When Word Sense Meets Jailbreak: A Three-Metric Framework (H,Q,R) for Multi-Stability Across Language Embeddings and LLM Behavioral Space

This release presents Phases 37 and 38 of the ESCT (Entropy-Signaling Convergence Theory) research program, introducing a unified three-metric framework—(H,Q,R)—tested across two distinct substrates via high-fidelity simulation: (1) language embedding spaces (768-dimensional synthetic BERT-like vectors, 12 pressure tests on polysemy, noise, and detector failure), and (2) AI large-model behavioral spaces (12-dimensional feature vectors, 8 pressure tests on temperature, safety alignment, and jailbreak resistance).Key simulation findings include: (i) perturbation stability Q is the most domain-invariant metric, collapsing under adversarial attack while basin entropy H remains constant—a reframing of jailbreak robustness from mode elimination to mode destabilization; (ii) basin entropy H behaves as a categorical step function with a geometric activation threshold in both domains; (iii) separation R shows domain-dependent correlation structure (orthogonal in embedding space, strongly anti-correlated in behavioral space); and (iv) unsupervised model selection (GMM+BIC) fails universally under small-sample high-dimension conditions, establishing supervised labeling as a methodological boundary condition.All findings are simulation-supported and await real-data validation. The release includes complete Python engines for both domains, real-execution protocols (BERT and LLM API), 20 integrated pressure tests, and visualization pipelines. The framework provides the first quantitative language distinguishing "how many modes" (H), "how stably distinguishable" (Q), and "how distinct" (R) across semantic and AI domains. We introduce a unified three-metric framework — H (basin entropy), Q (perturbation stability), and R (mode separation) — and test it across two structurally distinct substrates via high-fidelity simulation: 768-dimensional synthetic BERT-like word embedding spaces (Phase 37, 12 pressure tests) and 12-dimensional AI large-model behavioral feature spaces (Phase 38, 8 pressure tests). Five cross-domain findings emerge. First, Q is the most domain-invariant metric: it collapses under adversarial attack in both word disambiguation (−40%) and LLM jailbreaking (−71%), while H remains constant — reframing jailbreak robustness from mode elimination to mode destabilization. Second, H behaves as a threshold-gated categorical step-function in both domains rather than a continuous richness measure, with domain-specific geometric activation thresholds (cosine distance 0.35 for embeddings; standardized Euclidean 0.25 for behavioral vectors). Third, R shows domain-dependent correlation structure: weakly positive with H in embedding space (r = 0.53) but strongly anti-correlated in behavioral space (r = −0.982), revealing a fundamental capacity constraint in generalist AI design — more behavioral modes necessarily reduces each mode’s distinctiveness. Fourth, unsupervised model selection (GMM+BIC) fails universally under small-sample high-dimension conditions (n = 40, d = 768), establishing supervised labeling as a non-negotiable methodological requirement for (H,Q,R) computation. Fifth, over-alignment and jailbreak produce an identical (H,Q,R) signature — preserved H, collapsed Q — suggesting they are mechanistically opposite causes of the same structural fragility. All findings are simulation-grounded and await real BERT embedding and LLM API validation. Complete Python execution engines and protocols are provided. The framework offers the first quantitative language that separates how many behavioral modes exist (H), how reliably each mode is accessible (Q), and how geometrically distinct the modes are (R) — applicable to both NLP polysemy detection and AI safety auditing. Status: Simulation complete. Real-data execution pending (phase37_real_embeddings.py / phase38_real_llm.py provided). basin entropy, multi-stability, polysemy detection, LLM behavioral analysis, perturbation stability, jailbreak robustness, AI alignment, BERT embeddings, adversarial robustness, small-sample high-dimension statistics, mode destabilization, safety alignment trade-off, over-alignment, word sense disambiguation, cross-domain framework, ESCT, AI safety, dynamical systems, behavioral manifold, reproducible simulation