47-PTN Template v1.0 | Chapter 7 — Error Budget & Uncertainty

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Chapter 7 — Error Budget & Uncertainty

I. Error Sources Catalog

Uniform specification: each source includes symbol, unit, estimation method, distributional assumption, and correlation; type A (statistical) / B (non-statistical). Path-dependent terms explicitly declare gamma(ell) and measure d ell.

• Timing & synchronization: u(δt_abs) (s, A/B), u(Δτ_ch) (s, A), u(jitter) (s, A), u(σ_y(τ)) (dimensionless, A).
• Path & geometry: u(gamma(ell)) (m, B), u(d ell) (m, B), occlusion/distortion u(D) (dimensionless, B).
• Medium & dispersion: u(n_eff(ell)) (dimensionless, A/B), u(n_eff(λ)) (dimensionless, B).
• Reference constants: u(c_ref) (m/s, B), u(λ_ref) (m, B), u(k_ref) (1/m, B).
• Instrument & calibration: u(θ_k) (per parameter unit, B), u(g) (gain, B), u(σ_ro) (e⁻, A).
• Sampling & quantization: u(Δt) (s, B), u(Δell) (m, B), u(q_bits) (quantization step, B).
• Modeling & numerics: u(model) (residual scale, A), u(discretization) (discretization error, B), u(screen) (thin-screen approx., B).
• Environment & drift: u(α_T·ΔT) (per object unit, B), u(Δq_att) (attitude, B).
• Cleaning & selection: u(cleaning) (bias from exclusion rules, B), u(missing) (missingness pattern, A/B).

II. Uncertainty Propagation Models

Three tracks in parallel: linearization, covariance integration, and Monte Carlo. For robust losses, provide a second-order surrogate for propagation.

• Delta method (linearization): for y = f(x), x ∈ ℝ^p, Jacobian J = ∂f/∂x |_{x̂},
  u^2(y) ≈ J · Cov(x) · Jᵀ.
• Arrival time T_arr (first-order, explicit path & measure):
  T_arr = ( ∫ ( n_eff / c_ref ) d ell ), let w(ell) = ( 1 / c_ref ),
  u^2(T_arr) = ∬ w(ell₁) w(ell₂) · Cov[ n_eff(ell₁), n_eff(ell₂) ] d ell₁ d ell₂ + ( ∂T_arr/∂c_ref )^2 u^2(c_ref),
  where ∂T_arr/∂c_ref = - ( 1 / c_ref^2 ) ∫ n_eff d ell. Discrete form:
  u^2(T_arr) ≈ ∑_{i,j} w_i w_j · Cov(n_i, n_j) · d ell_i · d ell_j + (∂T_arr/∂c_ref)^2 u^2(c_ref).
• Phase accumulation Phi:
  Phi = ( 2π / λ_ref ) ( ∫ n_eff d ell ),
  u^2(Phi) = ( 2π / λ_ref )^2 ∬ Cov(n_eff(ell₁), n_eff(ell₂)) d ell₁ d ell₂ + ( ∂Phi/∂λ_ref )^2 u^2(λ_ref),
  ∂Phi/∂λ_ref = - ( 2π / λ_ref^2 ) ∫ n_eff d ell.
• Continuity/flux-type metrics (examples): for ε_flux or ΔM, use linearization or bootstrap:
  u(ε_flux) ≈ √{ Var(ε_flux_residuals) };
  u(ΔM) ≈ √{ u^2(∫ρ dV|_{t2}) + u^2(∫ρ dV|_{t1}) − 2·Cov }.
• Correlation structure: if n_eff along the path has correlation length L_c, model Cov(n_i,n_j) with kernel K(Δell) = exp( −|Δell| / L_c ); set L_c by fit or prior.
• Monte Carlo: sample x ~ 𝒩(x̂, Cov) (or robust family), B ≥ 10^4; compute y_b = f(x_b); take u(y) = std(y_b); for robust cases, report median and quantile band.

III. Composition & Intervals

• Combined standard uncertainty: for uncorrelated terms, u_c = √( ∑ u_i^2 ); with correlations, u_c^2 = ∑ u_i^2 + 2∑_{i<j} ρ_{ij} u_i u_j.
• Expanded uncertainty: U = k · u_c, with coverage factor k = 2 (~95%) unless specified otherwise.
• Interval types: report confidence intervals (frequentist) or credible intervals (Bayesian); robust pipelines provide median ± quantile spread (e.g., P2.5–P97.5).
• Threshold alignment: align U with Chapter 4 thresholds (τ_T, τ_phi, τ_M); positive/negative decisions are made on interval comparisons.
• Units & dimensions: every interval carries units; include a check_dim report showing dimensional closure per quantity.

IV. Monitoring & Alerts

• Online metrics: Q_res (residual robustness), p_dim (dimension-check pass rate), σ_y(τ), SNR, drift_rate.
• Control charts: EWMA/Shewhart for T_arr, Phi, ε_flux; out-of-control adds quality.flags += {"uncert_alert"}.
• Freshness gate: acq.ts_start − calib.timestamp ≤ τ_calib; if expired, enter restricted mode and elevate the reporting level of U.
• Recalibration trigger: force recalibration when any u(θ_k) exceeds threshold or p_dim < 1; re-evaluate u_c and U.

V. Error-Budget Card (publication format)

Fields: source, symbol, unit, type(A/B), estimate, distribution, correlation, note, see[].

Deliverables: provide error_budget.csv with a flattened see[] list (volume + version + anchor).

VI. Reporting & Records

• Uncertainty summary table: for each key quantity (T_arr, Phi, ε_flux, ΔM, Q), report x̂ ± u_c (k=1) and x̂ ± U (k).
• Method statement: specify propagation route (delta/Monte Carlo/bootstrap) and kernel/correlation length if used.
• Reproducibility: record random seeds, sample sizes, B, kernel params, estimator choices, and versions in audit.jsonl.

VII. Normative Examples (drop-in)

• Arrival-time uncertainty (discrete path)

Given: T_arr = ∑_i ( n_i / c_ref ) · d ell_i

u^2(T_arr) = ∑_{i,j} ( d ell_i d ell_j / c_ref^2 ) · Cov(n_i, n_j)

+ ( ∑_i n_i d ell_i / c_ref^2 )^2 · u^2(c_ref)

Dims: [1]/[m·s^-1]*[m] = [s] ✅

• Phase uncertainty (with λ_ref)

Phi = ( 2π / λ_ref ) ∑_i n_i d ell_i

u^2(Phi) = ( 2π / λ_ref )^2 ∑_{i,j} d ell_i d ell_j · Cov(n_i, n_j)

+ ( 2π ∑_i n_i d ell_i / λ_ref^2 )^2 · u^2(λ_ref)

Unit: [rad] ✅

• Robust intervals (bootstrap)

Draw B=10000 bootstrap replicates of residuals → refit → collect T_arr^*.

Report median [P2.5, P97.5]; compare to thresholds τ_T; mark pass/fail.

VIII. Machine-Readable Template (ready to commit)

version: "1.0.0"

uncertainty:

targets: ["T_arr","Phi","ε_flux","ΔM","Q"]

methods:

T_arr: ["delta","mc"]

Phi: ["delta","mc"]

ε_flux: ["bootstrap","delta"]

delta:

jacobian: "auto"

cov_model:

n_eff:

kernel: "exp"

L_c_m: 25.0

mc:

draws: 10000

seed: 20250924

coverage:

k: 2

type: "confidence" # or "credible" / "quantile"

report:

export: ["error_budget.csv","uncertainty.md","check_dim_report.json"]

see:

- "EFT.WP.Core.Equations v1.1:S20-1"

- "EFT.WP.Core.Metrology v1.0:check_dim"

- "Methods.Cleaning v1.0"

- "Methods.SimStack v1.0"

- "Core.DataSpec v1.0:TARR"

IX. Alignment with Quality Gates (as in Chapter 5)

• G4 | Dimensional closure: p_dim = 1.0.
• G6 | Noise-residual gate: Q_res within admissible band.
• Failure triggers: p_dim < 1 or U beyond threshold raises uncert_alert and initiates recalibration workflow.