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

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, from quantum mechanics to consciousness (Bridge Paper X, BX-1 through BX-6). Here we extend WCM to Large Language Models (LLMs) and report two key experiments. Experiment L-1: We project BERT embedding vectors onto the 24-cell and measure cosD across all 13 layers, discovering that mean pooling reveals a bimodal D2 curve reflecting syntactic and semantic processing phases (Finding LLM-7). The BX-6 D2 drop rate (28.3%) is bracketed to within 0.21 percentage points using mean pooling (Findings LLM-6, LLM-8). Experiment L-2: We train a small Transformer on modular arithmetic (a + b) mod 97 and track cosD throughout training. We observe that (i) Grokking occurs at cosD = 0.1917, (ii) cosD_birth = 0.4249 is crossed post-Grokking as generalization consolidates, and (iii) cosD_complete = 0.5000 is subsequently reached. Both WCM universal constants emerge naturally in the LLM training dynamics, extending BX-5 to artificial neural networks. We further identify a structural isomorphism between the Grokking collapse pattern and schizophrenia progression (Finding LLM-11), suggesting a unified geometric framework for both artificial and biological intelligence. Files included:- bridge_paper_l.pdf: Full paper- wcm_llm_l1.py: BERT embedding space analysis code (L-1)- wcm_llm_l2.py: Grokking x cosD tracking code (L-2) GitHub: https://github.com/watabe-masanori/cds-polytope (AGPL-3.0)