49-Error Budget Card Template v1.0 | Chapter 6 — Propagation Methods (Delta / MC / Bootstrap)

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Chapter 6 — Propagation Methods (Delta / MC / Bootstrap)

I. Purpose & Scope

II. Prerequisites & Inputs

• Control equations & explicit path: the text shows both gamma(ell) and d ell; data/metadata record delta_form ∈ {general, factored}.
• Metrological closure & sync: p_dim = 1.0; clock_state = locked; |ts_start − calib.timestamp| ≤ τ_calib.
• Covariance config: kernels/structure and parameters per Chapter 5: Σ = Cov(x); segment by domain if needed.
• Statistical settings: significance α, power target 1−β, number of draws B (MC/Bootstrap), robust-surrogate switches.

III. Delta Method (linearization)

1. Formulation: for target y = f(x), linearize at x̂, Jacobian J = ∂f/∂x |_{x̂}, then
   u^2(y) ≈ J · Σ · Jᵀ.
2. Arrival time / phase sensitivities (examples)
   • T_arr = ( ∫ ( n_eff / c_ref ) d ell ): ∂T_arr/∂c_ref = − (1 / c_ref^2) ∫ n_eff d ell; path covariance for n_eff(ell) modeled by Ch.5 kernels.
   • Phi = ( 2π / λ_ref ) ( ∫ n_eff d ell ): ∂Phi/∂λ_ref = − ( 2π / λ_ref^2 ) ∫ n_eff d ell.
3. Correlations & cross-terms: include cross-partials in J if couplings exist (e.g., Cov(n_eff, α_T·ΔT)).
4. Applicability: locally near-linear f(·); residuals near-Gaussian or well represented by a robust second-order surrogate.
5. Outputs: u(y), u_c, and, if needed, U = k·u_c; export component contributions (variance share per factor).

IV. Monte Carlo (MC)

• Formulation: sample x ~ 𝒩(x̂, Σ) or a robust family (t/mixtures), compute y_b = f(x_b), b=1..B;
  u(y) = std({y_b}), intervals reported by type (confidence/credible/quantile).
• Samplers: spectral/Cholesky/state-space; for scale, use low-rank + Woodbury, block Toeplitz/FFT acceleration.
• Robustification: for heavy tails/outliers, report median + quantile band (e.g., P2.5–P97.5) and differences vs mean ± U.
• Convergence & complexity: default B ≥ 10^4; supply convergence diagnostics (variance vs B curve).
• Outputs: point estimate (mean/median), std/quantile bands, power and sampling configuration.

V. Bootstrap

• Formulation: resample residuals or observation pairs (paired/stratified), preserving {batch/device/region} strata; for phase metrics, align within the reference window before bootstrapping.
• Statistics: build bootstrap distributions for ΔT_arr, r_phi, ε_flux, etc.; report quantile intervals and bias-correction (BCa optional).
• Robust flow: apply Huber/quantile losses to residuals, then bootstrap; output robust intervals.
• B value: B ≥ 10^4; provide interval-stability diagnostics.

VI. Method Selection & Switching

• Priority: locally linear and performance-sensitive → Delta; strong nonlinearity/closed-form sensitivities hard → MC; distribution unknown/complex dependence → Bootstrap.
• Switching rules: if Delta residual diagnostics fail or Q_res out of band → switch to MC/Bootstrap and report differences; if MC convergence insufficient → increase B or use Bootstrap.
• Restricted mode: if core assumptions fail (paraxial/coherence/dimensional closure) → [Restricted], publish qualitative intervals and diagnostics only.

VII. Numerics & Stability

• PD & jitter: ensure Σ is PD using Σ ← Σ + εI (ε ≈ 1e−6~1e−3 of target variance).
• Scale normalization: normalize units/scales before computation; restore physical units on output and validate via check_dim.
• Complexity control: low-rank approximations, block factorizations, Kronecker/Toeplitz & FFT; log wall-time and memory peaks.

VIII. Outputs & Compliance

• Required: uncertainty.md (methods & parameters), check_dim_report.json, mc_config.yaml/bootstrap_config.yaml (if used), sampling seed, and convergence diagnostics.
• Gate mapping: Q_res within band (G6); p_dim = 1.0 (G4); conservation diagnostics ε_flux (G7).
• Citations & versioning: text and see[]/references[] consistent; “volume + version + anchor (P/S/M/I)”; anchor coverage ≥ 90%, public v1.* only.

IX. Machine-Readable Templates

A. delta_config.yaml

version: "1.0.0"

model:

f: "T_arr = ∫ ( n_eff / c_ref ) d ell"

jacobian: "auto" # or "manual"

covariance:

kernel: "exp"

params: { sigma2: 3.0e-3, L_c_m: 25.0 }

coverage: { k: 2, type: "confidence" }

see:

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

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

B. mc_config.yaml

version: "1.0.0"

draws: 10000

sampler: "chol" # chol|spectral|state-space

seed: 20250924

covariance: { from: "cov_config.yaml" }

targets: ["T_arr","Phi","ε_flux"]

summaries: ["mean","std","p2.5","p50","p97.5"]

C. bootstrap_config.yaml

version: "1.0.0"

B: 10000

scheme: "stratified" # paired|stratified|residual

align_phase_window: true

strata: ["batch","device","region"]

robust: { loss: "huber", delta: 1.345 }

targets: ["DeltaT_arr","r_phi","ε_flux"]

X. Cross-References

• Inputs & sensitivities: see Chapter 3.
• Covariance construction & samplers: see Chapter 5.
• Gate mapping & stop criteria: see Chapter 8.
• Results & scoring: see Chapter 11.

XI. Checklist

• gamma(ell)/d ell explicit; delta_form recorded; p_dim = 1.0.
• Rationale, parameters, and convergence diagnostics for Delta/MC/Bootstrap recorded in uncertainty.md.
• MC/Bootstrap B, seed, sampler, and covariance source (cov_config.yaml) logged.
• Outputs include point estimate, u(y)/u_c, U = k•u_c or quantile bands, compared to gate thresholds.
• Release includes uncertainty.md / check_dim_report.json /(mc|bootstrap)_config.yaml; citations compliant, versions locked.