49-Error Budget Card Template v1.0 | Chapter 4 — Taxonomy & Definitions of Error Sources

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Chapter 4 — Taxonomy & Definitions of Error Sources

I. Purpose & Scope

• Establish a protocol-level taxonomy and unified nomenclature for error sources, covering definitions, units, Type-A/Type-B evaluation, correlation structure, and mapping to quality gates for propagation and release compliance.
• When path quantities (arrival time/phase) are involved, explicitly show gamma(ell) and the measure d ell in text, and record delta_form ∈ {general, factored} in data/metadata; publication requires p_dim = 1.0.

II. Taxonomy Principles & Coding Rules

• Categories (8 groups): Timing/Sync (TIM), Path/Geometry (PTH), Medium/Dispersion (MED), Instrument/Calibration (INS), Sampling/Quantization (SMP), Model/Numerics (MOD), Cleaning/Missingness (CLN), Environment/Drift (ENV).
• Item code: <CAT>-<two-digit index> (e.g., TIM-01); standard fields: source, symbol, unit, type(A/B), estimate, distribution, correlation, gate, see[].
• Type A/B: repeated-observation statistical assessment = A; specification/prior/handbook = B; mixed cases mark A/B and state weights or precedence.

III. Master Catalog (summary)

1. TIM | Timing & Synchronization

• TIM-01 Absolute timing offset: δt_abs (s, A/B) — from reference sync & loopback; approx. normal; correlated with Δτ_ch.
• TIM-02 Inter-channel skew: Δτ_ch (s, A) — pulser/loopback differential; intra-channel correlation.
• TIM-03 Allan deviation: σ_y(τ) (1, A) — timebase stability; constrains coherence-window choice.

2. PTH | Path & Geometry

• PTH-01 Path-measure error: u(d ell) (m, B) — ruler/integration step uncertainty; paired with gamma(ell).
• PTH-02 Path parameterization error: u(gamma(ell)) (m, B) — registration/distortion residuals; spatial correlation.
• PTH-03 Occlusion/mask error: u(M_occ) (1, B) — effective-path segmentation bias.

3. MED | Medium & Dispersion

• MED-01 In-path index: u(n_eff(ell)) (1, A/B) — spline/kernel estimation variance; correlation length L_c.
• MED-02 Dispersion model: u(n_eff(λ)) (1, B) — sub-band fitting/parametric priors.

4. INS | Instrument & Calibration

• INS-01 Calibration params: u(θ_k) (by param unit, B) — Bias/Dark/Flat/Gain/PSF/Delay.
• INS-02 Readout noise: u(σ_ro) (e⁻, A) — readout chain, near-Gaussian.
• INS-03 System delay: u(τ_sys) (s, A/B) — drive/link & thermal drift.

5. SMP | Sampling & Quantization

• SMP-01 Time sampling: u(Δt) (s, B) — jitter/time resolution.
• SMP-02 Path step: u(Δell) (m, B) — grid discretization; tied to leakage.
• SMP-03 Quantization step: u(q_bits) (step, B) — effective ADC bits.

6. MOD | Modeling & Numerics

• MOD-01 Model mismatch: u(model) (target unit, A) — residual scale/robust surrogate.
• MOD-02 Discretization error: u(discretization) (target unit, B) — integration/differencing truncation.
• MOD-03 Thin-screen/paraxial approx.: u(screen/paraxial) (1, B) — systematic term from threshold violation.

7. CLN | Cleaning & Missingness

• CLN-01 Cleaning rules: u(cleaning) (target unit, B) — selection bias; must be pre-registered.
• CLN-02 Missingness pattern: u(missing) (target unit, A/B) — MCAR/MAR/MNAR assessment.

8. ENV | Environment & Drift

• ENV-01 Thermal coefficient: u(α_T·ΔT) (by quantity unit, B) — thermo-sensitivity propagation.
• ENV-02 Attitude drift: u(Δq_att) (angle/quaternion, B) — pointing/geometric coupling.

Gate mapping (examples): G5 (freshness), G6 (residual band), G7 (conservation), G4 (dimensional closure); triggers of S1–S5 block release.

IV. Definitions & Estimation Conventions

• TIM-01/02: repeated loopback → differential mean = bias, std = Type A; datasheet/spec = Type B; combine as u_c = √(u_A^2 + u_B^2).
• PTH-01/02: geometric calibration/registration residuals (RMSE on grid); build Σ via spatial kernel K(Δr).
• MED-01: fit n_eff(ell) with exponential kernel K(Δℓ)=σ_n^2·exp(−|Δℓ|/L_c) or Matérn; report σ_n, L_c.
• INS-01: calibration outputs θ_k ± u(θ_k); propagate to target y by Delta method: u^2(y) ≈ J Cov(θ) Jᵀ.
• SMP-03: uniform quantization: u(q_bits) = Δ_q / √12.
• MOD-02: truncation with grid step h, order p: u(discretization)=O(h^p); estimate via grid refinement.
• CLN-01: preregister exclusion rate/rules; assess bias intervals via permutation/bootstrap.
• ENV-01: linear thermo: u(α_T·ΔT) = |α_T|·u(ΔT) ⊕ |ΔT|·u(α_T) (⊕ = root-sum-square).

V. Correlation Structure & Covariance Modeling

• Path correlation: Cov(n_i,n_j)=σ_n^2·exp(−|Δℓ|/L_c); use piecewise L_c if needed.
• Channel correlation: block-diagonal by channel with intra-block Toeplitz; Corr(Δτ_ch^a, Δτ_ch^b)=ρ_ab.
• Cross terms: if n_eff couples with thermal drift, include Cov(n, α_T·ΔT)=κ·σ_n·σ_T and cross-partials in the Delta method.

VI. Type A/B Assessment & Traceability

• Type A: repeated observations/residual statistics; report sample size N, robustness (e.g., Huber/quantile), and confidence intervals.
• Type B: specifications/priors/numerical bounds; state source and coverage (e.g., 95%).
• Combination: u_c = √(∑ u_i^2 + 2∑_{i<j} ρ_{ij} u_i u_j); expanded U = k·u_c (default k = 2).

VII. Gate Mapping & Stops

• G4: dimensional closure failure (p_dim < 1) → S1 stop.
• G5: freshness expired → trigger recalibration.
• G6: Q_res out of band → switch robust substitute and report second-order surrogate.
• G7: ε_flux non-conservation → model switch or segmented coherence.
• G2: citation non-compliance/low anchor coverage → S5 block release.

VIII. Machine-Readable Templates (drop-in)

A. error_budget.csv (headers & sample rows)

source,symbol,unit,type,estimate,distribution,correlation,gate,see[]

TIM-01,δt_abs,s,A,2.5e-8,normal,"corr:with Δτ_ch",G5,"Metrology.Sync v1.0:PPS"

PTH-01,u(d ell),m,B,5.0e-4,uniform,"kernel:exp(Lc=25m)",G3,"Core.DataSpec v1.0:TARR"

MED-01,u(n_eff),1,A/B,3.0e-3,gaussian,"kernel:exp(Lc=50m)",G4,"Core.Equations v1.1:S20-1"

INS-01,u(θ_gain),1,B,0.8e-2,normal,"block:channel",G6,"Metrology.Instrument v1.0:GAIN"

SMP-03,u(q_bits),e-,B,Δq/√12,uniform,independent,G6,"Methods.Repro v1.0:ADC"

B. error_sources.yaml (registry example)

version: "1.0.0"

sources:

TIM-01: { symbol: "δt_abs", unit: "s", type: "A/B", desc: "absolute timing", gate: ["G5"], see: ["Metrology.Sync v1.0:PPS"] }

PTH-01: { symbol: "u(d ell)", unit: "m", type: "B", desc: "path measure", gate: ["G3"], see: ["Core.DataSpec v1.0:TARR"] }

MED-01: { symbol: "u(n_eff)", unit: "1", type: "A/B", desc: "medium index along path", corr: "exp(Lc)", see: ["Core.Equations v1.1:S20-1"] }

INS-01: { symbol: "u(θ_k)", unit: "<by param>", type: "B", desc: "calibration params", gate: ["G6"], see: ["Metrology.Instrument v1.0"] }

IX. Exports & Compliance

• Required: error_budget.csv, uncertainty.md, check_dim_report.json; for path quantities, explicitly show gamma(ell)/d ell and record delta_form.
• Citations & versions: text and see[]/references[] consistent; use “volume + version + anchor (P/S/M/I)”, anchor coverage ≥ 90%, public releases v1.* only.

X. Checklist

• Each error source reports symbol/unit/type/estimate/distribution/correlation/gate/see[].
• Path quantities explicitly show gamma(ell) and d ell; len(path) ≥ 2, Δell compliant.
• Correlation structure and L_c declared; cross-terms included in the Delta method.
• Combined u_c and expanded U = k•u_c computed and mapped to thresholds.
• Release bundle includes error_budget.csv / uncertainty.md / check_dim_report.json; citations compliant, versions locked.