WCM Universal Constants in Large Language Models: Grokking as 24-cell Geometric Phase Transition

Version 1.1 (2026-06-28) — Added L-3, L-6, L-5 experiments (Findings LLM-12 through LLM-23) The Watabe-Claude Method (WCM) is a four-dimensional analysis framework based on 24-cell polytope geometry (F4 symmetry, self-dual in 4D) that has revealed two universal constants — cosD_birth = 0.4249 and cosD_complete = 0.5000 — across 43 orders of magnitude (Bridge Paper X, BX-1 through BX-6). This paper (v1.1) extends WCM to Large Language Models through five experiments and establishes 23 findings (LLM-1 through LLM-23). Experiment L-1: BERT embedding analysis with mean pooling reveals bimodal D2 curve (LLM-7) and brackets BX-6 to within 0.21 pp (LLM-8). Experiment L-2: Grokking training on modular arithmetic shows cosD_birth and cosD_complete are crossed sequentially post-Grokking. Structural isomorphism with schizophrenia identified (LLM-11). Experiment L-3 [NEW]: BX-6 D2 drop rate (28.3%) reproduced collectively across BERT/RoBERTa/GPT-2 (collective mean 25.89%, diff 2.41 pp). D2 bimodality universal across all 3 architectures (LLM-13). Experiment L-6 [NEW]: ToolD-Nest recursive 24-cell scanning achieves purity=1.000 for 4 semantic categories without human labels. Vertex-level semantic directionality discovered (LLM-18: V5=AI, V14=Arts, V18=Social). Experiment L-5 [NEW]: Pythia models (70M-410M) show Peak cosD converging to cosD_birth with scale. All models below cosD_birth show zero capability (LLM-23). Causal LM D2 curves are inverse of Encoder curves (LLM-20). Files included:- bridge_paper_l_v11.pdf: Full paper (v1.1, 9 sections, 6 figures, 23 findings)- wcm_llm_l1.py: BERT embedding space analysis (L-1, mean pooling)- wcm_llm_l2.py: Grokking x cosD tracking (L-2)- wcm_llm_l3.py: BX-6 universality across architectures (L-3) [NEW]- wcm_llm_l6.py: ToolD-Nest semantic nesting (L-6) [NEW]- wcm_llm_l5.py: Pythia scale analysis (L-5) [NEW] GitHub: https://github.com/watabe-masanori/cds-polytope (AGPL-3.0)