Triadic Neurosymbolic Engine

A deterministic algebraic framework for neurosymbolic validation, semantic projection, and AI model auditing.

Cosine similarity tells you "King and Queen are 0.87 similar" — a black-box number.

The Triadic Engine tells you "King = 2×3×5 and Queen = 2×5×7. They share {2,5} (Royalty). King has {3} (Male) that Queen lacks. Queen has {7} (Female) that King lacks." — fully transparent, deterministic decomposition.

Why not cosine similarity?

Install

pip install triadic-engine

# With optional extras
pip install "triadic-engine[dashboard]"  # Streamlit dashboard
pip install "triadic-engine[api]"        # FastAPI server

Quickstart

from neurosym import ContinuousEncoder, DiscreteMapper, DiscreteValidator

encoder = ContinuousEncoder("all-MiniLM-L6-v2")

# Choose a projection mode:
mapper = DiscreteMapper(n_bits=8, projection="pca")        # Deterministic, corpus-adapted
# mapper = DiscreteMapper(n_bits=8, projection="random")    # Classic LSH
# mapper = DiscreteMapper(n_bits=8, projection="consensus") # Multi-seed noise filtering
# mapper = DiscreteMapper(n_bits=8, projection="contrastive",   # Supervised
#                         hypernym_pairs=[("Animal","Dog"), ("Vehicle","Car")])

concepts = ["King", "Queen", "Man", "Woman"]
embeddings = encoder.encode(concepts)
prime_map = mapper.fit_transform(concepts, embeddings)

validator = DiscreteValidator()

print(validator.subsumes(prime_map["King"], prime_map["Queen"]))
# → False (King does not contain ALL features of Queen)

print(validator.explain_gap(prime_map["King"], prime_map["Queen"]))
# → {"shared": 10, "only_in_a": 3, "only_in_b": 7, "a_contains_b": False, "b_contains_a": False}

print(validator.compose(prime_map["King"], prime_map["Queen"]))
# → LCM of both — a new integer containing all features of King AND Queen

# Analogy: King:Man :: Queen:?
result = validator.analogy_prediction(prime_map["King"], prime_map["Man"], prime_map["Queen"])
print(result.output_value)  # → predicted integer for "Woman"

How It Works

Text → Neural Embedding → LSH Hyperplanes → Composite Prime Integer
         (R^384)            (k projections)      (Φ(x) = ∏ pᵢ)

Each concept becomes a single integer whose prime factors are its semantic features. This enables three operations impossible under cosine similarity:

Projection Modes

Core Modules

Use Cases

Explainable RAG — Instead of returning top-k by cosine score, return documents whose prime signatures subsume the query signature. Every result is provably relevant.

AI Model Auditing — Detect when two LLMs structure the same concept differently. The engine found 108,694 discrepancies auditing 2M semantic chains across two embedding models.

Semantic Deduplication — Two records are semantically duplicate if Φ(A) mod Φ(B) == 0. Exact, not probabilistic.

Compliance Validation — Verify that "GDPR" subsumes "consent" and "data-subject-rights" in your ontology. Machine-checkable, not fuzzy.

Anomaly Detection — Tabular rows that break the multiplicative patterns of their peers are flagged as anomalies — with a proof, not just a score.

Interactive Dashboard

pip install "triadic-engine[dashboard]"
triadic-dashboard

Six tabs: Ingestion & Encoding, Semantic Graph, Logic & Search, AI Auditor, Anomaly Detection, Benchmarks

The AI Auditor compares how different embedding models structure the same concepts using topological shortest-path differencing — finding exact structural discrepancies between models.

REST API

pip install "triadic-engine[api]"
uvicorn api.server:app --host 0.0.0.0 --port 8000

Interactive docs at http://localhost:8000/docs (Swagger UI).

CLI Tools

# Massive topological audit (model vs model)
python scripts/triadic_auditor.py --input examples/data/wordnet_2k.csv --col concept --output reports/audit.csv

# PCA vs Random vs Consensus vs Contrastive benchmark
python scripts/benchmark_pca.py

Benchmarks

Academic Paper

Full paper with 9 experiments: paper/

make paper   # requires pdflatex + bibtex

Citation

Paper

Ornelas Brand, J. A. (2026). Triadic Neurosymbolic Engine: Prime Factorization as a Neurosymbolic Bridge: Projecting Continuous Embeddings into Discrete Algebraic Space for Deterministic Verification. Zenodo. https://doi.org/10.5281/zenodo.19205805

@article{ornelas2026prime,
  author       = {Ornelas Brand, J. Arturo},
  title        = {Triadic Neurosymbolic Engine: Prime Factorization as a
                  Neurosymbolic Bridge: Projecting Continuous Embeddings
                  into Discrete Algebraic Space for Deterministic Verification},
  year         = 2026,
  month        = mar,
  doi          = {10.5281/zenodo.19205805},
  url          = {https://doi.org/10.5281/zenodo.19205805}
}

Repository

Ornelas Brand, J. A. (2026). Prime Factorization as a Neurosymbolic Bridge: Projecting Continuous Embeddings into Discrete Algebraic Space for Deterministic Verification (Repository) (0.1.0). Zenodo. https://doi.org/10.5281/zenodo.18748671

@software{ornelas2026triadic,
  author       = {Ornelas Brand, J. Arturo},
  title        = {Prime Factorization as a Neurosymbolic Bridge: Projecting
                  Continuous Embeddings into Discrete Algebraic Space
                  for Deterministic Verification (Repository)},
  year         = 2026,
  month        = feb,
  version      = {0.1.0},
  doi          = {10.5281/zenodo.18748671},
  url          = {https://doi.org/10.5281/zenodo.18748671}
}

Companion Work

This engine is the first in a series of four publications on triadic neurosymbolic methods:

triadic-microgpt — End-to-End Prime Factorization in a Generative Language Model

A 40M-parameter GPT learns composite prime signatures alongside language modeling, achieving 98% analogy verification and 100% signature uniqueness.

	DOI
Paper
Repository

reptimeline — Tracking Discrete Representation Evolution During Neural Network Training

Tooling for tracking how discrete representations evolve during training — lifecycle events, phase transitions, bottom-up ontology discovery, and causal intervention verification.

	DOI
Paper
Repository

Triadic Emergent Duality — 14+ Candidate Dualities Across 6 Algebraic Layers

14+ candidate dualities across 6 algebraic layers, evaluated through domain analysis (8 disciplines + 3 negative controls) and neurosymbolic learning (GPT-2 + 72-bit triadic head).

	DOI
Paper
Repository

Project Structure

├── src/neurosym/          ← Core Python package (pip installable)
├── api/                   ← FastAPI REST server
├── app.py                 ← Streamlit interactive dashboard
├── paper/                 ← Academic paper (LaTeX, 12 pages)
├── scripts/               ← CLI auditing & benchmark tools
├── tests/                 ← Test suite
├── notebooks/             ← Reproducibility demo (Jupyter)
├── examples/              ← Sample datasets (WordNet, e-commerce)
└── pyproject.toml         ← Package metadata & dependencies

License

Business Source License 1.1 (BUSL-1.1)

All users must contribute improvements back. See TERMS.md. Companies: see COMMERCIAL.md for the consortium participation model.

Change Date: 2030-03-21 — auto-converts to AGPL-3.0.

Contact: arturoornelas62@gmail.com

© 2026 J. Arturo Ornelas Brand