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Research

Paper

End-to-End Prime Factorization in a Generative Language Model: Emergent Algebraic Semantics from Joint Training

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

DOI Paper DOI Software

Key Results

Metric Value Notes
Language cost Zero PPL 7.69 vs 7.56 ablation (+1.7%)
Analogy verification 98% 51 analogies on XL model
Subsumption accuracy 98.3% Held-out, 158 supervised anchors
Domain separation 1.21 mean 12 domains, sentence-level
GPT-2 transfer gap 48% closure Toward Engine PCA upper bound
Signature uniqueness 100% All evaluated concepts
Scale crossover ~20M params Semantic ordering emergence

Pre-trained Models

Model Params Description Link
triadic-gpt-40m 40M Production model (Run 15, v1.4-strongalign) HuggingFace
triadic-gpt2-medium 355M GPT-2 medium with triadic head (Experiment 10) HuggingFace

Benchmark Suite

12 evaluation scripts covering scaling, analogies, subsumption, entropy, language quality, interpretability, topology, and engine comparison.

Script What it measures
scaling_study.py 4-point scaling study (1.3M--40M), PPL + triadic quality
analogy_benchmark.py Analogy verification accuracy (51 analogies)
subsumption_benchmark.py Taxonomic subsumption accuracy
bit_entropy.py Per-bit entropy analysis (dead bit detection)
language_quality.py Perplexity, Distinct-n, repetition, MAUVE
engine_comparison.py TriadicGPT vs Triadic Engine (5 projection methods)

Citation

Paper

@article{ornelas2026triadicgpt,
  author       = {Ornelas Brand, J. Arturo},
  title        = {End-to-End Prime Factorization in a Generative Language
                  Model: Emergent Algebraic Semantics from Joint Training},
  year         = 2026,
  month        = mar,
  doi          = {10.5281/zenodo.19206545},
  url          = {https://doi.org/10.5281/zenodo.19206545}
}

Software

@software{ornelas2026triadicgpt_repo,
  author       = {Ornelas Brand, J. Arturo},
  title        = {TriadicGPT: End-to-End Prime Factorization in a
                  Generative Language Model (Repository)},
  year         = 2026,
  month        = mar,
  doi          = {10.5281/zenodo.19207845},
  url          = {https://doi.org/10.5281/zenodo.19207845}
}
Project Role Link
Triadic Engine Mathematical foundation (post-hoc projection) PyPI
reptimeline Training analysis tools Docs

License

BUSL-1.1 (Business Source License 1.1). Same terms as the Engine.