23-EFT.WP.Metrology.PathCorrection v1.0 | Chapter 13 — Uncertainty and Guardband (GUM / MC)

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Chapter 13 — Uncertainty and Guardband (GUM / MC)

One-Sentence Goal
Under a unified measurement model, propagate uncertainties from geometry / media / instruments and numerical integration into T_arr / T_corr, provide two complementary approaches—GUM linearization and distributional Monte Carlo (MC)—and configure guardbands and compliance decisions accordingly.

I. Scope and Objects

1. Inputs
   • Measurement model: T_arr = ( 1 / c_ref ) * ( ∫_{gamma} n_eff d ell ) + T_inst + T_proc together with the component models in Chapters 5–12.
   • Variables & covariance: x = [x_geom, x_tropo, x_iono, x_fiber, x_inst, x_num], prior V_x, and cross-correlations.
   • Numerical-integration quality: u_q (quadrature), u_interp (interpolation), u_geom (geometry), delta_form (two-form discrepancy).
   • Specifications / decisions: tol_Tarr, compliance-risk parameters { alpha_consumer, alpha_producer }.
2. Outputs
   • Combined standard uncertainty u_c(T_arr), coverage interval U = k * u_c or quantile band [q_L, q_U].
   • Guardband g and a compliance decision report.
   • Dominant-term breakdown and sensitivities J; persisted to manifest.path.u.
3. Constraints
   All variables must explicitly declare unit/dim and pass check_dim; RefCond and tau_mono/ts must be consistent.

II. Terms and Variables

• u(x): standard uncertainty; U = k * u_c: coverage uncertainty; nu_eff: effective degrees of freedom.
• J = ∂f/∂x: Jacobian / sensitivity matrix, unit(J_ij) = unit(y)/unit(x_j).
• V_x: input covariance matrix; rho_ij: correlation coefficient; V_x[i,j] = rho_ij * u(x_i) * u(x_j).
• u_num: synthesized standard uncertainty of numerical terms (u_q, u_interp, u_geom, u_form).
• u_form: numerical-model term for the two-form difference, u_form = delta_form / sqrt(3) (uniform-bound assumption).
• T_corr: total path correction (see Chapter 12 composition interface).
• Distribution assumptions: N(μ,σ^2) (Normal), U[a,b] (Uniform), T (Student), LN (Lognormal), etc.

III. Axioms P813-*

• P813-1 (Unified Measurement Model) — Uncertainty propagation is based on y = f(x), where y ∈ { T_arr, T_corr } and x includes all influence quantities and numerical terms.
• P813-2 (Two-Form Accounting) — delta_form must enter the budget as u_form, on equal footing with u_q, u_interp, u_geom.
• P813-3 (Explicit Correlation) — Shared sources across components (e.g., the same met_3D or temperature sensor) must be entered with rho_ij ≠ 0; independence defaults are forbidden.
• P813-4 (Unit Consistency) — Apply check_dim to J, V_x, and results; check_dim( u_c ) = "[T]".
• P813-5 (Dual-Form Gate) — Publish uncertainty only when delta_form ≤ tol_Tarr and u_form has been included.
• P813-6 (GUM/MC Cross-Validation) — For strong nonlinearity or non-Gaussian inputs, MC is the primary approach; verify first-order consistency with GUM linearization.
• P813-7 (Traceability) — Budget sources, distribution assumptions, correlation structure, MC sample size N_mc, and random seeds must be persisted.

IV. Minimal Equations S813-*

1. S813-1 (GUM Linearized Combination)
   u_c^2(y) = J V_x J^T + u_num^2, where u_num^2 = u_q^2 + u_interp^2 + u_geom^2 + u_form^2.
   check_dim( J V_x J^T ) = "[T]^2", check_dim( u_num ) = "[T]".
2. S813-2 (Jacobian Decomposition and Path Measure)
   For T_med = ( 1 / c_ref ) * ( ∫_{gamma} n_eff d ell ):
   ∂ T_med / ∂ n_eff(x) = ( 1 / c_ref ) * w(x), where w(x) is the measure weight along gamma(ell);
   for c_ref: ∂ T_med / ∂ c_ref = - ( 1 / c_ref^2 ) * ( ∫_{gamma} n_eff d ell ).
3. S813-3 (Welch–Satterthwaite)
   nu_eff = ( u_c^4 ) / ( ∑_i ( c_i^4 * u_i^4 / nu_i ) ), where c_i are sensitivity coefficients.
   Coverage factor k = t_{nu_eff, 1-α/2} (when using GUM).
4. S813-4 (MC Coverage Quantiles)
   Sample x^(k) ~ P_x, compute y^(k) = f( x^(k) ), k = 1…N_mc;
   take quantiles [q_L, q_U] = [ Q(α/2), Q(1-α/2) ] and define U_MC = max( ŷ - q_L, q_U - ŷ ), where ŷ is the sample median or mean.
5. S813-5 (Dominant-Term Contribution Ratio)
   For term i, variance share η_i = ( (J_i)^2 * Var(x_i) ) / u_c^2 (or MC-based Shapley/regression alternatives) to rank dashboard contributors.
6. S813-6 (Rules → Guardband)
   For a two-sided spec |y| ≤ T_spec:
   • GUM: g = z_{β} * u_c or g = t_{nu_eff,β} * u_c;
   • MC: with consumer risk α_consumer, choose g such that Pr( |y| ≤ T_spec - g ) ≥ 1 - α_consumer.
7. S813-7 (Decision Function)
   Acceptance region: A = { |y_meas| ≤ T_spec - g }; gray zone triggers re-measurement or uncertainty inflation; rejection region: |y_meas| > T_spec - g.
8. S813-8 (Numerical Noise Floor)
   If cumulative rounding noise is u_round, the minimal attainable floor is u_floor = max( u_round, ε_machine * |y| ), constraining u_num ≥ u_floor.

V. Metrological Workflow M80-13

1. Ready — Collect u(x), distributions, and rho_ij for inputs x from Chapters 5–12; synchronize RefCond and tau_mono.
2. Sensitivities
   • Obtain J via analytic / automatic differentiation / complex-step;
   • From the integration layer, ingest u_q, u_interp, u_geom, delta_form (Chapter 10) and synthesize u_num.
3. GUM Combination — Compute u_c, nu_eff, k, and U = k * u_c; output dominant shares η_i.
4. MC Check / Primary Route
   • Sample x^(k) respecting distributions and correlation;
   • Compute y^(k), derive [q_L,q_U], U_MC;
   • If |U_MC - U| / U > thr_diff, use MC as the publication route and tag accordingly.
5. Guardband — From target risks { alpha_consumer, alpha_producer } and the chosen route (GUM/MC), compute g and define decision regions.
6. Compliance — Compare T_arr / T_corr to SLOs; produce pass / marginal / fail and remediation (retest / down-weight / fallback).
7. Persist —
   manifest.path.u = { u_c, U, nu_eff, method:{GUM|MC}, U_MC?, q:[q_L,q_U], J.hash, dominant:{η_top3}, u_num:{u_q,u_interp,u_geom,u_form}, guardband:g, risk:{alpha_consumer,alpha_producer}, seeds, N_mc, tags }.
8. Dashboard & Feedback — Display η_i ranking, nu_eff, U vs U_MC, delta_form, and u_num drifts; when thresholds are exceeded, rebuild V_x or densify integration.

VI. Contracts and Assertions (C80-13xx)

• C80-1301 Two-Form Accounting — u_form = delta_form / sqrt(3) is mandatory and logged alongside u_num.
• C80-1302 MC Sample Size — N_mc ≥ max( 10^4, ceil( z_{0.995}^2 / ε_rel^2 ) ), default ε_rel = 0.05 for the relative error of U.
• C80-1303 Degrees-of-Freedom Gate — If nu_eff < 20 or the model is strongly nonlinear (diagnosed by |U_MC - U|/U > 0.1), MC must be the publication route.
• C80-1304 Correlation Structure — Where shared sources exist (same met_3D, same thermal loop), |rho_ij| must not default to zero; if missing, tag rho_missing and inflate U.
• C80-1305 Numerical Floor — Enforce u_num ≥ u_floor; if violated, raise numerical precision or enable compensated summation (Chapter 10).
• C80-1306 Dominant-Coverage — The top K contributors (default 3) must explain ∑_{i=1..K} η_i ≥ 0.8; otherwise refine budget granularity.
• C80-1307 Guardband Risk — Explicitly state { alpha_consumer, alpha_producer } at the use-case level; defaults 2.5% / 2.5% (two-sided 95%).
• C80-1308 Unit Consistency — check_dim( u_c ) = "[T]", check_dim( g ) = "[T]".
• C80-1309 Traceability — J.hash, seeds, N_mc, and V_x.hash are mandatory; if missing, tag trace_missing and refuse publication.

VII. Implementation Bindings I80-*

• I80-131 build_uncertainty_budget(parts, cov_spec) -> { x, V_x, u_num, meta }
  Invariant: dim(V_x) = dim(x), PSD.
• I80-132 compute_jacobian(f, x, mode) -> { J, method } (mode ∈ { analytic, autodiff, complex-step, finite-diff }).
• I80-133 propagate_gum(J, V_x, u_num) -> { u_c, nu_eff, k, U, eta }
• I80-134 propagate_mc(f, Px, N_mc, seeds) -> { q_L, q_U, U_MC, y_hat }
• I80-135 choose_guardband(U_or_q, risk) -> { g, rule }
• I80-136 assert_uncertainty_contracts(payload, rules) -> report
• I80-137 emit_path_manifest_uncertainty(payload, policy) -> manifest.path.u

VIII. Cross-References

• Path integration and numerical error sources: Chapter 10.
• Environmental components and state covariance sources: Chapter 11.
• Instrument / processing-chain uncertainty and drift: Chapter 12.
• Time-base / sync and uncertainty accounting of offset/skew/J: EFT.WP.Metrology.TimeBase v1.0, …Sync v1.0.
• Cleaning and dual-form contracts: EFT.WP.Methods.Cleaning v1.0.

IX. Quality and Risk Control

• SLO — Suggested p95( U ) ≤ 1.0 ns for free-space links (10–50 km), ≤ 0.3 ns for indoor fiber; raise the p99 tier by one.
• Monitoring — Time series of U, U_MC, nu_eff, η_top3, delta_form, and u_num on dashboards; exceedances trigger recomputation and re-calibration.
• Fallback — When rho_missing / trace_missing occurs or N_mc is insufficient, revert to a conservative guardband g = z_{0.999} * u_c and apply downgrade tags.

Summary
This chapter establishes a unified GUM-and-MC framework for uncertainty propagation and consistency, mandates accounting for two-form and numerical terms, enforces explicit correlations, and provides risk-informed guardband configuration.
Key persisted object:
manifest.path.u = { u_c, U, nu_eff, method, U_MC?, q:[q_L,q_U], dominant:{η_*}, u_num:{u_q,u_interp,u_geom,u_form}, guardband:g, risk, seeds, N_mc, J.hash, V_x.hash, tags }.
Together with Chapters 5–12 (component modeling / integration / instrumentation) and the TimeBase/Sync/Cleaning volumes, publication of T_arr / T_corr carries auditable uncertainty and compliance determinations.