triadic-engine 0.4.0

Deterministic AI auditing and semantic validation via prime factorization — 30.9x faster than cosine, fully explainable

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?

	Cosine Similarity	Triadic Engine
Speed (50K pairs)	baseline	30.9× faster
Explainability	Black box	✅ Prime factor proof
Subsumption (A ⊆ B?)	❌ Approximation	✅ Exact (Φ(A) mod Φ(B) == 0)
Composition (A ∪ B)	❌ Geometric average	✅ lcm(Φ(A), Φ(B))
Gap analysis	❌ Not possible	✅ gcd + quotient decomposition
Determinism	❌ Seed-dependent	✅ PCA / contrastive modes
AI model audit	❌ Not supported	✅ Topological discrepancy

---

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:

Operation	Math	What it answers
Subsumption	Φ(A) mod Φ(B) == 0	"Does A contain every feature of B?"
Composition	lcm(Φ(A), Φ(B))	"What concept has all features of both A and B?"
Gap Analysis	gcd(Φ(A), Φ(B)) + quotients	"Which features do they share? Which are unique?"

---

Projection Modes

Mode	Deterministic	Requires Labels	Best For
random	✗ (seed-dependent)	✗	Baseline, exploration
pca	✓	✗	Production, reproducibility
consensus	✓	✗	Noise filtering, stability analysis
contrastive	✓	✓ (hypernym pairs)	Maximum accuracy (up to 100% TP at k=6; 96.2% deterministic)

---

Core Modules

Module	Description
neurosym.encoder	Multi-backend embedding encoder (HuggingFace, OpenAI, Cohere) + 4-mode LSH→Prime projection
neurosym.triadic	Algebraic validation: subsumption, composition, abductive gap analysis
neurosym.graph	Scalable graph builder with inverted prime index (avoids O(N²))
neurosym.storage	SQLite persistence for prime indices and audit results
neurosym.reports	Exportable reports in HTML, JSON, and CSV formats
neurosym.ingest	DataFrame ingestion with inverted prime index and semantic search
neurosym.anomaly	Multiplicative anomaly detection for tabular data

---

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

Endpoint	Method	Description
/health	GET	Engine status and loaded concepts count
/encode	POST	Encode concepts into composite prime integers
/audit	POST	Compare two embedding models topologically
/search	POST	GCD-based semantic search over indexed concepts
/subsumes	POST	Directional subsumption (A ⊇ B ↔ Φ(A) mod Φ(B) == 0)
/compose	POST	LCM composition — concept containing all input features
/gap	POST	Abductive gap analysis (shared GCD + unique factors)
/analogy	POST	Resolve A:B :: C:? analogies in prime space
/report	GET	Export engine state as HTML, JSON, or CSV

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

Metric	Result
Pairwise verification speed	30.9× faster than cosine (50K operations); 5.4× vs pre-normalized cosine
Composition guarantee	100% verified across 5,671 word pairs
Hypernym detection accuracy	up to 100% TP with contrastive projection at k=6 (96.2% deterministic)
Model audit scale	108,694 discrepancies in 1,999,000 semantic chains (WordNet 2K, 2 models)

---

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)

	Allowed
Individuals / personal projects / freelancing	✅ Free
Academic / research institutions	✅ Free
Non-profit organizations	✅ Free
For-profit companies (production use)	❌ Requires participation agreement

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: [e-mail]

© 2026 J. Arturo Ornelas Brand