Chapter 2 Symbols, Terms & Units
I. Abstract & Scope
This chapter unifies the volume’s symbols, terminology, and SI unit conventions. It fixes notation and dimensional rules for orientation quantities (n_hat, f_orient(n_hat,r,t), Q_ij) and the oriented-tension family (T_fil/T_fil_vec/T_fil_ij). All equations and symbols use English notation wrapped in backticks, with dimensional closure mandatory. If time-of-arrival (ToA) appears, both forms must be recorded in parallel with explicit path gamma(ell) and measure d ell.
II. Dependencies & References
- Layout hierarchy, lists/indentation, numbering, and cross-volume references: EFT Technical Whitepaper & Memo Template – Comprehensive Checklist v0.1.
- Cross-volume phrasing: “see companion whitepaper Energy Filaments Chapter x S/P/M/I…”.
- In-volume cross-refs: Ch.3 (orientation geometry & distributions S80-1/2), Ch.4 (axioms & minimal equations P80-2/3, S80-3/4), Ch.5 (metrology flows M80-1…4), Ch.7 (EDX accounting S80-7/8), Ch.12 (implementation bindings & APIs).
III. Normative Anchors (added in this chapter, P80-/M80-)
- P80-1 (Notation & Units Axiom): all symbols use English notation wrapped in backticks; SI units; dimensional closure required.
- P80-4 (Orientation Notation Axiom): unit orientation vector n_hat; orientation distribution f_orient(n_hat,r,t) satisfies ∫_{S^2} f_orient dΩ = 1; order tensor Q_ij is the second moment on S^2 (see S80-1/2).
- P80-5 (Do-Not-Confuse Axiom): T_fil (tension) ≠ T_trans (transmittance); n (number density) ≠ n_eff (effective refractive index); Gamma (Lorentz factor) ≠ gamma(ell) (path); Q_ij (order tensor) ≠ Q0_Z (injection normalization).
- P80-6 (Dual-Form ToA Axiom): if ToA appears, record in parallel:
T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) and T_arr = ( ∫ ( n_eff / c_ref ) d ell ), with delta_form and {gamma(ell), d ell}. - M80-12 (Symbol & Unit Audit Flow): SymbolUnitAudit checks {backtick, unit, dim, forbidden-pairs, ToA-dualform} and emits UnitsAudit.log plus a diff report.
IV. Body Structure
I. Terminology Domain & Naming Rules
- Orientation & order: n_hat (unit vector), f_orient(n_hat,r,t) (orientation distribution), Q_ij (order tensor), S_orient (orientation source term).
- Tension & energy: T_fil (scalar) / T_fil_vec (vector) / T_fil_ij (tensor), W_orient (orientation energy density).
- Coupling & transport: A_cpl (coupling rate), D_Q (order-parameter diffusivity), u_vec (flow), E_vec/B_vec (EM fields).
- Suffixes & reserved words: vectors _vec, fields _fld; frame tags (_flow/_sheet/_shear) only when needed; any custom abbreviation must be added to the glossary at first use.
II. Notation & Layout Rules
- Inline symbols: always wrapped in backticks (e.g., Q_ij, Phi(E), D_Q).
- Fractions/integrals/composite operators: parenthesize and state path/measure explicitly (e.g., ∫_{gamma(ell)} (…) d ell).
- Lists & numbering: bullets “- ”; numbered lists “1. ”; subsection titles use bold Roman numerals.
- Name collisions: resolve forbidden pairs by renaming or scoping (e.g., q_heat for heat flux to avoid confusion with Q_ij).
III. Units & Dimensional Rules (minimum set)
- Phi(E): m^-2·s^-1·sr^-1·eV^-1, dimension L^-2 T^-1 Ω^-1 E^-1.
- N(E): 1, dimension 1; dN/dE: eV^-1, dimension E^-1.
- T_fil: if scalar stress, unit Pa; tensor T_fil_ij: Pa; vector projections consistent with Pa.
- Q_ij: 1; f_orient: 1, with ∫_{S^2} f_orient dΩ = 1.
- D_Q: m^2·s^-1; A_cpl, relaxation rates: s^-1; tau_relax: s.
- EM & mechanics (SI): E_vec (V·m^-1), B_vec (T), u_vec (m·s^-1).
IV. ToA-Related Fields (iff present)
- Recorded items: T_arr^A, T_arr^B, delta_form, gamma(ell), d ell, n_eff(ν,r), c_ref.
- Expressions & units: T_arr in s; n_eff in 1; d ell in m.
- Dataset-card interface: in DatasetCard.columns, ToA fields MUST include both forms and delta_form; integrals MUST provide {path, measure}.
V. Validation, Criteria & Contrasts
- Positive criteria:
- Every numeric column carries unit/dim; UnitsAudit.log passes dimensional closure.
- Orientation relations satisfy ∫_{S^2} f_orient dΩ = 1.
- If ToA appears, both forms with delta_form and {gamma(ell), d ell} are present.
- Negative criteria:
- Mixing T_fil with T_trans; conflating n with n_eff.
- Phi(E) missing sr or energy dimension.
- Only one T_arr form provided without the counterpart and path/measure.
- Contrasts:
- {with/without sr} audit for Phi(E).
- {normalized / unnormalized} f_orient effects on Q_ij.
- {ToA Form A, Form B, A+B} audit differences in the same dataset card.
VI. Deliverables & Figure List
- Deliverables: Glossary.md (terms & do-not-confuse list), SymbolRegistry.csv (central symbols—units—dimensions), UnitsAudit.log (from M80-12), ToA.dualform.log (if applicable).
- Tables/Figures (this chapter):
- Tab. 2-1 Minimal symbol table (Orientation / Tension / Coupling / Transport).
- Tab. 2-2 Do-not-confuse list.
- Tab. 2-3 Key observables with units & dimensions (Phi(E), N(E), T_fil_ij, D_Q, tau_relax).
- Tab. 2-4 Suffixes & reserved words.
- Tab. 2-5 DatasetCard ToA field template (if applicable).
Tab. 2-1 Minimal Symbol Table
Symbol | Meaning | Unit (SI) | Dim | See |
|---|---|---|---|---|
n_hat | unit orientation vector | 1 | 1 | Ch.3 |
f_orient(n_hat,r,t) | orientation distribution function | 1 | 1 | Ch.3 |
Q_ij | order tensor | 1 | 1 | Ch.3 |
T_fil / T_fil_vec / T_fil_ij | oriented tension (scalar/vector/tensor) | Pa | M L^-1 T^-2 | Ch.4 |
W_orient | orientation energy density | J·m^-3 | M L^-1 T^-2 | Ch.4/7 |
A_cpl | coupling rate | s^-1 | T^-1 | Ch.6 |
D_Q | order-parameter diffusivity | m^2·s^-1 | L^2 T^-1 | Ch.6 |
u_vec | flow speed | m·s^-1 | L T^-1 | Ch.6 |
Phi(E) | differential flux | m^-2·s^-1·sr^-1·eV^-1 | L^-2 T^-1 Ω^-1 E^-1 | Ch.7 |
tau_relax | orientation relaxation time | s | T | Ch.4/5 |
Tab. 2-2 Do-Not-Confuse List
Conflict | Correct usage | Forbidden usage | Rule |
|---|---|---|---|
tension vs transmittance | T_fil | T_trans | never use T_trans for tension |
number density vs effective index | n | n_eff | distinct physics; never interchangeable |
Lorentz factor vs path | Gamma | gamma(ell) | case-sensitive; gamma(ell) is for paths |
order tensor vs injection normalization | Q_ij | Q0_Z | scope/rename to avoid ambiguity |
Tab. 2-3 Key Observables (Units & Dimensions)
Quantity | Unit | Dim | Audit focus |
|---|---|---|---|
Phi(E) | m^-2·s^-1·sr^-1·eV^-1 | L^-2 T^-1 Ω^-1 E^-1 | must include sr and E^-1 |
T_fil_ij | Pa | M L^-1 T^-2 | same as stress tensor |
D_Q | m^2·s^-1 | L^2 T^-1 | diffusivity |
tau_relax | s | T | time quantity |
N(E) | 1 | 1 | pure count |
dN/dE | eV^-1 | E^-1 | inverse energy |
Tab. 2-4 Suffixes & Reserved Words
Class | Convention | Example |
|---|---|---|
vector/field | _vec / _fld | u_vec, B_vec |
reference-frame tag | (_flow/_sheet/_shear) | Q_ij(flow) |
path & measure | gamma(ell), d ell | ∫_{gamma(ell)} (…) d ell |
ToA markers | T_arr^A, T_arr^B, delta_form | record in parallel |
Tab. 2-5 DatasetCard ToA Field Template (if applicable)
Field | Unit | Dim | Note |
|---|---|---|---|
T_arr^A | s | T | constant-factored form |
T_arr^B | s | T | general form |
delta_form | — | — | selected form flag |
integrals.path | — | — | gamma(ell) |
integrals.measure | m | L | d ell |