28-EFT.WP.Propagation.PathRedshift v1.0 | Appendix D — Metrics & Drift Measures (Redshift-Specific)

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Appendix D — Metrics & Drift Measures (Redshift-Specific)

One-sentence goal: Define a unified family of metrics and drift measures for the entire PathRedshift chain—kin/grav/med/cos → dispersion mapping → ray integration → observation → fusion/calibration → uncertainty → runtime—with computable formulas, windowing and threshold conventions, and tight coupling to the manifest/contract loop.

I. Scope & Objects

• Objects: z_parts = { z_kin, z_grav, z_med, z_cos, z_inst, z_proc }, composite z_path; observed z_meas; arrival times T_arr^{form1/form2}, harmonized T_arr*; mapping gap ΔT_map; online dual-form gap delta_form_rt.
• Inputs: windowed data W = [ ts − Δt, ts ], RefCond.hash and source hashes (ephemerides / gravity / media / timebase), two-form products and uncertainties u/U, panel/evidence URIs.
• Outputs: metric objects metrics.redshift.*, drift decomposition drift.* and aggregate drift_score, C65-* determinations and threshold references.
• Constraints: all metrics declare unit / dim and pass check_dim( y − f(x) ); two-form metrics must be replayable.

II. Terms & Variables

• quantile(z,p): p-quantile; EWMA_α(z): exponential smoothing; ⊕: conservative sum/convolution (state convention).
• Baselines are noted as ref@ts0: z_ref, T_arr_ref, RefCond_ref.
• Dimension examples: unit(z)=1, unit(T)=[T], unit(age)=[T], unit(coverage)=1.

III. Postulates P65D-*

• P65D-1 (Two-form coexistence): Within any publication window, record T_arr^{form1/form2} and delta_form; compute z_path vs z_meas residuals on the same window.
• P65D-2 (Explicit measures): For every metric, declare time/frequency/set domains: ( ∫_{t∈W} ), ( ∫_{f∈B} ), ( ∑_{k∈S} ).
• P65D-3 (Dimensions & provenance): All numeric fields pass check_dim; bind RefCond.hash and source hashes.
• P65D-4 (Attributable drift): Overall drift must be decomposable into kin / grav / med / cos / mapping / ray / observation / fusion / calibration / uncertainty / runtime sub-parts.

IV. Minimal Equations S65D-*

1. Two forms & arrival-time consistency

• S65D-01: delta_form(W) = | T_arr^{form1}(W) − T_arr^{form2}(W) |.
• S65D-02: alignment gap ΔT_obs(W) = | T_arr*(W) − t̂_cont(W) | (Chs. 8/9).

2. Analytic vs observation redshift

• S65D-11: resid_z(t) = z_meas(t) − z_path(t); window quantile resid_z,p95 = quantile( |resid_z|, 0.95 ).
• S65D-12: bias/drift: z_bias(W) = ⟨resid_z⟩_W, z_drift = d⟨resid_z⟩/dt (first-order regression).

3. Component consistency & conservation (composition check)

• S65D-21: E_balance_z(W) = | z_path − compose( z_kin, z_grav, z_med, z_cos, z_inst, z_proc ) |.
• S65D-22: convention consistency (phase↔group): ΔT_map_p95 = quantile( | T_g − T_phi |, 0.95 ) (Ch. 7).

4. Dispersion / rays & geometry

• S65D-31: curvature_max = max_ell || d^2 gamma / d ell^2 ||;
  snell_resid_p95 = quantile( || k_t^+ − k_t^- ||, 0.95 ).
• S65D-32: bend_radius_min = min_ell ( 1 / || d^2 gamma / d ell^2 || ) (fiber constraint).

5. Source freshness & coverage

• S65D-41: age(src) = ts_now − ts_src; coverage = |samples_valid| / |samples_total|.
• S65D-42: Params_cons = 1_{ age ≤ Δt_max ∧ coverage ≥ cov_min }.

6. Uncertainty & onboarding gates

• S65D-51: U = k•u_c (Ch. 13), gate: metric + k•u(metric) ≤ threshold.
• S65D-52: GUM/MC consistency ρ = u_c^{GUM} / u_c^{MC}.

7. Runtime & cache

• S65D-61: delta_form_rt_z(W) = | z_path^{config}(W) − z_meas(W) |.
• S65D-62: hit, stale_ratio, latency_p{q}, drop_rate, ρ_p95 (per Ch. 14).

8. Drift decomposition & score

• S65D-71: sub-drift (vs baseline)
  • drift_kin = ⟨ z_kin(W) − z_kin(ref) ⟩, drift_grav = ⟨ z_grav − z_grav(ref) ⟩,
  • drift_med = ⟨ z_med − z_med(ref) ⟩, drift_cos = ⟨ z_cos − z_cos(ref) ⟩,
  • drift_map = ⟨ ΔT_map − ΔT_map(ref) ⟩, drift_ray = curvature_max − curvature_ref,
  • drift_obs = ⟨ z_meas − z_meas(ref) ⟩, drift_sync = ⟨ offset/skew/J − ref ⟩.
• S65D-72 (Normalization & aggregation):
  drift_score = Σ_i w_i • norm_i(drift_i), with Σ w_i = 1; choose norm_i as z-score or threshold-normalized scale per metric units.

V. Workflow M65-D* (Ready → Compute → Aggregate → Decide → Persist)

• Ready: lock RefCond, window Δt, quantiles { p50, p95, p99 }, and weights w_i; load manifests and evidence URIs.
• Compute: evaluate all metrics by S65D-* across domains; compute two-form / mapping / online gaps in parallel.
• Aggregate: smooth via EWMA_α and compute quantiles; produce drift_* and drift_score.
• Decide: apply C65-* (Appendix B) and onboarding gates; issue strategy cards and record evidence_uri for failures.
• Persist:
  manifest.redshift.metrics = { window, metrics:{ Q, Int, Sec }, drift:{ parts, score }, thresholds, RefCond.hash, evidence_uri[], signature }.

VI. Contract & Threshold Suggestions C65D-* (aligned with Ch. 12 / Appendix B)

• Two forms & alignment: delta_form_p95 ≤ tol_Tarr; ΔT_obs_p95 ≤ tol_align.
• Analytic vs observation: resid_z_p95 ≤ tol_z; | z_bias | ≤ z_bias_max; | z_drift | ≤ z_drift_max.
• Dispersion / rays: ΔT_map_p95 ≤ tol_map; snell_resid_p95 ≤ tol_snell; curvature_max ≤ tol_curv.
• Sources & runtime: age(src) ≤ Δt_max; coverage ≥ cov_min; hit ≥ 0.8, stale_ratio ≤ 0.02.
• Uncertainty: ρ(GUM/MC) ∈ [0.8, 1.25]; coverage(U) ≥ 95%.
• Dimensions: all metrics pass check_dim.

VII. Implementation Bindings I65-* (Metrics API)

• I65-D1 compute_redshift_metrics(manifests, refs, windows) -> { metrics, drift_* }
• I65-D2 estimate_spectral_response(path, models) -> { snell_resid, curvature_max }
• I65-D3 evaluate_twoform_and_align(T_forms, t_cont) -> { delta_form, ΔT_obs }
• I65-D4 residuals_and_bias(z_meas, z_path) -> { resid_z_stats, z_bias, z_drift }
• I65-D5 source_freshness_and_coverage(sources) -> { age_stats, coverage }
• I65-D6 aggregate_runtime(stream) -> { delta_form_rt, hit, stale_ratio, latency_p{q} }
• I65-D7 compose_drift_score(parts, weights) -> { drift_score }
  Invariants: two_forms_present = true; unit/dim checks; RefCond.hash / URI traceable.

VIII. Cross-References

IX. Quality & Risk Control

• Panel keys: delta_form_p95, resid_z_p95, ΔT_obs_p95, ΔT_map_p95, age(src)_p95, coverage, hit/stale_ratio, ρ(GUM/MC), drift_score.
• Disposition ladder: guardband ↑ → window ↓ → re-estimate model/mapping/ray → adjust RefCond sources → bypass/rollback (all actions logged as strategy cards).
• Audit: threshold tables & versions, evidence URIs, metric & drift plots, signature chain, and replay scripts.

Summary

• This appendix unifies the two-form / mapping / residual / geometry / source / runtime metrics and drifts into a computable, judgeable, replayable system.
• With manifest.redshift.*, C65-*, and I65-*, teams can continuously monitor and govern the quality, integrity, and security of z_path / T_arr* in real-world engineering environments.