RMT ε‑Band Acceptance (Edge Risk Score)

Plain language: The RMT guard limits how much the activation edge risk can grow beyond its baseline, ensuring structural shifts trigger a failure while expected noise passes.

Overview

Claim

The Random Matrix Theory (RMT) guard accepts an edit when the activation edge risk score stays within the calibrated ε‑band for each family.

Let $r_f^{\text{base}}$ be the baseline edge risk score and $r_f^{\text{cur}}$ the current score for family $f$. The guard accepts if:

$$r_f^{\text{cur}} \le (1+\epsilon_f)\, r_f^{\text{base}}$$

with $\epsilon_f$ calibrated from null runs (e.g., 95th–99th percentile of $r_f^{\text{cur}}/r_f^{\text{base}} - 1$).

What is the edge risk score?

For a (token×hidden) activation matrix, the guard forms a whitened centered and standardized matrix, estimates its top singular value via a deterministic matvec estimator, and normalizes by the Marchenko–Pastur edge for the same shape:

$$r = \frac{\hat{\sigma}_{\max}(A')}{\sigma_{\mathrm{MP}}(m,n)}$$

The contract fixes the estimator budget and the activation sampling policy; those knobs are recorded in the report.

This note documents the runtime report contract for the activation edge-risk mode surfaced in reports; it does not catalog the full implementation surface inside src/invarlock/guards/rmt.py.

Derivation (sketch)

• Edge risk fluctuates under null due to finite‑sample deviations from the Marchenko–Pastur edge and estimator noise.
• The ε‑band permits expected null drift, flagging structural increases.
• Large edge risk indicates concentration of activation energy along a small number of directions beyond random‑matrix expectations.

Assumptions & Scope

• Null calibration must cover each family {ffn, attn, embed, other}; default ε values are exposed whenever data is sparse.
• Baseline and current scores use identical activation sampling and token‑weighted aggregation.
• CI/release and activation-required evidence require activation-based scoring; if activation batches are missing in those paths, the RMT guard fails closed.

Calibration (pilot-derived)

• Balanced tier uses $\epsilon_f = \{0.01, 0.01, 0.01, 0.01\}$ for {ffn, attn, embed, other} respectively (q95–q97 of null deltas).
• Conservative uses the same per-family ε defaults: $\epsilon_f = \{0.01, 0.01, 0.01, 0.01\}$. Values are recorded in the packaged tiers.yaml (runtime/tiers.yaml) and surfaced in reports. Provide overrides via INVARLOCK_CONFIG_ROOT/runtime/tiers.yaml when needed.

Example: with r_base = 1.20 and ε = 0.01, the guard allows r_cur ≤ (1+0.01) × 1.20 = 1.212.

Recalibration

Calibration values are derived from null-sweep runs and stored in the packaged runtime/tiers.yaml. See the full calibration methodology in 09-tier-v1-calibration.md.

To recalibrate, run null baselines (no edit) and compute per-family deltas Δ(f) = r_cur(f)/r_base(f) − 1 (skip cases with missing or zero baseline). Set ε(f) to the q95–q99 quantile of Δ(f). For small families or tiny sample sizes, use a slightly larger ε to avoid spurious failures.

Runtime Contract (report)

• report records rmt.{mode,edge_risk_by_family_base,edge_risk_by_family,epsilon_default,epsilon_by_family,epsilon_violations,stable,status}.
• Per-family details for rendering live under rmt.families.*.{edge_base,edge_cur,epsilon,allowed,ratio,delta}.
• rmt.measurement_contract.kind = "activation_edge_risk" records which RMT measurement path produced the evidence.
• report lint verifies the inequality and marks violations; validation.rmt_stable reflects the ε‑band gate.

Observability

• rmt.edge_risk_by_family_base.* and rmt.edge_risk_by_family.*.
• rmt.epsilon_default and rmt.epsilon_by_family.*.
• rmt.status / rmt.stable and rmt.epsilon_violations for pass/fail context.
• resolved_policy.rmt.{margin,deadband,epsilon_by_family} — resolved thresholds archived with the report bundle.

Edge cases

• Small samples: estimator variance dominates; increase activation sample count or widen ε for tiny families.

Background reading

• Marchenko, V. A., & Pastur, L. A. (1967). “Distribution of eigenvalues for some sets of random matrices.” Mathematics of the USSR-Sbornik, 1(4), 457–483.
• Bai, Z. D., & Silverstein, J. W. (2010). Spectral Analysis of Large Dimensional Random Matrices (2nd ed.). Springer.
• Pennington, J., & Worah, P. (2017). “Nonlinear Random Matrix Theory for Deep Learning.” Advances in Neural Information Processing Systems (NeurIPS). https://papers.nips.cc/paper/6857-nonlinear-random-matrix-theory-for-deep-learning
• Martin, C. H., & Mahoney, M. W. (2021). “Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning.” Journal of Machine Learning Research, 22(165), 1–73. Preprint: https://arxiv.org/abs/1810.01075