Module foundations 

Chapter 0: Foundations

This chapter builds all mathematical prerequisites from first principles, assuming no prior knowledge beyond elementary set theory.

Purpose

The Atlas of Resonance Classes and its embedding into E₈ is a deep mathematical structure. To understand it rigorously, we must establish foundations carefully.

This chapter provides:

• Basic definitions: Graphs, groups, vector spaces
• Exact arithmetic: Rational numbers, no floating point
• Action principles: Variational calculus and stationary configurations
• Resonance classes: The 96 equivalence classes forming Atlas vertices
• Category theory: The language for describing Atlas initiality (future section)

Reading Guide

For mathematicians: This chapter is elementary but establishes notation and conventions. Skim §0.1-0.2, focus on §0.3 (resonance classes) and §0.4 (categories).

For physicists: The action functional in §0.2 will be familiar. The 12,288-cell complex may be new—it’s a discrete analog of field theory configuration spaces.

For computer scientists: Note the emphasis on exact arithmetic (§0.1.2) and discrete optimization (§0.2.4). The code serves as executable definitions.

Chapter Organization

• §0.1 Primitive Concepts: Graphs, arithmetic, groups, vectors
• §0.2 Action Functionals: Variational calculus, the 12,288-cell complex
• §0.3 Resonance Classes: The 96 equivalence classes, label system
• §0.4 Categorical Preliminaries: Categories, functors, initial objects

Main Results

This chapter establishes:

1. Exact arithmetic framework (§0.1.2): All computations use rationals, no floats
2. Action functional (§0.2.3): Defined on 12,288-cell boundary complex
3. 96 resonance classes (§0.3.2): Stationary configuration partitions into exactly 96 classes
4. Label system (§0.3.3): Each class labeled by 6-tuple (e₁,e₂,e₃,d₄₅,e₆,e₇)
5. 8D extension (§0.3.4): Labels extend uniquely to E₈ coordinates

These results are computational discoveries, not assumptions. The number 96 emerges from optimization, not by design.

Connection to Main Theorem

The Atlas initiality theorem (proved in later chapters) states:

The Atlas is the initial object in the category of resonance graphs, from which all exceptional Lie groups emerge through categorical operations.

This chapter provides:

• The Atlas itself (96 resonance classes from §0.3)
• The categorical language to state initiality (§0.4)
• The first-principles approach ensuring no circular reasoning

Historical Context

The Atlas emerged from research by the UOR Foundation into invariant properties of software systems under the Universal Object Reference (UOR) framework. The discovery that fundamental mathematical structures arise from informational action principles was unexpected.

This chapter reconstructs the discovery path: starting only with an action functional, we derive the 96-vertex structure without assuming Lie theory.

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Navigation:

• Next: §0.1 Primitive Concepts
• Up: Main Page

Re-exports

pub use action::ActionFunctional;

pub use action::Complex12288;

pub use action::Configuration;

pub use categories::Functor;

pub use categories::Morphism;

pub use categories::Product;

pub use categories::Quotient;

pub use categories::ResonanceGraph;

pub use primitives::KleinElement;

pub use primitives::SimpleGraph;

pub use resgraph::ResGraphMorphism;

pub use resgraph::ResGraphObject;

pub use resonance::extend_to_8d;

pub use resonance::generate_all_labels;

pub use resonance::AtlasLabel;

pub use resonance::ResonanceClass;

Modules

action

Chapter 0.2: The Principle of Least Action

categories

Chapter 0.4: Categorical Preliminaries

primitives

Chapter 0.1: Primitive Concepts

resgraph

ResGraph Category: Formalization of Resonance Graphs

resonance

Chapter 0.3: Resonance and Equivalence