47-PTN Template v1.0 | Chapter 2 — Terms & Symbols (Minimal Set)

Home ／ Docs-Technical WhitePaper ／ 47-PTN Template v1.0

Chapter 2 — Terms & Symbols (Minimal Set)

I. Variables
Use backticks for all inline symbols. For path-dependent quantities, explicitly declare gamma(ell) and the measure d ell.

• phi(x,t): Scalar field or scalar observable; domain x ∈ ℝ^3, t ∈ ℝ; unit: context-dependent; used as baseline signal or residual field.
• Xi(x,t): Placeholder for vector/tensor field; unit: per component; used for instrument orientation, polarization, or stress/strain tensors.
• T_arr: Arrival time; unit s; path quantity; must co-appear with gamma(ell), n_eff, c_ref.
• gamma(ell): Parametrized propagation path; unit m; paired with measure d ell.
• n_eff: Effective refractive index; dimensionless; medium term in path integrals.
• T_fil: Tension; unit per metrology definition (subject to dimension check); must not be confused with T_trans.
• Q: Dimensionless quality/score metric; specify range and thresholds at point of use.
• K: Kernel (convolution) or curvature placeholder; declare domain and normalization.
• rho(x): Number or mass density (state explicitly); units 1/m^3 or kg/m^3; distinguish from n.
• Delta(·): Finite-difference/increment operator; pair with sampling steps Δt, Δell.
• sigma_x: Standard deviation of x; unit follows x.
• r_hat: Unit direction vector; dimensionless.

II. Constants
Constants must carry clear provenance and a dimension-check entry; bare usage is prohibited.

• c_ref: Reference propagation limit; unit m/s; baseline for arrival-time and phase integrals.
• k_B: Boltzmann constant; unit J/K; for statistical-temperature dimensional work.
• mu_0 / epsilon_0: Vacuum constants; units N/A^2 and F/m; for electromagnetic derivations.
• pi: Circle constant; dimensionless.

III. Units & Dimensions

• Base units: m, kg, s, A, K, mol, cd.
• Examples of derived units: N = kg·m/s^2, Pa = N/m^2, J = N·m, W = J/s, Hz = 1/s.
• Dimension checks: Every formula must pass dimension checking; provide a check_dim report in appendices or accompanying data.
• Arrival-time consistency: T_arr in s; path element d ell in m; n_eff dimensionless; c_ref in m/s.
• Statistics: Mean/median inherit the source unit; variance uses squared unit; standard deviation matches the source unit.
• Sampling & windows: Declare sampling steps Δt, Δell and the window function (rectangular/Hann/Blackman, etc.).

IV. Conflicts & Constraints

• T_fil (tension) ≠ T_trans (transmittance): never interchangeable; never abbreviated into each other.
• n (number density) ≠ n_eff (effective refractive index): use n_eff in path integrals; use n or rho for statistical densities with units stated.
• Prohibited bare symbols: c, T, n; replace with c_ref, the specific quantity name (e.g., T_fil), and n_eff (or n).
• Subscripts & modifiers: Use modifiers like ref/eff/fil/trans for physical roles only; when combined with indices, use layered notation, e.g., T_fil,i, n_eff(ell).
• Operation disambiguation: Any expression with division, integrals, or composite operators must use parentheses; e.g., T_arr = ( ∫ ( n_eff / c_ref ) d ell ).
• Formula language: Variable names, operators, units, and dimensions use English and international symbols only.

Appendix: Minimal Symbol Registry (ready to use, example)

version: "1.0.0"

symbols:

- name: T_arr

kind: variable

unit: s

desc: arrival time

requires:

path: "gamma(ell)"

measure: "d ell"

see:

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

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

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

- name: c_ref

kind: constant

unit: m/s

desc: reference propagation limit

see:

- "EFT.WP.Core.Terms v1.0:P10-3"

- name: n_eff

kind: variable

unit: "1"

desc: effective refractive index

see:

- "EFT.WP.Core.Terms v1.0:P10-3"

- name: T_fil

kind: variable

unit: "<by definition>"

desc: tension (not transmittance)

note: "must not be confused with T_trans"

- name: T_trans

kind: variable

unit: "1"

desc: transmittance (not tension)

- name: gamma(ell)

kind: path

unit: m

desc: propagation path parameterization

measure: "d ell"

- name: phi(x,t)

kind: field

unit: "<by context>"

desc: scalar field or observable

- name: Xi(x,t)

kind: field

unit: "<by context>"

desc: vector/tensor placeholder

Canonical Writing Examples

• Arrival time (constant factored): T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
• Arrival time (general form): T_arr = ( ∫ ( n_eff / c_ref ) d ell )
• Related operations: ∇·( n_eff v ), exp(·), ln(·), log10(·) (avoid ambiguous log).
• Sampling notation: ⟨x⟩ = (1/N) ∑_{i=1}^{N} x_i; distinguish “mean of ratios” vs “ratio of means” in text with rationale and applicability.