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

We present TriadicGPT, a 40M-parameter GPT language model augmented with a triadic projection head that produces discrete prime-factor signatures alongside standard nexttoken predictions. Unlike the post-hoc approach of the Triadic-Neurosymbolic-Engine [Ornelas Brand, 2026], which projects frozen sentence embeddings into prime composites, TriadicGPT learns triadic representations end-to-end through a dual-objective training lossthat combines language modeling with a novel embedding alignment objective.Across 29+ training runs and systematic ablation studies, we demonstrate eight principal ndings: (1) the triadic head adds negligible cost to language quality (perplexity7.69 vs. 7.56 ablation baseline, +1.7%); (2) semantic ordering emerges gradually withscalethe gap between related and unrelated concept similarity crosses zero around 20Mparameters, with a smooth crossover rather than a sharp phase transition; (3) a bits sweepover k ∈ {8, 16, 32, 48, 64, 128} reveals an optimal regime at k = 3264, shifted upward fromthe k = 612 range reported for post-hoc projection; (4) a transfer experiment attaching the triadic head to pre-trained GPT-2 with an InfoNCE alignment loss closes 48% ofthe gap to the Triadic Engine's post-hoc projection; (5) a subsumption loss recovers100% held-out subsumption at k = 64, resolving the primary limitation at high bitcounts; (6) an iFSQ activation (2σ(1.6x) − 1) resolves the subsumptionlanguage tradeoentirely: language quality is preserved (loss 0.9240.951 vs. baseline 0.946) while achieving up to 87.1% held-out subsumption, compared to +47% perplexity degradation undertanh; (7) compositional analysis reveals that the bit space functions as a computationalsubstrateround-trip accuracy (98.1%) far exceeds the multiplicative prediction (81.9%),two-step chains show sub-linear error accumulation, and fork analysis conrms that themechanism is ontologically categorical, not vectorial; and (8) a discovery loop expandingfrom 50 hand-labeled anchors to 158 improves holdout accuracy from 87% to 93% and subsumption from 90.7% to 98.3%, demonstrating that the traindiscovercorrectretrain cyclescales semantic knowledge beyond initial supervision.TriadicGPT achieves 98% analogy verication (50/51 analogies), 100% signature uniqueness, and reproducible semantic ordering (+0.038 ± 0.005 gap, n = 3, 95% CI positive)allwithin a single forward pass. Critically, the triadic head's value lies in algebraic operations:100% analogy verication, 87.1% held-out subsumption, and sub-linear compositional chaining, none of which random projections can provide.