Lectures on Algorithmic Geometrodynamics

A Two-Volume Study of Bounded Intelligence and Interacting Particles

by Guillem Duran-Ballester and Sergio Hernández Cerezo

An agent has limited memory, incomplete observations, and a finite budget for computation. It must decide what to represent, how to predict, and where to spend its next step of search. These lectures study mathematical models of those choices and the geometry used to describe them.

Volume I, Fragile Mechanics, develops an agent architecture through representation, belief dynamics, control, and runtime diagnostics. Volume II, The Fractal Gas, develops interacting-particle algorithms, their convergence analysis, and geometric constructions from their histories.

You can begin with the question that interests you. The agent chapters explain how representations and control fit together. The particle chapters start from an explicit update rule and ask what can be established about the resulting population. Each volume has its own definitions, reading guide, and detailed mathematical arguments.

• Volume I: Fragile Mechanics

• Volume II: The Fractal Gas

The hands-on Control Laboratory guide is a separate documentation site with its own navigation. It accompanies the lectures; it is not a third volume.

What this book studies

A geometry tells us which states are close and what it costs to move between them. For an agent, those choices affect representation, prediction, and control. For a swarm, they affect companion selection, motion, and the interpretation of its recorded interactions. The term algorithmic geometrodynamics names the study of these relationships between computation, dynamics, and geometry.

A concrete example is the separation between moving a walker and copying one. Motion transports an existing state. Cloning changes the population allocated to that state. A mathematical description of the swarm must account for both. Likewise, an agent’s belief update can change both the location of a hypothesis and the probability assigned to it. Transport and information geometry provide ways to express these changes precisely.

The book contains definitions, model-specific theorems, proposed constructions, and numerical studies. Each claim has the scope of its stated assumptions. Connections with field theory require the variables, dynamics, and limiting regime specified in the relevant chapter. The Fractal Gas introduction distinguishes finite-particle relaxation, mean-field approximation, and geometric continuum limits before presenting those connections.

The two volumes

Volume I: Fragile Mechanics

On Geometry, Thermodynamics, and Bounded Intelligence

Volume I develops a proposed architecture for agents under partial observability and finite capacity. Its components include a structured latent representation, a world model, belief updates, a critic, a policy, and runtime monitoring. The Sieve organizes diagnostics and interventions around conditions such as stability, capacity, and grounding; the associated analyses state the conditions under which each diagnostic or intervention applies.

Topic	Entry point
State, observations, and the control loop	Foundations
Runtime diagnostics and interventions	The Sieve
Representation and implementation architecture	Architecture at a glance
Exploration and belief dynamics	Control and belief
Metrics, transport, and equations of motion	Geometric dynamics
Boundary interfaces and reward fields	Holography and field theory
Memory, adaptation, and other cognitive extensions	Cognitive extensions
Multi-agent interactions and gauge formulations	Multi-agent gauge theory
Proof of Useful Work	Economics
Encoder, world model, and planning implementation	Implementation

The volume introduction provides the full map. Its FAQ, derivations, and loss reference support detailed reading and implementation.

Volume II: The Fractal Gas

Interacting Particles, Convergence, and Emergent Geometry

Volume II starts with populations of walkers that select companions, compute fitness, clone, and undergo kinetic motion. The analysis follows the specified transition kernel through operator estimates, conditional relaxation, mean-field limits, and stronger entropy or regularity bounds. Later chapters construct the Fractal Set from walker histories and examine geometric and field models associated with that record.

Part	Subject	Central task
I	Algorithms and foundations	Define the framework, Euclidean and latent variants, and single-particle dynamics
II	Finite-particle convergence	Combine cloning, Wasserstein, and kinetic estimates for the full update
III	Mean-field limits and equilibrium	Relate population laws, marginals, quasi-stationarity, and exchangeability
IV	Entropy, regularity, and bounds	Establish stronger estimates with explicit parameter and population dependence
V	Fractal Set and continuum limits	Construct the discrete record and examine conditions for geometric limits
VI	Fields and emergent physics	Develop field constructions and their conditional physical interpretations
VII	Computation and experiments	Define observables, inspect simulations, and perform calibration
Reference	FAQ and literature	Locate explanations, historical sources, and related studies

The volume introduction explains the dependencies between these parts. The analytic arguments appear as chapters in the main reading sequence.

Choose a reading path

Start with a model you can describe operationally. For the agent, trace one observation through the representation, belief update, and action. For the gas, trace one swarm update through companion selection, cloning, and motion. Once you know what changes in a step, the estimates have something concrete to refer to.

The Full Mode view includes explanatory prose and examples. Expert Mode hides the marked explanatory additions so you can concentrate on definitions, statements, and proofs. The mathematical hypotheses are the same in both views.

Your interest	Suggested route
Agent architecture	Volume I foundations, architecture, control, and runtime diagnostics; then the implementation chapters
Optimization and sampling	Volume II algorithm introduction and Euclidean model; then finite-particle convergence and parameter constraints
Probability and analysis	Volume II Parts I–IV, keeping track of the state space, conditioning, metric, and order of limits
Geometry and fields	Volume I geometry and multi-agent chapters; Volume II Parts V–VI together with their analytic prerequisites
Numerical assessment	Volume II Part VII, returning to the definitions of the measured quantities and the hypotheses of the comparison

Direct entry points

• Agent architecture overview

• Wasserstein–Fisher–Rao geometry

• Fractal Gas algorithm introduction

• Finite-particle convergence

• Fractal Set construction

• Computation and experiments

Using the volumes together

One possible integration uses the agent’s learned representation as the space in which a swarm searches. The agent supplies a model, a metric, or a score; the swarm produces candidate trajectories and measurements. Those outputs can then inform planning or learning.

This connection has mathematical consequences. A changing learned metric changes the swarm’s transition rule. A model-generated trajectory carries the errors of that model. To apply a convergence result to the combined system, identify which quantities are held fixed, which are updated, and whether the estimates remain valid during those updates.

Volume I supplies the agent’s representation and control setting. Volume II supplies particle models whose assumptions can be checked in that setting. A result established for a fixed Euclidean transition kernel applies to an adaptive latent-space implementation only when the required hypotheses are verified for that implementation.

Following a mathematical claim

The linked chapters contain the formal statements and arguments. To assess a claim, identify four things:

1. The object. Determine whether the claim concerns an agent, an entire swarm, a one-particle marginal, an empirical measure, or a continuum field.

2. The hypotheses. Locate assumptions about the transition rule, boundary, noise, regularity, survival, and parameter regime.

3. The conclusion. Record the precise topology, metric, rate, or observable controlled by the result.

4. The dependencies. Follow the estimates used in the proof and check their scope before transferring the conclusion to another model.

For the Fractal Gas, convergence at fixed population size, propagation of chaos, and convergence of geometric operators are distinct statements. Uniformity in population size or time requires its own estimates. Conditional results retain their hypotheses, and conjectural physical identifications require further argument and empirical assessment.

Common questions

Where should I start if I want the proofs?

For particle dynamics, begin with Volume II foundations and finite-particle convergence. For agent geometry, begin with the definitions supporting the metric law and WFR geometry.

How does the agent relate to reinforcement learning?

Volume I formulates representation, belief dynamics, and control together with capacity and geometric constraints. Its introduction discusses comparisons and limits connecting this formulation to reinforcement-learning models.

What do the physics connections establish?

Their scope depends on the specified construction and argument. A symmetry or a geometric observable must be defined before its dynamics or physical interpretation can be assessed. Volume II’s field chapters should be read with their continuum conditions and mathematical status in view.

Where are detailed objections and limitations discussed?

Consult the Fragile Mechanics FAQ and the Fractal Gas FAQ, then follow the references to the relevant definitions and results.

Downloads and assisted reading

The Download prompt menu exports a volume for offline reading or use as context in an assistant. Its choices are:

Setting	Choices
Volume	Vol 1 - Agent (Fragile Mechanics) or Vol 2 - Fractal Gas
Proofs	With proofs or Without proofs
Format	Markdown (.md) or Text (.txt)

Choose With proofs when following an argument. The shorter export is suited to orientation and locating topics; consult the chapter or full export when the proof is needed. File sizes depend on the generated version.

An assistant is easiest to check when you give it a bounded task. Supply the relevant chapter and ask it to identify a definition, explain one estimate, or trace the hypotheses of a particular theorem. Keep the source open beside the answer so that you can compare the explanation with the statement.

For example:

• “Explain one Fractal Gas update using the definitions in this chapter.”

• “Which hypotheses are used when the cloning and kinetic estimates are combined?”

• “Distinguish the finite-particle QSD from the mean-field stationary law.”

• “Identify the geometric and sampling assumptions in this continuum result.”

• “For this calibration report, separate the fixed inputs from the quantities being compared.”

Citation

The DOI and citation below identify the project’s published record.

How to Cite This Work

DOI: 10.5281/zenodo.18237451

Online version: https://fragiletech.github.io/fragile/

Preferred citation:

Duran Ballester, G. (2025). Lectures on Algorithmic Geometrodynamics. Zenodo. https://doi.org/10.5281/zenodo.18237451

BibTeX:

@book{duranballester2025lag,
  author    = {Duran Ballester, Guillem and Hernández Cerezo, Sergio },
  title     = {Lectures on Algorithmic Geometrodynamics},
  year      = {2025},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.18237451},
  url       = {https://fragiletech.github.io/fragile/}
}

Note on naming: The author’s name follows Spanish naming conventions. Duran Ballester is the complete surname (paternal + maternal). Please index under D for Duran, not B for Ballester. The hyphenated form Duran-Ballester is used in English contexts to prevent parser errors. The same logic applies to Sergio Hernández Cerezo.