# Apoth3osis - Full Documentation

> Formal verification research platform bridging mathematical proofs in Lean 4 to executable computational models.

## Mission

We develop machine-verified mathematical frameworks for complex systems - biological, physical, and computational. Every theorem has a complete proof chain checked by Lean 4's type system. No hand-waving, no gaps.

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## Verified Research Projects

### 1. Semantic Closure in Lean 4

**Central Question**: What makes living systems different from machines?

A car engine converts fuel to motion but cannot repair itself. A computer executes programs but cannot rewrite its own compiler while running. Living systems do both - they maintain and reproduce themselves using processes that are themselves products of the system.

**Mathematical Framework**:
Robert Rosen's (M,R)-systems formalize this as "closure to efficient causation":
- Metabolism (M): Transforms inputs to outputs
- Repair (R): Maintains the metabolic machinery
- Replication: Produces the repair system itself

We prove in Lean 4 that these form a closed causal loop using category theory:
- Morphisms represent causal processes
- Composition captures sequential causation
- The closure condition becomes a fixed-point theorem

**Key Theorems Verified**:
1. Existence of closure morphism
2. Uniqueness under composition
3. Categorical naturality conditions

**Repository**: https://github.com/Abraxas1010/semantic-closure-lean
**Status**: VERIFIED (Lean 4 + Mathlib)

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### 2. Causal Confluence in Wolfram Physics

**Central Question**: Are determinism and relativity the same thing?

In the Wolfram Physics Project, the universe is modeled as a hypergraph evolving via local rewriting rules. Different evaluation orders create different "reference frames" - but physical predictions must be invariant.

**Mathematical Framework**:
We prove that causal invariance is equivalent to confluence from abstract rewriting theory:
- **Causal Invariance**: All evaluation orders yield isomorphic causal graphs
- **Confluence**: All reduction paths eventually converge (Church-Rosser property)

This connects Wolfram's physics model to 60 years of term rewriting research.

**Key Theorems Verified**:
1. Causal invariance implies local confluence
2. Newman's Lemma adaptation for hypergraphs
3. Equivalence of global and local properties

**Repository**: https://github.com/Abraxas1010/causal-confluence-wolfram-lean
**Status**: VERIFIED (Lean 4 + Wolfram Language)

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## Visual Models

Proof-of-concept 3D visualizations showing how verified Lean theorems translate to computational models. These demonstrate the Lean -> Lambda IR -> executable code pipeline.

### Nucleus Koopman Visualization
Spectral decomposition of dynamical systems using Koopman operator theory. Shows eigenfunction evolution on attractors.

### SKY Multiway Visualization
Causal invariant graph structures from the Wolfram model. Demonstrates branchial space and multiway evolution.

**Note**: These are interactive WebGL visualizations. For LLM consumption, the mathematical content is available in the Lean repositories above.

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## Technology Stack

- **Proof Language**: Lean 4 with Mathlib
- **Intermediate Representation**: Lambda IR (custom)
- **Target Languages**: C, WebGL/GLSL
- **Frontend**: Next.js 15, React 19, TypeScript
- **Visualization**: Three.js, WebGL

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## Academic Publications

Research papers are available at /research on the main website. Topics include:
- Formal methods in systems biology
- Category-theoretic approaches to self-reference
- Discrete physics and causal structures
- Verified scientific computing

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## API Endpoints (Machine-Readable)

For programmatic access to research data:

- `GET /api/docs/fetch` - All published content
- `GET /api/docs/search?query=<term>` - Search papers and projects

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## Repository Structure

```
apoth3osis.io/
  /research                    - Research overview page
  /research/projects/          - Individual project pages
    semantic-closure-lean/
    causal-confluence-wolfram/
  /models/                     - Visual model demonstrations
    semantic-closure/
    causal-confluence/
    nucleus-koopman/
    sky-multiway/
```

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## Contact and Collaboration

Research inquiries: https://www.apoth3osis.io

GitHub Organization: https://github.com/Abraxas1010

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## License

Research code is open source. See individual repositories for specific licenses.

## Last Updated

2025-01