25-EFT.WP.STG.Dynamics v1.0 | Appendix D — Metrics and Drift Measures (STG-Exclusive)

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Appendix D — Metrics and Drift Measures (STG-Exclusive)

I. Scope and Objects

• Objects: graph(V,E), operators {L, A, H}, kernel K = g(L), propagation & stability, filtering statistics, causal invariance, uncertainty & guardband, dual-form gaps, runtime SLI/SLO.
• Inputs: windowed data, spectral and residual statistics, manifests & hashes, RefCond.
• Outputs: metrics.*, drift.*, alarms.*, dashboard subscription streams, and manifest.stg.runtime metric fragments.
• Constraints: every metric must declare unit(metric) and dim(metric); compute over window W = [ t − Δt_win, t ] and persist.

II. Terms and Variables

• Structure & operators: L, A, H, λ_i(L), λ_max, λ_2 (algebraic connectivity).
• Kernel & stability: K, ρ(K), Δt, γ_growth, ε_approx.
• Filtering & residuals: x̂, P, Q, R, r = y - H x̂, NIS, ESS/Np.
• Uncertainty & dual forms: u_num, u_model, u_c, g = k * u_c, delta_form_stream, delta_form_unc.
• Drift: D_L, D_spec, D_H, D_K, D_res_KS, D_θ, D_inv.
• Time: ts, tau_mono, wm(t).

III. Postulates P70D-*

• P70D-1: All metrics are computed on a sliding window; wm(t) is non-decreasing: wm(t_{i+1}) ≥ wm(t_i).
• P70D-2: Spectral/residual metrics must provide a baseline baseline@RefCond and windowed quantiles.
• P70D-3: Maintain dual forms in parallel—event time and processing time; notarize the gap between statistical and uncertainty formulations separately.
• P70D-4: Drift detection follows a two-stage procedure “metric → normalization → statistical test / cumulative sum”, and test parameters are persisted.

IV. Minimal Equations S70D-*

• Structural spectrum bounds: λ_min = min_i λ_i(L), λ_max = max_i λ_i(L), connectivity λ_2 = nth(λ(L), 2).
• Kernel spectral radius & stability margin: ρ(K) = max |eig(K)|, m_stab = 1 - ρ(K) (discrete linear approximation).
• Kernel approximation error: ε_approx = ( ||K - K_approx||_2 / ||K||_2 ).
• Propagation behavior: γ_growth = ( ||x̂_t||_2 / ||x̂_{t-1}||_2 ) (normalized within the window).
• KF consistency: NIS = r^T S^{-1} r, S = H P H^T + R.
• PF degeneracy: ESS/Np = 1 / ( ∑ w_i^2 ).
• Combined uncertainty: u_c^2 = J V_x J^T + u_num^2 + u_model^2, g = k * u_c.
• Spectral/operator drift: D_L = ( ||L_t - L_ref||_F / ||L_ref||_F ); D_spec = ( ||λ(L_t) - λ(L_ref)||_2 / ||λ(L_ref)||_2 ); D_K = ( ||K_t - K_ref||_2 / ||K_ref||_2 ).
• Observation & parameter drift: D_H = ( ||H_t - H_ref||_F / ||H_ref||_F ); D_θ = ( ||θ_t - θ_ref||_2 / ||θ_ref||_2 ).
• Residual distribution drift: D_res_KS = sup_z |F_t(z) - F_ref(z)|.
• Invariance drift (across environments): D_inv = ( ||β_e - β_{e'}||_2 / ||β_ref||_2 ) (invariant-risk parameters compared).
• Dual-form gaps: delta_form_stream = | metric_evt - metric_proc |; delta_form_unc = | u_c^{LPU} - u_c^{MC} |.
• CUSUM (standardized statistic z_t): S_t = max( 0, S_{t-1} + z_t - κ ), trigger when S_t ≥ h.

V. Metrology Pipeline M70-4 (Ready → Compute → Test → Persist)

1. Ready: load RefCond and baseline baseline.metrics; construct W and align ts / tau_mono.
2. Compute:
   • Structural spectrum: derive λ_min, λ_2, λ_max, then D_L, D_spec;
   • Kernel & propagation: ρ(K), m_stab, ε_approx, γ_growth;
   • Filtering statistics: NIS quantiles, ESS/Np, residual quantiles;
   • Uncertainty & dual forms: u_c, g, delta_form_*.
3. Test: apply KS/χ^2 or CUSUM / Page-Hinkley to D_* and key metrics; produce alarms.*.
4. Persist: write metrics.*, drift.*, thresholds & test parameters, alarms.* with signatures; push dashboard feed.

VI. Contracts & Assertions C70D-* (threshold guidelines)

1. Structural spectrum:
   • C70D-01: λ_2 ≥ λ2_min; λ_max ≤ λ_cap.
   • C70D-02: D_L_p95 ≤ tol_L and D_spec_p95 ≤ tol_spec.
2. Kernel & propagation:
   C70D-11: m_stab ≥ m_min; ε_approx ≤ ε_cap; γ_growth_p95 ≤ γ_cap.
3. Filtering statistics:
   C70D-21: NIS_p95 ∈ [ ν_low, ν_high ]; ESS/Np ≥ ess_min; residual_p95 ≤ r_cap.
4. Uncertainty & dual forms:
   C70D-31: coverage ≥ 1 − α − ε_cov; delta_form_unc_p95 ≤ tol_unc; delta_form_stream_p95 ≤ tol_stream.
5. Drift testing:
   C70D-41: if D_res_KS > ks_thr or CUSUM ≥ h for k consecutive windows → alarm = major and trigger the baseline-reestimation strategy card.

VII. Implementation Bindings I70-* (metric API prototypes)

• compute_graph_spectrum(L) -> { λ_min, λ_2, λ_max, spectrum }
• metric_operator(K, Δt) -> { ρ, m_stab, ε_approx }
• metric_residual(y, H, x_hat, P, R) -> { NIS_stats, residual_stats }
• metric_uncertainty(J, Vx, u_num, u_model, k) -> { u_c, g }
• metric_drift(curr, ref, kind) -> D_kind ( kind ∈ { L, spec, H, K, res_KS, θ, inv } )
• detect_change(series, method, params) -> { alarms, scores } ( method ∈ { CUSUM, PH, KS, ADWIN } )
• emit_metric_panel(metrics, policy) -> feed_id
• Invariants: non_decreasing(wm); valid dim(metric); input hashes and RefCond are traceable.

VIII. Cross-References

• Stability & integration: see Chapter 9 (rigidity / splitting / events).
• Filter consistency: see Chapter 12 (KF / UKF / PF on Graphs).
• Uncertainty & guardband: see Chapter 13.
• Runtime windows & dashboards: see Chapter 14.
• Manifest field mapping: see Appendix C (embedding of metrics.* and contracts.*).

IX. Quality & Risk Control

1. SLOs: metric computation latency_p95 ≤ B_mod; drop rate ≤ drop_cap; dashboard lag ≤ Δt_panel_max.
2. Drift response strategy cards (summary):
   • S-Drift-Graph: if D_L exceeds limit → freeze anomalous subgraph; roll back to last graph.hash;
   • S-Drift-Residual: if D_res_KS exceeds limit → tune Q/R and re-estimate H;
   • S-Drift-Stability: if m_stab is low → spectral trimming or reduce Δt; switch solver = BDF if needed.
3. Audit: retain series.z, thresholds κ / h, test p-values, and alarms.* evidence pointers.

Summary

4. This appendix specifies STG-exclusive definitions for metric and drift measurement: scope, equations, pipeline, contracts, and implementation prototypes.
5. Artifacts persist to metrics.* / drift.* / alarms.*, and manifest.stg.runtime binds RefCond and hashes, enabling system-level monitoring, replay, and comparison.