Geometric Locality as the Primary Driver of Macroscopic Basin Diversity: A Four-Step Mechanistic Chain from Anti-Physical Curvature to Spectral Phase Transition

We report a complete mechanistic investigation of anti-physical curvature as a driver of macroscopic basin diversity in delay-coupled complex networks, integrating results from four sequential experimental steps (Steps 1–4) and revising claims made in three prior versions (v1.0–v1.2). The central contribution is a verified causal chain: anti-physical curvature increases kernel locality, which collapses the spectral gap at a critical threshold (κ≈2.0), triggering a phase transition from global synchrony to local multi-stability and producing high macroscopic basin entropy (H up to 3.9 bits, N_eff up to 15.0). Temporal delay acts as a conditional amplifier (+0.382 bits, 1.31× N_eff) rather than a necessary condition. Systematic comparison against Modern Hopfield networks (polynomial p=2) reveals that quantitative H equivalence under coarse detectors masks a qualitative divergence: ESCT basins are deep dynamical attractors (Q=95–97.5% recovery at σ=0.5), while Modern HF basins are shallow statistical aggregates (Q=22.5%). Modern HF further exhibits a sharp first-order-like phase transition at p=2→3 (H collapses from 1.32 to 0), contrasting with ESCT’s robust saturation platform across κ≥2.0. Hybrid systems combining both mechanisms underperform both pure systems (H=1.0 vs 3.6/1.3), confirming mechanistic non-additivity. Cross-system analysis identifies kernel locality (r=0.841) and spectral gap (r=0.738) as universal predictors of macro-order basin entropy across different coupling architectures, generalizing the finding beyond ESCT. A falsification test confirms that no tested non-geometric system matches ESCT in both H and stability simultaneously. We propose the (H, Q) dual-metric framework as the necessary reporting standard for any claim about basin diversity or semantic capacity: H measures diversity quantity, Q measures diversity quality, and neither alone is sufficient. This framework maps directly to AI alignment — current LLMs may occupy a low-H, high-Q regime, while curvature-delay geometry offers a theoretical path toward high H with controllable Q. All prior overclaims are documented and revised transparently, demonstrating the scientific value of iterative self-correction. Keywords anti-physical curvature, basin entropy, spectral gap, phase transition, macroscopic diversity, locality index, Hopfield control, (H, Q) dual-metric, basin stability, delay coupling, Frustration Zone, ESCT-CSI, attractor landscape, emergent multi-stability