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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 |
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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