The Structural Coherence Bound — Dirichlet Energy Monitoring for Neural Network Representational Health

This specification defines the Structural Coherence Bound (Clause AI-7), the fifth and final mandatory invariant in the Model Attestation Interface (MAI-1). AI-7 introduces Dirichlet energy monitoring — adapted from spectral graph theory and geometric deep learning — as a governance-grade metric for neural network representational health. The invariant certifies that a model's internal geometry has not entered either of two known-pathological regimes: representational collapse (where token representations converge, destroying the model's capacity to distinguish inputs) or representational fragmentation (where representations become chaotically sensitive, topologically bottlenecked, or spectrally noisy). A certified coherence band [E_min, E_max], calibrated per model via distribution-free conformal prediction, provides binary pass/fail compliance determination at each attested measurement point.The specification synthesizes over a decade of convergent research — oversmoothing in graph neural networks (Li 2018, Oono & Suzuki 2020), doubly exponential rank collapse in pure self-attention (Dong et al. 2021), neural collapse geometry (Papyan et al. 2020), dimensional collapse in self-supervised learning (Jing et al. 2022), the tunnel effect (Masarczyk et al. 2023), oversquashing via negative Ricci curvature (Topping et al. 2022), loss of plasticity in continual learning (Dohare et al. 2024, Nature), and the strict generality of rank collapse over energy collapse (Zhang et al. 2025) — into a unified, deployable monitoring framework. A dual-diagnostic architecture combines Dirichlet energy (measuring spatial smoothness on the attention graph) with effective rank (measuring representational dimensionality), grounded in the theoretical result that rank collapse is strictly more general than energy collapse and therefore energy monitoring alone is insufficient.Six computational optimizations — Top-K attention sparsification, randomized SVD (Halko et al. 2011), request sampling, asynchronous computation, incremental generation updates, and Frobenius proxy screening — achieve less than 2% inference latency overhead across model sizes from 7B to 405B parameters. The calibration procedure provides a finite-sample conformal coverage guarantee without distributional assumptions, with governance risk margins scaled by deployment context. A four-state compliance state machine (GREEN/AMBER/RED/UNKNOWN) with deterministic transition rules and recovery hysteresis produces the attestation signal carried in the MAI-1 Layer 2 payload.The invariant is mapped to five regulatory frameworks — EU AI Act Articles 15 and 72, Federal Reserve SR 11-7, FDA Predetermined Change Control Plans, NIST AI RMF 1.0, and Solvency II — demonstrating that representational health monitoring fills an open implementation gap in frameworks that mandate continuous monitoring but prescribe no specific internal metrics. An insurability framework demonstrates how the certified coherence band transforms representational health into a parametric insurance trigger with four severity classes, enabling the insurance-procurement feedback loop that drives market adoption.With AI-7, the five mandatory MAI-1 invariants — entropy floor, gradient stability, SRAM thermal integrity, distribution drift, and structural coherence — are complete. The Auburn Governance Stack's Layer 2 (Model State Health) is fully specified.