48-Experimental Protocol Card Template v1.0 | Chapter 9 — Error Budget & Uncertainty (Protocol Level)

Home ／ Docs-Technical WhitePaper ／ 48-Experimental Protocol Card Template v1.0

Chapter 9 — Error Budget & Uncertainty (Protocol Level)

I. Purpose & Scope

• Provide a protocol-level minimal yet sufficient set of error sources, propagation routes, composition, and interval reporting conventions for runtime validation and release compliance.
• When arrival time/phase (path quantities) are involved, explicitly show gamma(ell) and the measure d ell in text, and record delta_form ∈ {general, factored} in data/metadata.

II. Prerequisites & Inputs

• Dimensional closure: p_dim = 1.0; exports include check_dim_report.json.
• Path consistency: len(gamma_ell)=len(d_ell)=len(n_eff)≥2; Δell meets sampling constraints.
• Sync & calibration: clock_state = locked; |ts_start − calib.timestamp| ≤ τ_calib.
• Statistical settings: preregister significance α, target power 1−β, and multiple-testing control (e.g., FDR).

III. Error Sources (protocol minimal set)

• Timing & synchronization: u(δt_abs), u(Δτ_ch), u(σ_y(τ)).
• Path & geometry: u(gamma(ell)), u(d ell), occlusion/distortion u(D).
• Medium & dispersion: u(n_eff(ell)), and (wideband) u(n_eff(λ)).
• Reference quantities: u(c_ref), u(λ_ref), u(k_ref).
• Instrument & calibration: u(θ_k), u(σ_ro), u(g).
• Sampling & quantization: u(Δt), u(Δell), step u(q_bits).
• Modeling & numerics: u(model), u(discretization), thin-screen/paraxial u(screen/paraxial).
• Cleaning & missingness: u(cleaning), u(missing) (preregister exclusion rules).

IV. Propagation Models

• Delta method (linearization): for y=f(x), with J=∂f/∂x|_{x̂}, u^2(y) ≈ J·Cov(x)·Jᵀ.
• Arrival time T_arr (explicit path & measure):
  T_arr = ( ∫ ( n_eff / c_ref ) d ell ) = ( 1 / c_ref ) * ( ∫ n_eff d ell )
  Let w(ell) = (1/c_ref), then
  u^2(T_arr) = ∬ w(ℓ₁)w(ℓ₂)·Cov[n_eff(ℓ₁),n_eff(ℓ₂)] dℓ₁ dℓ₂ + (∂T_arr/∂c_ref)^2 u^2(c_ref),
  with ∂T_arr/∂c_ref = − (1/c_ref^2) ∫ n_eff d ell. Discrete approximation:
  u^2(T_arr) ≈ ∑_{i,j} (dℓ_i dℓ_j / c_ref^2)·Cov(n_i,n_j) + (∑_i n_i dℓ_i / c_ref^2 )^2 u^2(c_ref).
• Phase Phi:
  Phi = ( 2π / λ_ref ) ( ∫ n_eff d ell )
  u^2(Phi) = (2π/λ_ref)^2 ∬ Cov(n_eff(ℓ₁),n_eff(ℓ₂)) dℓ₁ dℓ₂ + (∂Phi/∂λ_ref)^2 u^2(λ_ref),
  ∂Phi/∂λ_ref = − (2π/λ_ref^2) ∫ n_eff d ell.
• Conservation/flux metrics (e.g., ε_flux, ΔM): linearization or bootstrap:
  u(ε_flux) ≈ √{Var(residuals)}; u(ΔM) ≈ √{ u^2(∫ρ dV|_{t2}) + u^2(∫ρ dV|_{t1}) − 2·Cov }.
• Correlation structure: for path correlation length L_c, model Cov(n_i,n_j) = σ_n^2·exp(−|Δℓ|/L_c); set L_c by fit/prior.
• Monte Carlo (MC): sample x ~ 𝒩(x̂, Cov) or robust family, B ≥ 10^4; u(y)=std({y_b}); for robust flows, report median and quantile band.

V. Composition & Intervals

• Combined standard uncertainty: uncorrelated u_c = √(∑ u_i^2); correlated u_c^2 = ∑ u_i^2 + 2∑_{i<j} ρ_{ij}u_i u_j.
• Expanded uncertainty: U = k·u_c (default k=2, ≈95% coverage); report k or confidence level.
• Interval types: frequentist confidence, Bayesian credible, or robust quantile bands (e.g., P2.5–P97.5).
• Threshold alignment: align with Chapter 8 gates (e.g., τ_T, r_phi_min, ε_flux guard); make positive/negative calls by interval–threshold comparison.
• Units & dimensions: attach units to all quantities; verify intervals and expressions via check_dim.

VI. Robust Options & Power

• Heavy tails/heteroscedasticity: switch to Huber/quantile metrics and provide second-order surrogates for propagation.
• Permutation/bootstrap: for phase metrics, align in the reference window before permutation; use stratified bootstrap aligned to {batch/device/region}.
• Power guidance: target 1−β = 0.9; core metrics α = 0.01, secondary α = 0.05; for correlation targets (e.g., r_target=0.6), use Fisher-z approximations for sample size.

VII. Reporting & Deliverables

• Uncertainty summary table: for T_arr, Phi, ε_flux, Q, p_dim, report x̂ ± u_c (k=1) and x̂ ± U (k).
• Methods statement: specify delta/MC/bootstrap, kernel/correlation length, and robust surrogate if used.
• Audit & reproducibility: record in audit.jsonl the random seeds, B, kernel params, estimator versions, and see[]/references[]/version.

VIII. Machine-Readable Template (minimal)

version: "1.0.0"

uncertainty:

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

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"

- "EFT.WP.Core.DataSpec v1.0:TARR"

IX. Alignment with Chapter 8

• G4 | Dimensional closure: p_dim = 1.0.
• G6 | Noise & residuals: Q_res within band; robust flows output surrogate.
• Triggering S1/S5 blocks release; only [Restricted] qualitative intervals and diagnostic plots may be provided.

X. Normative Examples

• Arrival-time uncertainty (discrete path)

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] ✅

XI. Checklist

• Error-source card and propagation route (delta/MC/bootstrap) selected and recorded.
• Path expressions explicitly show gamma(ell) and d ell; delta_form recorded; p_dim = 1.0.
• u_c and U = k·u_c computed; reported intervals aligned to Chapter 8 thresholds.
• Robust/permutation/bootstrap plans and sample-size/power settings logged in audit.jsonl.
• Exports include error_budget.csv / uncertainty.md / check_dim_report.json; see[]/references[]/version consistent.