arturoornelasb/Shadow-Engine: Cubical Triads Shadow Engine and Paper Draft

Abstract

Cubical Triads: A Homotopy-Type-Theoretic Foundation for Proportional Reasoning and Abductive Discovery 🌌 Overview Shadow Engine is the reference implementation of the Unified Holographic Resonance Theory (UHRT) and the core logic behind the paper, "Cubical Triads." Unlike statistical AI models (LLMs) that approximate logic via vector probability, this engine treats proportional reasoning as a strict Topological Necessity. It embeds the classical arithmetic of integer proportions into a set-truncated Higher Inductive Type framework from Homotopy Type Theory (HoTT), proving that valid semantic and physical laws are "path fillers" in a synthetic logarithmic space. Key Capabilities ** Topological Immunity:** The engine refuses to "hallucinate." If data is structurally degenerate (redundant or contradictory), it raises a Topological Obstruction rather than attempting a statistical fit or approximation. ** Abductive Discovery (New in v2.1):** It doesn't just reject errors; it rigorously diagnoses them. When an obstruction occurs, the engine mathematically calculates the Missing Integer Factor ($\delta$) required to restore "cubical resonance." This factor is a computable witness for a hidden variable. Physics Example: Predicts missing mass/constants (e.g., in the Degenerate Gravity Test). Security/Semantics: Detects structural impostors (spoofing) that mimic magnitude but lack a fundamental prime signature. ** Thermodynamics of Reason:** Defines Simplicity ($K$) not as a heuristic, but as a Boltzmann probability $K = e^{-E}$ derived from the minimal logarithmic path energy ($E$) in the fundamental $\infty$-groupoid of magnitudes ($\M$). Quick Start Prerequisites Python 3.8+ Installation git clone https://github.com/arturoornelasb/Shadow-Engine.git cd Shadow-Engine # Recommended: Create a virtual environment python3 -m venv venv source venv/bin/activate Usage Run the engine to witness the transition from Validation (Newton) to Discovery (Degenerate Gravity). python Python/shadow_Engine_v2.1.py Experiments: Topological Immunity in Action The file Python/shadow_Engine_v2.1.py contains the core logic (SyntheticShadow class) and two key experiments. 1. The Newton Test (Validation) Validates that fundamental laws ($F=ma$) correspond to identity paths ($E=0, K=1$) in the homotopy category, meaning the proportion is perfectly balanced in its simplest form. 2. The Degenerate Gravity Test (Abductive Discovery) Scenario: A triad is tested against the Gravity Law form $m_1 \cdot m_2 = G \cdot (r^2 F)$, where $G$ is an unknown integer factor $C_4'$. The inputs are structurally redundant: $r^2F=36, m_1=6, m_2=6$. Arithmetic: $6 \times 6 = 36$ is true. Shadow Engine: Detects that GCD normalization ($\gcd(36, 6, 6) = 6$) collapses the magnitude space to a point that requires a fractional solution in $\mathbb{Z}^+$. Internal Trace (Normalized): $1 \cdot 1 = 6 \cdot C_4'$ Output: [GLITCH DETECTED] Topological Obstruction. Prediction: Missing Factor: 6. Meaning: The system deduces a hidden variable (the factor of 6) is necessary to close the Kan cube and restore structural consistency. Repository Structure | Directory | Description | | :--- | :--- | | /Python | The Shadow Engine v2.1. A functional Python implementation of the core GCD-based discovery logic for empirical testing. | | /Paper | The latest $\LaTeX$ source (From GCD to Cubical Triads.tex) and PDF of the research paper. | | /Agda (Coming Soon) | Formal proofs in Cubical Agda or Lean 4 verifying the main Embedding Theorem and properties of the Higher Inductive Type $\M$. | | LICENSE | The license file (CC BY-NC 4.0). | Citation If you use this framework or theory in your research, please cite: Ornelas Brand, J. A. (2025). From GCD to Cubical Triads: A Homotopy-Type-Theoretic Reconstruction of Proportional Reasoning. Contributing This is a foundational zero-to-one project. We are looking for contributors in: Formal Verification: Porting the Python logic and theorems to a proof assistant like Lean 4 or Agda. Knowledge Graphs: Building the "Prime Dictionary" for richer semantic discovery beyond physics. Performance: Optimizing the $\gcd$ operations for massive datasets. ⚖️ License This work is licensed under the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0). See the LICENSE file for details. License Copyright © 2025 José Arturo Ornelas Brand. "Reality is the unique self-consistent configuration that does not raise a topological exception when asked to justify its own existence."