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Research

Paper

Triadic Neurosymbolic Engine: Prime Factorization as a Neurosymbolic Bridge: Projecting Continuous Embeddings into Discrete Algebraic Space for Deterministic Verification

J. Arturo Ornelas Brand (2026). Independent Researcher.

DOI Paper DOI Software

Benchmarks

Metric Result Notes
Pairwise verification speed 28.4x faster than cosine 50K operations
Composition guarantee 100% verified 5,671 word pairs
Hypernym detection accuracy 100% TP Contrastive projection at k=6
Model audit scale 108,694 discrepancies 2M semantic chains, 2 models

Key Contributions

  1. LSH-to-prime projection: Maps continuous embeddings to composite integers via locality sensitive hashing, enabling exact algebraic operations
  2. Four projection modes: Random, PCA, consensus, and contrastive — covering unsupervised through supervised settings
  3. Scalable graph building: Inverted prime index reduces pairwise comparison from O(N^2) to O(N x F x B)
  4. Topological model auditing: Compares how different embedding models structure semantic space using shortest-path discrepancy analysis

Known Limitations

  • Hash overlap is not semantic containment: Subsumption reflects LSH bucket overlap, not genuine semantic containment
  • Lossy projection: R^384 to Z is lossy; similar words may receive identical encodings
  • Narrow useful k range: k=6--12 is practical; no principled selection method exists
  • Low analogy accuracy: 2--10% for discovery (designed for verification, not discovery)

Citation

Paper

@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}
}

Software

@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      = {0.3.0},
  doi          = {10.5281/zenodo.18748671},
  url          = {https://doi.org/10.5281/zenodo.18748671}
}

Companion Work

This engine is the foundation of the Triadic research program:

Project What it does DOI
TriadicGPT 40M-parameter GPT that learns prime signatures end-to-end DOI
reptimeline Tracks how discrete representations evolve during training DOI

License

Business Source License 1.1 (BUSL-1.1)

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

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

Contact: [email protected]