28-EFT.WP.Propagation.PathRedshift v1.0 | Chapter 13 — Uncertainty & Guardbands (GUM / MC)

Home ／ Docs-Technical WhitePaper ／ 28-EFT.WP.Propagation.PathRedshift v1.0

Chapter 13 — Uncertainty & Guardbands (GUM / MC)

One-sentence goal: Establish a unified framework for propagating uncertainty in path redshift z_path and **arrival time T_arr* via GUM linearization (LPU) and Monte Carlo (MC); synthesize uncertainties for two-form gaps and phase/group mappings and design guardbands; persist all artifacts to enable auditable publication and runtime protection.

I. Scope & Objects

1. Inputs
   • Targets: z_parts = { z_kin, z_grav, z_med, z_cos, z_inst, z_proc }, composed z_path; T_arr^{form1/form2}, harmonized T_arr*.
   • Sources & priors: RefCond (ephemerides / potentials / media / meteorology / timebase / model version hashes), observations z_meas (Ch. 9), path & ray gamma(ell) (Ch. 8), phase/group mapping parameters (Ch. 7), clock parameters offset / skew / J (Ch. 10).
   • Covariance: V_ξ (measurement noise / model params / structural bounds / approximation errors / runtime drift).
2. Outputs
   Combined standard uncertainties u_c(z_path), u_c(T_arr*) and coverage U = k•u_c; uncertainties for two-form gaps and mappings u(delta_form), u(ΔT_map);
   guardbands for publication and the results of on-boarding gates;
   manifest manifest.redshift.u.* and contracts C65-13x.
3. Boundary
   Prefer GUM for weak nonlinearity/small perturbations; upon detecting strong nonlinearity / thresholds / discrete effects, switch to MC or piecewise linearization and record method and rationale.

II. Terms & Variables

• Source vector:
  ξ = [ x_geo, n_params, ephem, grav, iono/trop, TEC, osc(offset,skew,J), approx, map_params, … ]^T.
• Jacobians & covariance: J = ∂h/∂ξ |_{ξ̂}, V_ξ ⪰ 0; targets z = h(ξ) and T = h_T(ξ).
• Coverage: U = k•u_c (k ≈ 2 ↔ ~95%), effective dof nu_eff (Welch–Satterthwaite).
• Two-form & mappings: delta_form = | T_arr^{form1} − T_arr^{form2} |, ΔT_map = | T_g − T_phi | (Ch. 7), ΔT_obs = | T_arr* − t̂_cont | (Chs. 8/9).
• Composite redshift: 1+z_path = ∏_i (1+z_i); small-signal z_path ≈ ∑_i z_i.
• Dimensions: check_dim(z) = "1", check_dim(T) = "[T]".

III. Postulates P65-13x

• P65-1301 (Dual-track): For every published target z / T, provide both GUM and (as needed) MC results; record method, sample size, and consistency ρ = u_c^{GUM} / u_c^{MC}.
• P65-1302 (Two forms & mappings): Provide explicit uncertainties for delta_form / ΔT_map / ΔT_obs, and include them in go/no-go gates and guardbands.
• P65-1303 (Explicit measures): Any propagation integral states its domain/measure: ( ∫_{gamma(ell)} ), ( ∫_{t∈W} ), ( ∫_{f∈B} ).
• P65-1304 (Dimensions & RefCond): All inputs/outputs pass check_dim( y − f(x) ); RefCond.hash / versions and V_ξ sources are traceable.
• P65-1305 (Conservation-first): Apply constraint projection before publishing uncertainties and guardbands for constrained quantities (energy/boundaries/guard bits).

IV. Minimal Equations S65-13x

1. GUM linearization (LPU)

• S65-1301:
  u_c^2(z) = J_z V_ξ J_z^T, u_c^2(T) = J_T V_ξ J_T^T.
• S65-1302 (Welch–Satterthwaite):
• nu_eff ≈ ( ∑ u_i^4 / ν_i ) / ( ∑ u_i^2 )^2,
• U = k•u_c, k = t_{ nu_eff, 1−α/2 }.

2. Sensitivity of composite redshift

• S65-1303 (Product form):
  ln(1+z_path) = ∑ ln(1+z_i) ⇒ ∂z_path/∂z_i = ∏_{j≠i} (1+z_j) ≈ 1 (small-signal);
  rigorously propagate in log space with J_log = ∂ ln(1+z_path)/∂ z_i = 1/(1+z_i).
• S65-1304 (Linear-sum approximation):
  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

• S65-1305:
• u^2(delta_form) ≈ u^2(T_form1) + u^2(T_form2) − 2•cov(T_form1, T_form2)
• 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 uncertainty (weak dispersion)

• S65-1306: 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

(first-order retention).

5. Clock / sync uncertainty

• S65-1307: With T_arr* … + ΔT_sync (Ch. 10),
• u^2(ΔT_sync) ≈ u^2(offset) + ( T_win • u(skew) )^2 + u^2( E[J] )

where E[J] variance follows the PSD model.

6. Inclusion of approximation errors

• S65-1308: Add independent approximation/truncation terms:
• u_c^2 ← u_c^2 + u^2(ε_ray) + u^2(ε_disp) + u^2(ε_map) + u^2(ε_lin)

Include cross-terms if correlated.

7. Monte Carlo (MC)

• S65-1309: 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-distance threshold ≤ τ_KS.
• S65-1310 (GUM–MC consistency): Record ρ = u_c^{GUM} / u_c^{MC} for method selection and alerts.

8. Constraint projection (conservation)

• S65-1311: For linear constraints C y = d, project
• y' = y − C^T (C C^T)^{-1} ( C y − d ), V_{y'} = (I − C^T(CC^T)^{-1}C) V_y (I − …)^T

applied to T_arr* and guard-bit constraints (Chs. 3/10).

9. Guardband synthesis

• S65-1312: Base guardbands 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).

V. Metrology Pipeline M65-13 (Ready → Model → Propagate → Check → Persist)

1. Ready: freeze RefCond, unit/dimension maps, window W and quantiles; collect priors and V_ξ (including approximation errors); set method (GUM/MC), k, α, η, and constraints.
2. Model: build h(ξ) for z_path and T_arr* and Jacobians J; define targets and tolerances delta_form / ΔT_map / ΔT_obs.
3. Propagate: run GUM and/or MC to obtain u_c(z_path), u_c(T_arr*), U, u(delta_form), u(ΔT_map), u(ΔT_obs), and ρ.
4. Check:
   • Two-form gate: delta_form + k•u(delta_form) ≤ tol_Tarr;
   • Mapping gate: ΔT_map + k•u(ΔT_map) ≤ tol_map;
   • Analytic vs observation: | z_meas − z_path | + k•u(resid_z) ≤ tol_z;
   • Conservation: post-projection T_arr* meets guard-bit & boundary constraints;
   • Dimensions & provenance: all pass.
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_params },

constraints, RefCond, contracts.*, signature

}

VI. Contracts & Assertions C65-13x (suggested thresholds)

• C65-1301 (GUM/MC consistency): 0.8 ≤ ρ ≤ 1.25; otherwise prefer MC and annotate rationale.
• C65-1302 (Coverage publication): report U at ≥ 95% confidence (k ≈ 2 or empirical quantiles).
• C65-1303 (Two-form gate): delta_form + k•u(delta_form) ≤ tol_Tarr (Ch. 2).
• C65-1304 (Mapping gate): ΔT_map + k•u(ΔT_map) ≤ tol_map (Ch. 7).
• C65-1305 (Residual gate): | z_meas − z_path | + k•u(resid_z) ≤ tol_z (Ch. 12).
• C65-1306 (Approximation share): for any approximation term *, u(ε_*)/u_c ≤ 0.5; if exceeded, raise order or change models.
• C65-1307 (Freshness): age(RefCond) ≤ Δt_max; MC sample size satisfies stderr ≤ η•u_c or N ≥ N_min.

VII. Implementation Bindings I65-13* (interfaces, I/O, invariants)

• I65-131 build_sensitivity(models, gamma, refcond) -> { J_z, J_T, meta }
• I65-132 propagate_gum(J, V_ξ, constraints) -> { u_c, nu_eff, U }
• I65-133 propagate_mc(sampler, N, α, η, constraints) -> { stats, U, ρ }
• I65-134 compose_twoform_uncert(T_form1, T_form2, cov) -> { u(delta_form) }
• I65-135 map_uncert_phase_group(n_phi, n_g, band, z_φ_series) -> { u(ΔT_map), method }
• I65-136 design_guardband(metrics_u, drift_score, policy) -> { gb, gb', actions }
• I65-137 assert_uncert_contracts(u_report, rules) -> { report, pass }
• I65-138 emit_uncert_manifest(results, policy) -> manifest.redshift.u
  Invariants: two_forms_present = true; check_dim(*) passes; RefCond.hash / V_ξ / sampler traceable; apply constraint projection before publishing U and gb.

VIII. Cross-References

• Baseline & two forms: Ch. 2; components: Chs. 3–6; dispersion mapping: Ch. 7; rays & path integrals: Ch. 8; observations: Ch. 9; fusion & calibration: Chs. 10/11; contracts & metrics: Ch. 12.
• Manifests & interfaces: Appendix C (manifest.redshift), Appendix A (I65-*), Appendix D (metrics/drift), Appendix E (implementation details for uncertainty).

IX. Quality & Risk Control

• SLI / SLO: u_c(z_path)_p95, U/|z_path|, u(delta_form)_p95, ρ(GUM/MC), coverage, gb / tol_* ratio.
• Fallback ladder: raise approximation order / update models → increase MC samples & convergence monitoring → constraint projection / widen guardbands → degrade/bypass → rollback to the previous manifest.
• Audit: V_ξ provenance, Jacobians/samplers & convergence logs, guardband change records, and the manifest signature chain with replay scripts.

Summary

• This chapter unifies uncertainty for path redshift and arrival time into a GUM/MC parallel workflow with two-form gates, mapping gates, constraint projection, and guardband synthesis.
• With M65-13 / C65-13x / I65-13* and manifest.redshift.u.*, publication and runtime become measurable, auditable, and rollback-ready, setting the stage for Chapter 14 on runtime & streaming release.