31-EFT.WP.BH.TensionWall 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 an end-to-end error model and protection framework around the Tension Wall Sigma_TW. From measured arrival time T_arr through the full chain to n_eff = F( Phi_T, grad_Phi_T, … ) + H_TW, define GUM and MC uncertainty-propagation routes, gauge-specific errors, thin/thick-wall switching error, interface and energy-consistency errors, band-differencing leakage, directional and multi-path weight errors, together with safeguards and auditable procedures.
• II. Scope & Non-Goals
• Covered: error taxonomy and models; GUM/MC propagation formulae (for both gauges); consistency between ΔT_sigma and thick-wall volume integration; interface matching and energy-triplet errors; path discretization and quadrature errors; band differencing and out-of-band leakage; c_ref calibration and drift; gauge/boundary and coordinate/unit consistency; clamping/saturation and other nonlinearities; guardband strategy; logging and falsification lines.
• Not covered: equation derivations and implementation details from Chapters 3–9; device-level mechanical/electrical designs.
• 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, segments { ell_i }.
• Wall-related: TWProfile, W(r), Xi_TW(r), ΔT_sigma (zero-thickness correction), R_TW, T_trans, A_sigma with R_TW + T_trans + A_sigma = 1.
• Gauges: mode ∈ {constant, general}; constant-factored T_arr = (1/c_ref) * ∫ n_eff d ell, general T_arr = ∫ ( n_eff / c_ref ) d ell.
• Constraints: n_eff ≥ 1, T_arr ≥ L_path / c_ref (equivalently embedded in the integrand under the general gauge).
• IV. Error Taxonomy & Source Layers
• Metrology layer: u(T_arr_obs), u(c_ref), timebase & synchronization, coordinate/unit mappings, environment blocks (temperature/humidity, timebase source).
• Modeling layer: u(Phi_T) (including gauge & boundary), u(grad_Phi_T), u(n_eff) (parameters of F and H_TW), u(TWProfile) (r_H, Δw, sigma_w, a_{lm}, …).
• Interface layer: misclassification error among continuous/potential-jump/flux-jump; u(C_sigma), u(J_sigma); triplet-estimation errors u(R_TW), u(T_trans), u(A_sigma).
• Numerical layer: path discretization and quadrature errors; interface-endpoint solving and segmentation errors; interpolation errors; thin/thick switching error τ_switch.
• Band & directional layer: out-of-band leakage in ΔT_arr; regression uncertainty of directional terms (b1, b1_sigma); estimation error of multi-path weights w_m.
• Environment layer: injected TBN(x,t) and drift/aging.
• V. GUM Uncertainty 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 = − T_arr / c_ref
• ∂T_arr/∂n_eff[k] = Δell[k] / c_ref
• ∂T_arr/∂Δell[k] = n_eff[k] / c_ref
• Composition:
• 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)
• with q_i ∈ { c_ref, n_eff[*], Δell[*] }. The correlation coefficients ρ_ij must have a reported estimation method.
• VI. GUM Uncertainty 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]
• Correlations: the spatiotemporal correlation of c_ref[k] enters via a covariance function or block-constant approximation; correlations between n_eff[k] and c_ref[k] must be assessed during calibration.
• VII. MC Uncertainty Propagation (Primary Route for Nonlinearity & Clamping)
• Sampling flow
• Generate N samples { c_ref^(s), n_eff^(s)[k], Δell^(s)[k] }, preserving correlation structure between shared/common and path terms.
• Compute T_arr^(s) per sample to obtain distribution, quantiles, and tail metrics.
• Report median(T_arr) and mean ± k·std, to be compared with GUM.
• Applicability: when clamping n_eff ∈ [1, n_max] is active; when ΔT_sigma triggers discretely; when interface types switch or weights w_m induce strong nonlinearity—use MC as the primary reporting route.
• VIII. Gauge-Specific Errors & Consistency
• Selection criterion: use the constant-factored gauge if max_{ell} |δc_ref/c_ref| ≤ eta_c; otherwise use the general gauge and record c_ref(x,t,f) estimation and u(c_ref[k]).
• Consistency metric: eta_T = | T_arr^{const} − T_arr^{gen} | must be within threshold; exceedances trigger, in order, a revisit of c_ref calibration, n_eff decomposition, path & segmentation, and ΔT_sigma consistency.
• IX. Thin/Thick-Wall Switching Error & Remediation
• Definition: τ_switch = | T_arr^{thick} − ( T_arr^{thin} + ΔT_sigma ) |.
• Switch rule: use thin-wall when Δw / r_H ≤ eta_w; perform dual computations in that neighborhood and report τ_switch.
• Safeguards:
• Solve endpoints { ell_i } with convergence thresholds and symmetric-interval refinement.
• Force step reduction where |dW/dr| is large to avoid under-sampling the correction.
• If τ_switch > tau_switch, lock the thick-wall pipeline and revisit TWProfile or endpoint tolerances.
• X. Interface Matching & Energy-Consistency Errors
• Type misclassification: continuous/potential-jump/flux-jump errors bias n_eff^± side limits, entering u_sys.
• Triplet error: uncertainties in R_TW, T_trans, A_sigma affect weights w_m and T_arr_total.
• Safeguards:
• Audit R_TW + T_trans + A_sigma = 1 residual for every interface event.
• Test consistency between segmented integration and zero-thickness corrections.
• Enforce side-limit lower bounds n_eff^± ≥ 1 as hard constraints—violations are falsifications.
• XI. Band Differencing & Out-of-Band Leakage
• Differential forms (same path)
• Constant-factored: ΔT_arr(f1,f2) = (1/c_ref) ∫ [ n_path(f1) − n_path(f2) ] d ell
• General: ΔT_arr(f1,f2) = ∫ [ ( n_path(f1) − n_path(f2) ) / c_ref ] d ell
• Leakage & alignment: route out-of-band power into u_sys; reuse identical { gamma[k], Δell[k] } and identical ΔT_sigma settings at both frequencies.
• Safeguards: configure out-of-band suppression thresholds and frequency alignment policies; log leakage ratios with residual attribution.
• XII. Path Discretization, Quadrature & Interpolation Errors
• Step-size control: triple triggers on curvature ‖ d^2γ/dℓ^2 ‖, medium variation | d n_eff / dℓ |, and wall gradient | dW/dr |; include interface endpoints explicitly.
• Error estimation: estimate local error via two quadrature orders on the same segment; global error is the root-sum-square of locals, targeting | T_arr^{(fine)} − T_arr^{(coarse)} | ≤ eps_T.
• Interpolation: in grid-first methods fix interpolation order; use symmetric stencils near endpoints; never interpolate across interfaces—segment instead.
• XIII. c_ref Calibration & Drift Safeguards
• Calibration: c_ref = ( ∫ n_eff_ref d ell ) / T_arr_ref (constant-factored) or solve numerically under the general gauge; record environment blocks and u_stat, u_sys.
• Drift: maintain a drift curve c_ref(t); for cross-environment reuse include a drift_budget in u_sys(c_ref).
• Consistency: require eta_T to remain within threshold under cross-application across paths and bands.
• XIV. Gauge/Boundary & Coordinate/Unit Consistency
• Gauge & boundary: fix Phi_T(x_ref,t_ref) = 0 and boundary_config; if observables depend only on grad_Phi_T, they must be invariant under Phi_T → Phi_T + const—violations falsify the construction.
• Coordinates & units: coords_spec and units_spec are mandatory; ensure units of Δell and c_ref match; perform and record any transforms at ingress.
• Dimensional guarding: enforce check_dimension at ingress—dim(T_arr)=[T], dim(n_eff)=1, dim(c_ref)=[L][T^-1].
• XV. Clamping, Saturation & Nonlinearity
• Clamping rule: n_eff ∈ [1, n_max]; clamping reduces local sensitivities and introduces bias.
• Safeguards: use truncated or reflected sampling in MC; report clamping trigger rates and the induced shift in T_arr; if triggers exceed threshold, revisit modeling and data quality.
• XVI. Bias Detectors, Falsification Lines & Guardband
• Bias detectors
• Lower bound: T_arr_obs − L_path/c_ref < −k·u_c.
• Gauge: eta_T > threshold.
• Thin/thick: τ_switch > tau_switch.
• Differential: ΔT_arr nonlinearity or slope mismatch not attributable to out-of-band leakage.
• Energy: R_TW + T_trans + A_sigma ≠ 1.
• Side limits: any n_eff^± < 1.
• Falsification: if any detector fires and implementation errors are excluded, record as a falsification sample and route to Chapter 11 audit; three consecutive independent replications on the same dimension trigger review of Chapter 2 P40-* and Chapter 5 TWProfile.
• Guardband: GB = k_guard · u_c; send edge samples to the review queue.
• XVII. Logging & Audit (Minimal Fields)
• Physics & geometry: hash(Phi_T), hash(grad_Phi_T), hash(n_eff), hash(gamma), Sigma_TW labels and { ell_i } tolerances.
• Gauges & thresholds: mode, eps_T, eta_T, eta_c, eta_w, tau_switch, lower-bound margin T_arr − L_path/c_ref.
• Energy & differentials: residuals for R_TW,T_trans,A_sigma; ΔT_sigma trigger stats; ΔT_arr linear-region diagnostics and leakage ratios.
• Uncertainty: u_stat, u_sys, u_c, GUM/MC configs, k, seed.
• Clamping/saturation: n_eff clamping triggers and impact assessment.
• Audit bundle: data, code, parameters, SolverCfg, hash manifest, falsification samples, and replay entry.
• XVIII. Interfaces & Implementation Bindings (I40- Safety Checks)
• check_dimension( expr ) -> DimReport — dimensional/unit checks with anomaly reporting.
• propagate_uncertainty_GUM( inputs ) -> u_c, propagate_uncertainty_MC( inputs, Nsamples, seed ) -> dist(T_arr) — the two propagation routes.
• consistency_dual_mode_TW( inputs ) -> eta_T — two-gauge consistency.
• consistency_thin_vs_thick_TW( inputs ) -> tau_report — thin/thick consistency.
• rt_estimator_TW( data ) -> R_TW, T_trans, A_sigma — energy-consistency audit.
• log_tw_propagation( meta, hashes, metrics ) -> Log — logging & audit sealing.
• Constraints: at every entry, enforce n_eff ≥ 1 and the lower-bound audit; for cross-band differencing reuse the same { gamma[k], Δell[k] } and identical ΔT_sigma settings.
• XIX. Cross-References
• EFT.WP.BH.TensionWall v1.0 Chapters 3 (Minimal Equations & Structural Model), 5 (Wall Parameterization), 6 (Propagation Gauges), 7 (Metrology Flows), 8 (Interface Matching), 9 (Numerics), 11 (Validation & Benchmarks).
• EFT.WP.Propagation.TensionPotential v1.0 — the two gauges and data conventions.
• EFT.WP.Core.Metrology v1.0 M05-, M10-; EFT.WP.Core.Errors v1.0 M20-*.
• XX. Deliverables
• Error-budget checklist: templates for GUM/MC inputs, correlation assumptions, and output metrics.
• Safeguard playbook: operational points for gauges; path/segmentation; interface/energy; thin/thick; band differencing; c_ref; directional and multi-path terms; clamping/saturation.
• Audit templates: bias-detector dashboards; falsification cards; guardband configuration; τ_switch & energy-consistency reports; replay instructions and hash manifest.