49-Error Budget Card Template v1.0 | Chapter 2 — Terms, Symbols & Units (Minimal Uncertainty Set)

Home ／ Docs-Technical WhitePaper ／ 49-Error Budget Card Template v1.0

Chapter 2 — Terms, Symbols & Units (Minimal Uncertainty Set)

I. Variables & Symbols (minimal set)

Wrap all inline symbols in backticks; for any path-dependent quantity explicitly declare gamma(ell) and the measure d ell.

• T_arr: arrival time; unit s; path quantity—must co-appear with gamma(ell), d ell, n_eff, c_ref, and record delta_form.
• Phi: accumulated phase; unit rad; paired with lambda_ref.
• n_eff(ell): effective refractive index along path; dimensionless.
• gamma(ell): parametrized propagation path; unit m.
• d ell: path measure element; unit m.
• ε_flux: paraxial conservation error; dimensionless.
• ΔM: mass-conservation deviation; unit follows ρ.
• Q_res: robust residual metric; dimensionless.
• r_phi: correlation between Phi_ref and Phi_obs; dimensionless.
• Δell: maximum path discretization step; unit m.

II. Statistical & Uncertainty Notation (minimal UQ set)

• u(x): standard uncertainty.
• u_c: combined standard uncertainty.
• U = k · u_c: expanded uncertainty; k is the coverage factor.
• E[x], Var[x], Cov(x,y): expectation, variance, and covariance.
• ρ: correlation coefficient (or corr(x,y)).
• L_c: correlation length (path/time/frequency).
• B: number of MC/bootstrap draws.
• Interval types: confidence / credible / quantile (state the type and confidence/coverage level in text).

III. Constants & Reference Quantities

• c_ref: reference propagation limit; unit m/s.
• lambda_ref (λ_ref): reference wavelength; unit m.
• k_ref = 2π / λ_ref: reference wavenumber; unit 1/m.

IV. Units & Dimensions

• Base units: m, s, rad, 1.
• Arrival-time dimensional check:
  T_arr = ( ∫ ( n_eff / c_ref ) d ell ) ⇒ [1]/[m·s^-1]·[m] = [s].
• Phase dimensional check:
  Phi = ( 2π / λ_ref ) ( ∫ n_eff d ell ) ⇒ [1/m]·[m] = [rad].
• Reporting: results and intervals must carry units; exports include check_dim_report.json, requiring p_dim = 1.0.

V. Mandatory Conventions (writing & conflict names)

• Parentheses & precedence: any expression with division/integrals/composite operators must use parentheses; use ln(·), log10(·), exp(·) for logs/exp.
• Explicit path: whenever T_arr/Phi appear, the text must also show gamma(ell) and d ell, and data/metadata must record delta_form ∈ {general, factored}.
• Forbidden bare symbols: c, T, n; use c_ref, explicit quantity names (e.g., T_arr), and n_eff (or n with explicit unit).
• Conflict names: T_fil (tension) ≠ T_trans (transmittance); n (number density) ≠ n_eff (effective refractive index); do not mix.
• Language: formulas, symbols, and definitions must not use Chinese (use international notation for variables/operators/units/dimensions).

VI. Normative Formulae (paste-ready)

• Arrival time (two equivalent forms):
  T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
  T_arr = ( ∫ ( n_eff / c_ref ) d ell )
• Phase accumulation:
  Phi = ( 2π / λ_ref ) * ( ∫ n_eff d ell )
• Combined standard uncertainty (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 (default k = 2, ≈95% coverage).

VII. Machine-Readable Registry (YAML, minimal snippet)

version: "1.0.0"

symbols:

- name: T_arr

kind: variable

unit: s

desc: arrival time

requires: { path: "gamma(ell)", measure: "d ell", delta_form: "general|factored" }

- name: Phi

kind: variable

unit: rad

desc: accumulated phase

requires: { lambda_ref: m, path: "gamma(ell)", measure: "d ell" }

- name: n_eff

kind: variable

unit: "1"

desc: effective refractive index along path

- name: c_ref

kind: constant

unit: m/s

desc: reference propagation limit

- name: lambda_ref

kind: constant

unit: m

desc: reference wavelength

- name: u_c

kind: derived

unit: "<unit of target>"

desc: combined standard uncertainty

- name: U

kind: derived

unit: "<unit of target>"

desc: expanded uncertainty; U = k * u_c

meta: { k: 2 }

VIII. Publishing Consistency (results page & data side)

• Text and data see[]/references[] use “volume + version + anchor (P/S/M/I)”, with anchor coverage ≥ 90%.
• Path expressions explicitly show gamma(ell) and d ell, with delta_form recorded; len(path) ≥ 2; Δell meets sampling constraints.
• On the results page, list metrics with units and intervals: T_arr (s), Phi (rad), ε_flux (1), Q_res (1), r_phi (1), etc.