Set as Causal Set
Part VI — Fields and Emergent Physics
Direct Field Observables and Lattice QFT on the Fractal Set
Direct Observables and Standard Model Representations on the Fractal Set
Discrete Yang–Mills Actions, Noether Identities, and Quantum Reconstruction
Twistor Formulation of the Fractal Set
Curvature of the Algorithmic Fitness Metric
Field Equations and Pressure Dynamics
Algorithmic Thermodynamics, Graph Cuts, and Boundary Geometry
Expansion, Equilibrium, and Macroscopic Closure
Interactive experiments for Part VI: fields and algorithmic QFT
Part VII — Computation and Experiments
Measurement Operators on Scutoid Spacetime
Computational Proxies for Scutoid Geometry
Voronoi Wilson Loops on the Fitness Manifold
Empirical Validation of QFT Predictions
Fractal Gas QFT Calibration Notebook
QFT Calibration Report: Standard Model Parameter Mapping
QFT Calibration: Channel Knobs and Mass Plateaus
Reference Material
Frequently Asked Questions
FractalAI: Research Lineage and Empirical Comparisons
Conclusion
Index
