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Ultrametric AI — Proof of Concept

Token-level explainable attention via p-adic valuation trees and Spencer-Brown's Laws of Form.

A working web application demonstrating ultrametric attention, distinction calculus, and cocycle cognitive architecture — all in one interactive interface.

What This Demonstrates

Module Description
Ultrametric Attention Attention weights computed from p-adic valuation distances on the Bruhat-Tits tree. Fully auditable — every weight has a traceable LCA path.
Distinction Calculus Spencer-Brown's primitives (mark #, enclosure [ ], Calling ##→#, Crossing [[A]]→A) applied to attention. Pure syntactic computation.
Cocycle Auditor Verifies the strong triangle inequality across token sets. The cocycle condition must hold for cognitive consistency.
Particle Zoo Explores stable syntactic patterns mapped to Standard Model particles.

Quick Start

pip install -r requirements.txt
streamlit run app.py

Then open http://localhost:8501 in your browser.

How It Works

  1. Token Encoding: Each word is assigned semantic primes (good, bad, not, very, but) via WordNet hypernyms. The prime product encodes the word as an integer whose p-adic valuations form a vector.

  2. Ultrametric Distance: Distance between two tokens is max|v_p(a) - v_p(b)| across all primes — this satisfies the strong triangle inequality (ultrametric property).

  3. Attention: Similarity = exp(-distance / temperature). Normalized row-wise. No learned parameters — the attention pattern emerges purely from the tree geometry.

  4. Cocycle Check: For any three tokens, d(a,b) ≤ max(d(a,c), d(b,c)) must hold. Violations indicate inconsistency in the cognitive space.

Architecture

ultrametric-ai-poc/
├── app.py                    # Streamlit web application (4 modes)
├── model.py                  # Attention models (ultrametric + distinction)
├── distinction_calculus.py   # Spencer-Brown primitives + tree operations
├── cocycle.py                # Cocycle verification + valuation utils
├── requirements.txt          # Python dependencies
├── README.md                 # This file

Based On

  • Quantum Laws of Form — Syntactic Token Calculus generating the Standard Model
  • Laws of Form — George Spencer-Brown (1969)
  • Bruhat-Tits Tree — Ultrametric geometry from p-adic numbers
  • Cocycle Cognitive Architecture — Brain as maintaining cognitive consistency

License

Foundational research tool. Free. No capture.