[README.md]

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Toward Homological Efficiency: An Architecture for Hadamard Discovery

"Dedicated to the pursuit of absolute balance in the $n=140$ abyss."

Abstract

This project implements a multi-tiered computational framework for the discovery of Hadamard matrices. By treating the search as a mapping problem between Group Cohomology, Williamson Transports, and Diophantine Filters, we significantly mitigate search complexity. We report the rapid synthesis of the historic $n=92$ Hall-Baumert order in 1.40s.

Project Architecture

The engine is built on three fundamental pillars:

• Cocyclic Framework: Solving for 2-cocycles $\psi: G \times G \to \mathbb{Z}_2$ to reduce search space from $2^{n^2}$ to $2^k$.
• Williamson Transport: Mapping the problem onto four circulant matrices ($A, B, C, D$) and evaluating via the Periodic Autocorrelation Function (PAF).
• Arithmetic Pruning: An "Intelligence Layer" utilizing the Sum of Four Squares Theorem ($s_A^2 + s_B^2 + s_C^2 + s_D^2 = 4n$) to discard invalid candidates before expensive evaluation.

Repository Structure

Based on the:

├── Cargo.toml                # Rust dependencies (Rayon, Serde, etc.)
├── Toward_Homological_...pdf  # Final Academic Whitepaper
├── docs/                     # Research and Explainers
│   ├── Grok4-esplains-full.md # Symbolic derivation of Hadamard magic
│   └── WHITEPAPER.md         # Source markdown for the formal paper
├── src/                      # High-performance search engine (Rust)
│   ├── williamson92.rs       # n=92 Hall-Baumert solver
│   ├── williamson44.rs       # n=44 Optimized solver
│   └── n=140.rs              # The n=140 search manifold
└── src/py-visuals/           # Visualization layer (Python)
    ├── visualizer.py         # Heatmap generation
    └── result.json           # Discovered sequence data

Performance Benchmarks

Our engine demonstrates that algebraic selection outpaces brute force: Order (n)Primary SymmetrySearch TimeManifold Reduction32$H^2(D_{16}, \mathbb{Z}_2)$7.38s$2^{1024} \to 2^{32}$92Williamson MITM1.40s$O(N^4) \to O(N^2)$44Williamson + Filter2.44ms90% initial prune

Getting Started

Prerequisites

Rust: cargo 1.70+ Python: matplotlib, numpy, scipy (for visualization)

1. Run the Search EngineNavigate to the source directory and execute the $n=92$ solver:

cargo run --bin williamson92

This will output a result.json containing the discovered sequences.

2. Visualize Results

Render the success as a high-resolution heatmap:Bashpython src/py-visuals/visualizer.py

The $n=140$ Event Horizon At $n=140$, the search manifold ($N \approx 1.1 \times 10^5$) exceeds the memory limits for standard $O(N^2)$ Hash Maps on consumer-grade hardware (64GB RAM).Future work focuses on:GPU Kernels: Moving the "Square-Sum Sieve" to CUDA architectures.

Partitioned Sector Search: Further categorical reduction of the $n=140$ manifold.

Citation If you use this architecture in your research, please cite the included whitepaper: whisprer & Google GeminiPro3.0. (2025). Toward Homological Efficiency: An Architecture for Hadamard Discovery.

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