30-EFT.WP.Propagation.TensionPotential v1.0 | Chapter 12 — Error Budget & Systematic-Error Safeguards

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Chapter 12 — Error Budget & Systematic-Error Safeguards

• I. One-Sentence Aim
  Establish a full-chain error budget and systematic-error protection scheme from measured arrival time T_arr to the physical mapping n_eff = F( Phi_T, grad_Phi_T, … ). Provide two uncertainty-propagation paths (GUM and MC), covering gauge selection, path discretization, interface matching, band decomposition, c_ref calibration, TBN noise, and clamping/saturation. Fix acceptance criteria and falsification lines.
• II. Scope & Non-Goals
• Covered: error taxonomy and models; GUM and MC propagation; gauge-specific errors; n_common/n_path separation error; path discretization and quadrature error; inter-layer/interface error; c_ref and gauge errors; TBN injection; clamping and saturation; guardband strategy; audit and logging.
• Non-goals: no repetition of Chapter 4’s theory for Phi_T or Chapter 5’s n_eff; no replacement of Chapter 9’s numerical implementation details.
• III. Minimal Terms & Symbols
• Observations & model: T_arr_obs(f, gamma), T_arr_mod(f, gamma), Residual = T_arr_obs − T_arr_mod.
• Uncertainty: u_stat (statistical), u_sys (systematic), u_c (combined), coverage factor k, guardband GB = k_guard · u_c.
• Key quantities: c_ref, Phi_T(x,t), grad_Phi_T(x,t), n_eff(x,t,f); path gamma(ell), line element d ell, discretization { gamma[k], Δell[k] }.
• Decomposition: n_eff = n_common(x,t) + n_path(x,t,f); anisotropic term b1 · dot( grad_Phi_T , t_hat ).
• Gauges: mode ∈ {constant, general}; constant-factored T_arr = (1/c_ref) * ∫ n_eff d ell, general T_arr = ∫ ( n_eff / c_ref ) d ell.
• IV. Error Taxonomy & Models
• Source layering
• Metrology layer: u(c_ref), u(T_arr_obs), coordinate/unit mappings, timebase and synchronization.
• Modeling layer: u(Phi_T) (including gauge and boundaries), u(grad_Phi_T), u(n_eff) (including decomposition and anisotropy).
• Numerical layer: path discretization and quadrature error, interface segmentation and correction error, grid-sampling/interpolation error.
• Environment layer: TBN injection, drift and aging, out-of-band leakage.
• Correlations
• Same-band multi-path shares c_ref and n_common; same-path multi-band shares geometry and interfaces. Correlation coefficients must be explicit in propagation formulas or represented via MC sampling.
• V. GUM Propagation (Constant-Factored Gauge)
• Discrete form: T_arr ≈ (1/c_ref) * ∑_{k=0}^{N-1} n_eff[k] · Δell[k].
• First-order sensitivities
• ∂T_arr/∂c_ref = − (1/c_ref^2) * ∑ n_eff[k] · Δell[k] = − T_arr / c_ref
• ∂T_arr/∂n_eff[k] = (1/c_ref) · Δell[k]
• ∂T_arr/∂Δell[k] = (1/c_ref) · n_eff[k]
• Combined uncertainty
• u_c^2(T_arr) ≈ (∂T/∂c_ref)^2 u^2(c_ref) + ∑ ( (Δell[k]/c_ref)^2 u^2(n_eff[k]) ) + ∑ ( (n_eff[k]/c_ref)^2 u^2(Δell[k]) ) + 2∑∑ ρ_ij (∂T/∂q_i)(∂T/∂q_j) u(q_i) u(q_j)
• where q_i spans c_ref, n_eff[*], Δell[*] and required couplings; methods for estimating ρ_ij must be reported.
• VI. GUM Propagation (General Gauge)
• Discrete form: T_arr ≈ ∑ ( n_eff[k] / c_ref[k] ) · Δell[k].
• First-order sensitivities
• ∂T_arr/∂n_eff[k] = Δell[k] / c_ref[k]
• ∂T_arr/∂c_ref[k] = − n_eff[k] · Δell[k] / c_ref[k]^2
• ∂T_arr/∂Δell[k] = n_eff[k] / c_ref[k]
• Correlation modeling: spatiotemporal correlation of c_ref[k] enters via covariance functions or block-constant approximations; correlation with n_eff[k] must be quantified during calibration.
• VII. MC Uncertainty Propagation
• Sampling flow
• Generate N sample sets { c_ref^(s), n_eff^(s)[k], Δell^(s)[k] }, preserving estimated correlations (e.g., shared common terms and path-conformal weights).
• Compute T_arr^(s) per sample; obtain the distribution and quantiles.
• Report median(T_arr), mean ± k·std, and tail risk.
• Use cases: validate the linearity assumptions of GUM; act as the primary reporting method under strong nonlinearity (clamping, interface jumps) or strong correlation.
• VIII. Gauge-Specific Errors & Consistency
• Selection criterion: if max |δc_ref/c_ref| ≤ eta_c, use the constant-factored gauge; otherwise use the general gauge and record the estimation method and uncertainty of c_ref(x,t,f).
• Consistency metric: eta_T = | T_arr^{const} − T_arr^{gen} |; require eta_T ≤ threshold. If exceeded, revisit c_ref calibration or n_eff decomposition (see Chapter 7).
• IX. Errors in Separating n_common and n_path
• Funnel design: fit n_path on multi-band data along the same path; estimate n_common from low-frequency bands.
• Residual routing: out-of-band leakage, order mismatch, and frequency misalignment residuals are pooled into u_sys(n_path).
• Differencing robustness: when computing ΔT_arr(f1,f2), enforce identical { γ[k], Δell[k] } and the same step-size policy to minimize numerical contamination.
• X. Path Discretization & Quadrature Errors
• Step control: dual thresholds on geometric curvature and medium variation drive Δell; use higher-order quadrature locally and report local error estimates.
• Convergence tests: two-/three-level refinement with ratio r, target | T_arr^{(fine)} − T_arr^{(coarse)} | ≤ eps_T.
• Grid interpolation: when sampling n_eff(γ[k]) from a grid, record interpolation order and kernel; interpolation error enters u_sys.
• XI. Interface & Inter-Layer Errors
• Matching modes: continuous, potential-jump, flux-jump, and anisotropic interfaces (see Chapter 8).
• Segmented integration: split at { ell_i }; do not interpolate across interfaces. Zero-thickness correction terms ΔT_sigma must have trigger counts and amplitudes logged and budgeted into u_sys.
• Energy consistency: R_sigma + T_trans + A_sigma = 1; any side that implies n_eff < 1 is infeasible and must be recorded as a falsification sample.
• XII. c_ref Calibration & Gauge Errors
• c_ref: calibrate from gamma_ref, T_arr_ref; record environmental blocks and drift curves. When reused across environments, contribute u_sys(c_ref) = drift_budget.
• Gauge & boundaries: with Phi_T(x_ref,t_ref) = 0 and boundary_config fixed, observables depending only on grad_Phi_T must be invariant; measurable changes constitute a falsification line and require revisiting Chapter 4’s construction.
• XIII. TBN Noise & Environmental Drift
• Model: TBN(x,t) enters as a zero-mean broadband term in the statistics of n_eff or as a perturbation; SNR and out-of-band leakage are assessed via MC injection.
• Reporting: provide increments in u_stat and the clamping trigger rate for n_eff; if above threshold, list a dedicated “saturation-induced bias risk.”
• XIV. Clamping, Saturation & Constraint Errors
• Rule: enforce n_eff ∈ [1, n_max]. When clamping triggers, local sensitivities distort and GUM linearity may fail.
• Strategy: in MC, use truncated or reflected sampling; report the clamping trigger fraction and the induced shift in T_arr.
• XV. Bias Detection, Falsification & Guardband
• Bias detectors
• Lower bound: T_arr_obs − L_path/c_ref < −k·u_c.
• Gauge: eta_T > threshold.
• Differentials: ΔT_arr vs. model differentials non-linear in-band or slope outside threshold.
• Interfaces: energy-consistency violation or one-sided limits yielding n_eff < 1.
• Falsification line: if any detector fires and implementation errors are excluded, record as a falsification sample and route to Chapter 11 audit.
• Guardband: set k_guard per application risk to form GB = k_guard · u_c; send edge samples to a review queue.
• XVI. Logging & Audit (Minimal Fields)
• Metrology: hash(Phi_T), hash(grad_Phi_T), hash(n_eff), hash(gamma), mode, c_ref calibration details, coordinate/unit contracts.
• Numerics: SolverCfg, quadrature method, step policy and thresholds, interface marks { ell_i }, correction triggers.
• Errors: u_stat, u_sys, u_c, k, GB, clamping trigger rate, eta_T, lower-bound margin T_arr_obs − L_path/c_ref.
• MC: sample size N, RNG seed, correlation-sampling policy; output quantiles and tail metrics.
• Conclusions: pass/falsification tags with causes, rollback handles, and replay parameters.
• XVII. Acceptance Criteria
• Dimensions & lower bound: dim(T_arr) = [T], T_arr_obs ≥ L_path/c_ref.
• Consistency: two-gauge consistency within threshold; band differencing isolates the path term within the target band.
• Stability: refinement convergence achieved; c_ref drift within band; interface energy consistency holds.
• Transparency: logs complete; hashes consistent; replay succeeds.
• XVIII. Cross-References
• EFT.WP.Propagation.TensionPotential v1.0 Chapters 3, 5, 6, 7, 8, 9, 11
• EFT.WP.Core.Metrology v1.0 M05-, M10-
• EFT.WP.Core.Errors v1.0 M20-*
• EFT.WP.Core.Equations v1.1 S06-*
• XIX. Deliverables
• Error-budget checklist: inputs, correlation assumptions, and output metrics for GUM and MC.
• Safeguard playbook: operational manuals for gauges, paths, interfaces, term separation, c_ref, TBN, and clamping/saturation.
• Audit templates: bias-detection dashboards, falsification cards, guardband configuration, and replay instructions.