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.