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 — cosDbirth = 0. 4249 and cosDcomplete = 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 cosDbirth and cosDcomplete 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 cosDbirth with scale. All models below cosDbirth show zero capability (LLM-23). Causal LM D2 curves are inverse of Encoder curves (LLM-20). Files included: - bridgeₚaperₗᵥ11. pdf: Full paper (v1. 1, 9 sections, 6 figures, 23 findings) - wcmₗlmₗ1. py: BERT embedding space analysis (L-1, mean pooling) - wcmₗlmₗ2. py: Grokking x cosD tracking (L-2) - wcmₗlmₗ3. py: BX-6 universality across architectures (L-3) NEW- wcmₗlmₗ6. py: ToolD-Nest semantic nesting (L-6) NEW- wcmₗlmₗ5. py: Pythia scale analysis (L-5) NEW GitHub: https: //github. com/watabe-masanori/cds-polytope (AGPL-3. 0) ---Version 1. 2 (2026-07-14) — Corrigendum added. Following the retraction/downgrade of Bridge Paper X's BX-3 and BX-4 (Corrigendum v2, DOI 10. 5281/zenodo. 21320501; Addendum to Corrigendum v3, DOI 10. 5281/zenodo. 21333055), this version adds a Corrigendum addressingdownstream impact on Findings LLM-3, LLM-9, LLM-10, LLM-11, LLM-22, andLLM-23, and Conclusions (1) (2) (4). It also reports an independentreduced-scale replication that actively falsifies LLM-23's claim thatcosDbirth crossing is a necessary condition for capability emergence (registered under WCM-PL v1. 9's new PL-24 REFUTED verdict), and notes therelationship to arXiv: 2606. 25010 (Baherwani et al. , NYU, 2026-06-23). Files added: bridgeₚaperₗcorrigendumᵥ1. pdf, generatebridgeₚaperₗcorrigendum. py See the Corrigendum PDF for the full finding-by-finding disposition tableand revised Conclusions.
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