Spectral Guard False Positive Rate (FPR)

Plain language: The spectral guard keeps each layer’s singular values close to the baseline so the per-run false positive rate (FPR) stays within the calibrated WARN budget.

Claim

The spectral guard enforces the calibrated WARN budget by monitoring per-family z-scores and limiting the per-run FPR under the chosen null-modeling assumptions.

Derivation (sketch)

Per-family spectral monitoring computes z-scores

$$z = \frac{s - \mu_f}{\sigma_f}$$

for a spectral statistic $s$ (e.g., top singular value). A WARN is issued if $|z| \ge \kappa_f$. Under a modeled null where $z \approx \mathcal{N}(0,1)$, the per-module two-sided tail probability becomes

$$p_{\text{tail}} \approx 2\big(1 - \Phi(\kappa_f)\big).$$

Applying Bonferroni across the $m_f$ modules controls the family-wise error rate (FWER); applying Benjamini–Hochberg (BH) controls the expected false-discovery proportion (FDR). Balanced tiers choose BH (α=0.05, m=4); Conservative tiers choose Bonferroni (α=0.000625, m=4). Document the policy alongside $\kappa_f$ so auditors can recover the expected per-run WARN rate.

Assumptions & Scope

• Baseline runs provide $(\mu_f, \sigma_f)$ per family $f \in \{\text{ffn}, \text{attn}, \text{embed}, \text{other}\}$; when $\sigma_f = 0$ we fall back to the tier deadband δ.
• Only 2‑D weight matrices (FFN blocks, attention projections, embeddings) are evaluated; 1‑D LayerNorm parameters are explicitly excluded from spectral monitoring. LayerNorm coverage is provided by invariants (presence checks) and activation‑based RMT (CI/Release); VE captures any aggregate performance shift.
• Balanced tier uses the Benjamini–Hochberg procedure (method = "bh", α = 0.05, m = 4 families) with per-family caps {ffn: 3.849, attn: 3.018, embed: 1.05, other: 0.0}, sigma_quantile = 0.95, and max_caps = 5, yielding ≤5% WARN rate on null runs (stored in tiers.yaml). Scope is all, so FFN, attention, embeddings, and other 2‑D weights are all monitored.
• Conservative tier applies Bonferroni (method = "bonferroni", α = 0.000625, m = 4) with caps {ffn: 3.849, attn: 2.6, embed: 2.8, other: 2.8}, sigma_quantile = 0.90, and max_caps = 3, keeping WARNs within the calibrated budget. Scope is ffn in the shipped tier policies, so only FFN blocks are actively budgeted under the Conservative spectral guard.
• Deadband δ suppresses flicker around the cap: Balanced records δ = 0.10, Conservative δ = 0.05, surfaced in reports via spectral.summary.deadband.
• reports expose the calibrations under spectral.multiple_testing.{method,alpha,m}, spectral.summary.{sigma_quantile,max_caps,deadband}, and spectral.family_caps[*].kappa.
• The FPR story is a calibration assumption under the chosen null model, not a theorem about arbitrary transformer weights.
• Empirical histograms of $z$ should be approximately standard normal; heavy tails → raise $\kappa_f$ or use robust $\sigma$ (MAD-scaled).

The deadband δ is a guardrail against flicker: relative changes within ±δ are treated as neutral, so WARNs only fire when sustained growth exceeds both δ and the family κ cap. Auditors can confirm the chosen δ directly in the report summary.

Runtime Contract (report)

• report exposes spectral.summary.{sigma_quantile,deadband,modules_checked,max_caps,caps_exceeded}, spectral.family_caps, and spectral.families[family] with {max, mean, count, violations, kappa}. sigma_quantile is the calibrated baseline percentile used to derive the reference target.
• Tier files document FPR targets and mapping $\kappa_f \rightarrow$ expected WARNs.
• Policy metadata records the multiple-testing method (spectral.multiple_testing) and the cap limit (spectral.max_caps).

Observability

• spectral.summary.{sigma_quantile,deadband,modules_checked,max_caps,caps_exceeded}
• spectral.family_caps[*].kappa and spectral.families[*].{kappa,violations}
• spectral.multiple_testing.{method,alpha,m} and spectral.max_caps

Worked example (Balanced tier)

• For FFN modules, family_caps.ffn.kappa = 3.849. Suppose a layer reports $z = 3.90$.
• report records a WARN in spectral.families.ffn.violations += 1; spectral.caps_applied increments.
• Balanced max_caps = 5. After the fifth WARN the guard continues to WARN; the sixth triggers spectral.caps_exceeded=true and the run aborts.
• Multiple-testing metadata shows spectral.multiple_testing = {method: "bh", alpha: 0.05, m: 4} so reviewers can verify the tier-wide correction.

Calibration

Calibration values are derived from null-sweep runs using the order-statistic and parametric methods described in the tier calibration documentation (09-tier-v1-calibration.md). The calibrated κ values are stored in the packaged tiers.yaml (invarlock._data.runtime/tiers.yaml).

To recalibrate, run null baselines (no edit) and collect per-module z-scores. Allocate the WARN budget across families proportionally by module count, then set κ(f) via order-statistic (the B(f)-th largest |z| in that family) or parametric inversion of the tail probability. Add a small safety margin (η ≈ 0.05–0.10) and validate that subsequent null runs stay within the budget.

Basis column in Quality Gates tables: "point" = point estimate gate, "upper" = upper-bound gate, "point & upper" = both point and upper bounds must pass.

References

• Benjamini, Y., & Hochberg, Y. (1995). “Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing.” Journal of the Royal Statistical Society: Series B (Methodological), 57(1), 289–300. https://doi.org/10.1111/j.2517-6161.1995.tb02031.x