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Appendix A Unified Symbols & Dimensions (OT)
I. Abstract & Scope
This appendix consolidates the global and cross-chapter symbols, SI units, and dimensional rules for this volume (Oriented Tension, OT). All equations/symbols use English notation wrapped in backticks; any expression with integrals/divisions/composite operators must use parentheses and explicitly state the path gamma(ell) and measure d ell (when ToA appears). Dimensional closure is mandatory.
II. Dependencies & References
- Layout & numbering: EFT Technical Whitepaper & Memo Template – Comprehensive Checklist v0.1.
- In-volume: Ch.2 (notation & unit axioms P80-1/4/5/6), Ch.3 (S80-1/2), Ch.4 (S80-3/4), Ch.6 (S80-5/6), Ch.7 (S80-7/8), Ch.8 (S80-9/10/11), Ch.10 (S80-15…18), Ch.12 (I80-*).
III. Naming Conventions & Reserved Suffixes
- Vectors _vec, fields _fld; tensors use subscripts *_ij.
- Frame qualifiers (only when needed): (_flow/_sheet/_shear).
- Paths & measures: gamma(ell), d ell; spherical measure: dΩ.
- Do-not-confuse: T_fil (tension) ≠ T_trans (transmittance); n (number density) ≠ n_eff (effective refractive index); Gamma (Lorentz) ≠ gamma(ell) (path); Q_ij (order tensor) ≠ Q0_Z (injection normalization).
IV. Central Symbols & Units (grouped)
Tab. A-1 Constants & Baselines
Symbol | Meaning | Unit (SI) | Dim | See |
|---|---|---|---|---|
c_ref | reference speed | m·s^-1 | L·T^-1 | Ch.12 |
e_charge | elementary charge | C | I·T | — |
k_B | Boltzmann constant | J·K^-1 | M·L^2·T^-2·Θ^-1 | — |
z | redshift | 1 | 1 | — |
Tab. A-2 Orientation & Order
Symbol | Meaning | Unit | Dim | See |
|---|---|---|---|---|
n_hat | unit orientation vector | 1 | 1 | Ch.3 |
f_orient(n_hat,r,t) | orientation distribution | 1 | 1 | Ch.3 |
Q_ij | order tensor (symmetric, traceless) | 1 | 1 | Ch.3/4 |
tau_relax | relaxation time | s | T | Ch.4/5 |
D_Q | order-parameter diffusivity | m^2·s^-1 | L^2·T^-1 | Ch.4/5 |
W_orient | orientation energy density | J·m^-3 | M·L^-1·T^-2 | Ch.4/7 |
Tab. A-3 Tension & Coupling Parameters
Symbol | Meaning | Unit | Dim | See |
|---|---|---|---|---|
T_fil / T_fil_ij | oriented tension (scalar/tensor) | Pa | M·L^-1·T^-2 | Ch.4/5 |
Λ_{ijkl} | orientation→tension map | Pa | M·L^-1·T^-2 | Ch.4 |
W_cpl | coupling free energy | J·m^-3 | M·L^-1·T^-2 | Ch.6 |
χ_E, χ_B, χ_u | EM/strain couplings | see note | — | Ch.6 |
D1 | anisotropic diffusion gain | m^2·s^-1 | L^2·T^-1 | Ch.6 |
α_E, α_B | EM constitutive gains | 1 | 1 | Ch.6 |
κ_s | acoustic/elastic coupling gain | m^2·s^-2 | L^2·T^-2 | Ch.6 |
Note: units of χ_* depend on the coupled pair (e.g., in χ_E Q:E E, χ_E has units J·m^-3·(V·m^-1)^-2).
Tab. A-4 Media, Fields & Effective Tensors
Symbol | Meaning | Unit | Dim | See |
|---|---|---|---|---|
E_vec / B_vec | electric / magnetic field | V·m^-1 / T | — | Ch.6 |
u_vec | flow velocity | m·s^-1 | L·T^-1 | Ch.6 |
D_{ij} | rate-of-deformation tensor | s^-1 | T^-1 | Ch.6 |
ε_ij / μ_ij | permittivity / permeability | F·m^-1 / H·m^-1 | — | Ch.6 |
C_eff,ijkl | effective elastic tensor | Pa | M·L^-1·T^-2 | Ch.6 |
D_eff,ij | effective diffusion tensor | m^2·s^-1 | L^2·T^-1 | Ch.6 |
Tab. A-5 EDX Energy & Power Terms
Symbol | Meaning | Unit | Dim | See |
|---|---|---|---|---|
Φ_E | energy flux | W·m^-2 | M·T^-3 | Ch.7 |
𝒫_in, 𝒫_mech, 𝒫_cpl, 𝒫_diss | power densities | W·m^-3 | M·L^-1·T^-3 | Ch.7 |
P_in, P_trans, P_diss | control-volume powers | W | M·L^2·T^-3 | Ch.7 |
Tab. A-6 Interfaces/Defects & Anisotropy
Symbol | Meaning | Unit | Dim | See |
|---|---|---|---|---|
W_surf | surface/interface energy density | J·m^-2 | M·T^-2 | Ch.8 |
κ_a, κ_b | anchoring coefficients | J·m^-2 | M·T^-2 | Ch.8 |
K | gradient penalty | J·m^-1 | M·T^-2 | Ch.8 |
U_core(Q) | defect core potential | J·m^-3 | M·L^-1·T^-2 | Ch.8 |
n_b, t_hat | normal / tangent basis | 1 | 1 | Ch.8 |
S_dir(ê) | directional projection ê_i Q_ij ê_j | 1 | 1 | Ch.8 |
A_aniso | λ_max − λ_min of Q | 1 | 1 | Ch.8 |
R_aniso | ‖Q‖_F | 1 | 1 | Ch.8 |
Tab. A-7 Data, Instrument & Noise
Symbol | Meaning | Unit | Dim | See |
|---|---|---|---|---|
R_inst | instrument response operator | — | — | Ch.5/10/12 |
S_n(ω) | noise PSD | (unit of y)^2·Hz^-1 | — | Ch.5 |
Cov | covariance | (unit of y)^2 | — | Ch.5/9 |
SpecMAE / EDX-closure | metrics | 1 | 1 | Ch.10/11 |
Tab. A-8 ToA & Path/Measure (if used)
Symbol | Meaning | Unit | Dim | See |
|---|---|---|---|---|
T_arr | time of arrival | s | T | Ch.12 |
gamma(ell) | path curve | — | — | — |
d ell | line element | m | L | — |
n_eff(ν,r) | effective refractive index | 1 | 1 | — |
delta_form | ToA form flag (A/B) | — | — | — |
V. Unit & Dimensional Audit Rules (Enforcement)
- T_fil_ij in Pa (M L^-1 T^-2); Q_ij dimensionless with Tr(Q)=0.
- D_eff,ij and D_Q in m^2·s^-1; ε_ij/μ_ij as absolute or relative quantities with mapping notes.
- Φ_E in W·m^-2; 𝒫_* in W·m^-3; W_orient in J·m^-3.
- If spectral flux Phi(E) appears, it MUST be m^-2·s^-1·sr^-1·eV^-1 (dim L^-2 T^-1 Ω^-1 E^-1).
- Dimensional non-closure or Tr(Q)≠0 constitutes an audit failure (API returns 422 UNIT_MISMATCH).
VI. Dual-Form ToA Templates (if used, mandatory)
- Form A (constant factored): T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell )
- Form B (general): T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell )
Both forms MUST explicitly include gamma(ell) and d ell, and record delta_form.
VII. Statistical & Uncertainty Notation
Expectation E[·], variance Var[·], median Med[·], quantile Q_p[·]; Fisher information F and posterior covariance Cov; evidence Z and ratio K; coverage/credence bands for EDX ledgers and derived quantities.
VIII. Local Symbol-Table Template (per chapter)
Symbol | Meaning | Unit | Validity(Ch.) | Notes/See |
|---|---|---|---|---|
… | … | … | Ch.X | local-only / overrides |
— The tables and rules in this appendix constrain and audit Unit/Dim consistency and orientation-notation compliance for all data, models, and interfaces in this volume.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
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