Semantic Basin Structure in High-Dimensional Embedding Space: Synthetic Validation of the (H, Q, R) Tri-Metric Framework and Methodological Boundaries for Real Language Analysis

This paper is Phase 37 of the ESCT (Entropy-Signaling Convergence Theory) research framework. It provides synthetic validation of the (H, Q, R) tri-metric framework proposed in the ESCT Control Experiment v2.0, and documents four methodological warnings for real BERT/GloVe embedding analysis. All findings are synthetic. Real language validation remains an open task for external researchers. Protocol code is included for independent replication. We report a synthetic validation study of the (H, Q, R) tri-metric framework for characterizing semantic basin structure in high-dimensional embedding spaces, as a preparatory investigation for real BERT/GloVe embedding analysis. Eight experimental findings are documented, of which four concern the structural behavior of the metrics and four constitute methodological warnings for future empirical work. The central structural finding is that basin entropy H behaves as a step function rather than a continuous richness measure:义项 separation above a critical cosine distance threshold produces H = log₂(n_senses), while separation below the threshold causes discontinuous collapse to lower integer values. This threshold-dependent behavior is structurally analogous to the spectral phase transition at κ≈2.0 identified in the ESCT Frustration Zone experiments, suggesting that the same geometric transition principle may operate in both dynamical and semantic spaces. Basin stability Q shows linear correlation with semantic separation (r=0.54) and proves sensitive to spatial structure destruction under permutation testing (40% Q reduction under coordinate shuffle while H remains unchanged), establishing Q as a genuinely independent structural indicator. Mean-state range R provides a pure geometric anchor (inter-sense cosine distance) orthogonal to both H and Q. Pairwise correlations between all three metrics are weak (r=0.42–0.54), confirming that (H, Q, R) constitutes a mathematically non-redundant tri-dimensional characterization of semantic capacity. Four methodological warnings are identified. Unsupervised GMM+BIC model selection fails systematically in high-dimensional small-sample embedding contexts (n=40, d=768), always selecting k=1 regardless of true cluster structure due to complexity penalty dominance. H is label-dependent while Q is structure-dependent, meaning H alone cannot detect spatial degradation. The occupancy skew threshold effect shows that the rare-basin polysemy hypothesis holds only for extreme rarity ratios. The merger threshold constitutes a semantic phase transition point analogous to physical first-order transitions. All findings are derived from statistically generated 768-dimensional vectors, not real BERT data. The framework is internally consistent, but correspondence with real language requires external empirical validation using supervised sense labels such as WordNet. The phase37_real_embeddings.py protocol is provided for independent replication. semantic basin entropyembedding space geometry(H, Q, R) tri-metric frameworkpolysemy detectionbasin stabilityhigh-dimensional clusteringphase transitionstep function entropyWordNet sense labelsGMM BIC failureBERT embedding analysisESCT-CSIsynthetic validationmethodological auditunsupervised detector failuresemantic capacitycosine distance thresholdrare basin occupancy