28-EFT.WP.Propagation.PathRedshift v1.0 | Appendix E — Error & Uncertainty Propagation (Redshift Edition)

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Appendix E — Error & Uncertainty Propagation (Redshift Edition)

One-sentence goal: Establish a parallel GUM (LPU) and Monte Carlo (MC) framework for uncertainty propagation across PathRedshift—covering z_parts / z_path, T_arr^{form1/form2} / T_arr*, phase↔group mapping ΔT_map, and the online dual-form gap delta_form_rt—and produce coverage intervals and guardbands with manifest-grade traceability.

I. Scope & Objects

• Objects: z_parts = { z_kin, z_grav, z_med, z_cos, z_inst, z_proc }, composed z_path; T_arr^{form1/form2}, harmonized T_arr*; phase↔group mapping ΔT_map; observations z_meas and residuals resid_z = z_meas − z_path.
• Inputs: source vector and covariance V_ξ (geometry / ephemerides / gravity / media / meteorology / spectral lines / clocks / mappings / approximations / runtime drift), RefCond.hash and version provenance; window W = [ ts − Δt, ts ] and the relevant measures.
• Outputs: u_c(z_path), u_c(T_arr*), U = k•u_c, u(delta_form), u(ΔT_map), u(ΔT_obs); gate checks and guardbands; manifest.redshift.u.*.
• Boundary: Prefer GUM for weak nonlinearity / small perturbations; switch to MC or piecewise linearization under strong nonlinearity / thresholds / discrete effects; persist method selection and rationale.

II. Terms & Variables

• Source vector: ξ = [ x_geo, n_params, ephemeris, gravity, iono/trop, TEC, osc(offset,skew,J), obs, approx, map, … ]^T.
• Targets: z = h(ξ), T = h_T(ξ); covariance V_ξ ⪰ 0; Jacobian J = ∂h/∂ξ |_{ξ̂}.
• Two forms & mappings: delta_form = | T_form1 − T_form2 |; ΔT_map = | T_g − T_phi |; ΔT_obs = | T_arr* − t̂_cont |.
• Coverage: U = k•u_c (k ≈ 2 ↔ ~95%); effective dof nu_eff (Welch–Satterthwaite).
• Dimensions: check_dim(z) = "1", check_dim(T) = "[T]"; log↔linear conversions documented in scale.note.

III. Postulates P65E-*

• P65E-1 (Dual-track): Run GUM and MC in parallel; any published u/U must record method∈{GUM,MC}, sample size / convergence, and consistency ρ = u_c^{GUM} / u_c^{MC}.
• P65E-2 (Two forms & mappings): Provide uncertainties for delta_form / ΔT_map / ΔT_obs; include them in onboarding gates and guardbands.
• P65E-3 (Explicit measures): Any propagation must declare ( ∫_{gamma(ell)} ), ( ∫_{t∈W} ), ( ∫_{f∈B} ) as applicable.
• P65E-4 (Dimensions / RefCond): All I/O passes check_dim; RefCond.hash / versions and V_ξ provenance are traceable.
• P65E-5 (Constraint projection): Apply projection before publishing U / gB when guard-bit / boundary / energy constraints apply (see Ch. 13 S65-1311).

IV. Minimal Equations S65E-*

1. GUM linearization & coverage

• S65E-01: u_c^2(z) = J_z V_ξ J_z^T; u_c^2(T) = J_T V_ξ J_T^T.
• S65E-02: nu_eff ≈ ( ∑ u_i^4/ν_i ) / ( ∑ u_i^2 )^2; U = k•u_c, k = t_{nu_eff,1−α/2}.

2. Composite redshift sensitivity (product / linear-sum)

• S65E-03: ln(1+z_path) = ∑ ln(1+z_i) ⇒ ∂z_path/∂z_i = ∏_{j≠i}(1+z_j); small-signal ≈ 1.
• S65E-04: linear-sum z_path ≈ ∑ z_i ⇒ u_c^2(z_path) ≈ ∑ u_c^2(z_i) + 2∑_{i<j}cov(z_i,z_j).

3. Uncertainty of two-form & mapping gaps

• S65E-05:
  u^2(delta_form) ≈ u^2(T_form1) + u^2(T_form2) − 2•cov(T_form1,T_form2).
• S65E-06:
  u^2(ΔT_map) ≈ u^2(T_g) + u^2(T_phi) − 2•cov(T_g,T_phi);
  u^2(ΔT_obs) ≈ u^2(T_arr*) + u^2(t̂_cont).

4. Phase↔group mapping (weak dispersion)

• S65E-07: from Ch. 7 z_g ≈ z_φ − (1/n_g)( d n_g/d ln f ) z_φ:
• u^2(z_g) ≈ u^2(z_φ)
• + ( (z_φ/n_g) u(d n_g/d ln f) )^2
• + ( (d n_g/d ln f)/n_g • u(z_φ) )^2

(higher orders neglected).

5. Clock/sync uncertainty

• S65E-08:
  u^2(ΔT_sync) ≈ u^2(offset) + ( T_win • u(skew) )^2 + u^2( E[J] ), with E[J] variance from clock PSD/ARMA.

6. Ray-path & media parameters

• S65E-09:
  ∂T/∂n ≈ (1/c_ref) ∫ d ell, ∂T/∂x ≈ (1/c_ref) ∫ (∇ n • δx) d ell; discretization step Δell truncation u(ε_ray) enters V_ξ.

7. MC forward propagation

• S65E-10: sample ξ^{(m)} ~ N(ξ̂, V_ξ) or empirical/bootstrap; forward-evaluate z^{(m)} = h(ξ^{(m)}), T^{(m)} = h_T(ξ^{(m)});
  u_c = std(•), U = [ q_{α/2}, q_{1−α/2} ]; convergence via stderr ≤ η•u_c or KS ≤ τ_KS.
• S65E-11: record ρ = u_c^{GUM} / u_c^{MC} for method switching and alerting.

8. Incorporating approximation errors

• S65E-12: include ε_* ∈ { ε_ray, ε_disp, ε_map, ε_lin } as independent terms:
  u_c^2 ← u_c^2 + ∑ u^2(ε_*); add 2•cov(ε_i,ε_j) if correlated.

9. Guardband synthesis

• S65E-13: base gb(z) = k•u_c(z), gb(T) = k•u_c(T);
  runtime extension gb' = k•u_c + β•drift_score•range (with drift_score from Appendix D; range via historical quantiles).

V. Workflow M65-E* (Ready → Model → Propagate → Check → Persist)

1. Ready: freeze RefCond, unit/dimension maps, window and quantiles; assemble/build V_ξ (measurement / model / structural / approximation / drift); set k, α, η and constraints.
2. Model: define h(ξ) (for z_path, T_arr*) and J; specify targets & tolerances delta_form / ΔT_map / ΔT_obs.
3. Propagate: run GUM and/or MC to obtain u_c, U, u(delta_form), u(ΔT_map), u(ΔT_obs), ρ.
4. Check:
   • Two-form gate: delta_form + k•u(delta_form) ≤ tol_Tarr;
   • Mapping gate: ΔT_map + k•u(ΔT_map) ≤ tol_map;
   • Residual gate: | z_meas − z_path | + k•u(resid_z) ≤ tol_z;
   • Conservation: projected T_arr* respects guard-bit/boundary constraints (Chs. 3/10).
5. Persist:
   manifest.redshift.u = { targets:{ z_path, T_arr*, delta_form, ΔT_map, ΔT_obs }, u:{ u_c, U, nu_eff, method, ρ }, sources:{ V_ξ, approx, map }, constraints, RefCond, contracts.*, signature }.

VI. Contracts & Assertions C65E-* (suggested thresholds)

• C65E-01 (GUM/MC consistency): 0.8 ≤ ρ ≤ 1.25, else prefer MC with annotation.
• C65E-02 (Coverage): coverage(U) ≥ 95% (k≈2 or empirical).
• C65E-03 (Two-form gate): delta_form + k•u(delta_form) ≤ tol_Tarr (Ch. 2).
• C65E-04 (Mapping gate): ΔT_map + k•u(ΔT_map) ≤ tol_map (Ch. 7).
• C65E-05 (Residual gate): | z_meas − z_path | + k•u(resid_z) ≤ tol_z (Ch. 12).
• C65E-06 (Approximation share): u(ε_*) / u_c ≤ 0.5; exceedance requires higher order or model change.
• C65E-07 (Freshness): age(RefCond) ≤ Δt_max; MC samples meet stderr ≤ η•u_c or N ≥ N_min.

VII. Implementation Bindings I65-E* (interfaces & invariants)

• I65-E1 build_sensitivity(models, gamma, RefCond) -> { J_z, J_T, meta }
• I65-E2 propagate_gum(J, V_ξ, constraints) -> { u_c, nu_eff, U }
• I65-E3 propagate_mc(sampler, N, α, η, constraints) -> { stats, U, ρ }
• I65-E4 compose_twoform_uncert(T_form1, T_form2, cov) -> { u(delta_form) }
• I65-E5 map_uncert_phase_group(n_phi, n_g, band, z_φ_series) -> { u(ΔT_map), method }
• I65-E6 design_guardband(metrics_u, drift_score, policy) -> { gb, gb', actions }
• I65-E7 assert_uncert_contracts(u_report, rules) -> { report, pass }
• I65-E8 emit_uncert_manifest(results, policy) -> { uri, status }
  Invariants: V_ξ ⪰ 0; two_forms_present = true; RefCond.hash / units / dim consistent; publish U/gB after constraint projection.

VIII. Cross-References

• Baseline & two forms: Ch. 2; component models: Chs. 3–6; dispersion mapping: Ch. 7; paths & integrals: Ch. 8; observation: Ch. 9; fusion & calibration: Chs. 10/11; contracts & metrics: Ch. 12; runtime: Ch. 14.
• Manifests & interfaces: Appendix C (manifest), Appendix A (I65-*), Appendix D (metrics), and Chapter 13 (runtime guardianship).

IX. Quality & Risk Control

• Monitors: u_c(z_path)_p95, U/|z_path|, u(delta_form)_p95, u(ΔT_map)_p95, ρ(GUM/MC), coverage, gb / tol_*.
• Fallback ladder: increase approximation order / update models → increase MC samples / tighten convergence → apply constraint projection / widen guardbands → degrade / bypass → rollback.
• Audit: V_ξ provenance, Jacobians/samplers & convergence logs, guardband change log, signature chain, and replay scripts.

Summary

• This appendix unifies PathRedshift uncertainty into a GUM/MC parallel workflow with two-form and mapping gates, constraint projection, and guardband synthesis.
• With manifest.redshift.u.* and C65E-*, redshift and arrival-time publications become measurable, auditable, and rollback-ready.