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I. Conventions & Dialect (Mandatory)
- All symbols/formulae are in English and wrapped in backticks (e.g., I(t), J(x,t), T_fil).
- Any division/integral/composed operator must use parentheses, with explicit path gamma(ell) and measure d ell.
- Time-of-arrival (two equivalent dialects; record delta_form):
- Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
- General: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
- Two-dialect agreement: On the same data, the two T_arr results must agree within u(T_arr).
- Forbidden aliases & conflicts: T_fil (tension) ≠ T_trans (transmittance); n (number density) ≠ n_eff (effective refractive index); bare c is forbidden—use c_ref; when sigma is ambiguous use sigma_cond (conductivity) / sigma_std (standard deviation).
- Units: SI. By default report 3 significant digits (tighten as metrology gates require).
II. Observables & Derived Quantities
Symbol (inline) | Name | Type/Domain | SI Unit | Definition/Notes |
|---|---|---|---|---|
I(t) | loop current | scalar / real | A | port-level observable |
J(x,t) | current density | vector / real or complex | A·m⁻² | field; steady-state ∇·J=0 |
V(t) / V(omega) | voltage (time/freq) | scalar / real or complex | V | use omega as frequency variable |
Z(omega) | port impedance | complex scalar | Ω | Z_eft = Z_ref + ΔZ_T (+ ΔZ_rad) |
` | Z(omega) | ` | impedance magnitude | scalar |
arg Z(omega) | impedance phase | scalar | rad | in window ≈ omega·T_arr + φ_T |
T_group(omega) | group delay | scalar | s | d/domega( arg Z_eft ) |
III. Paths & Time-of-Arrival
Symbol | Name | Type/Domain | SI Unit | Definition/Notes |
|---|---|---|---|---|
gamma(ell) | propagation path | geometric curve | — | ell is arclength; piecewise smooth |
d ell | path measure | differential | m | paired with gamma(ell) |
T_arr | shared arrival term | scalar | s | two dialects; record delta_form |
c_ref | reference propagation limit | scalar | m·s⁻¹ | unifies dialect & dimensions |
delta_form | arrival dialect tag | enum | — | "n_over_c" or "one_over_c_times_n" |
w_p(omega) | path weight | scalar | — | 0…1, Σ_p w_p ≤ 1 |
ΔT_arr | differential arrival | scalar | s | across paths/layouts |
IV. Tension & Kernels
Symbol | Name | Type/Domain | SI Unit | Definition/Notes |
|---|---|---|---|---|
T_fil(x,t) | tension (EFT intrinsic) | scalar field | — | ontology defined in the text |
∇T_fil | tension gradient | vector field | — | direction biases relay |
K_s(x,t) | static kernel | scalar/tensor kernel | chosen so J in A·m⁻² | causal, t<0 → 0 |
K_t(x,t) | dynamic kernel | scalar/tensor kernel | chosen so J in A·m⁻² | causal, t<0 → 0 |
J_T | tension-induced current | vector | A·m⁻² | J_T = ( K_s ⊛ ∇T_fil ) + ( K_t ⊛ ∂T_fil/∂t ) |
φ_T(omega) | tension extra phase | scalar | rad | originates from ΔZ_T |
V. Material & Equivalent Parameters
Symbol | Name | Type/Domain | SI Unit | Definition/Notes |
|---|---|---|---|---|
n_eff(x,t) / n_eff(omega) | effective refractive index | scalar | — | bound to geometry/boundaries/material |
sigma_eff(x,t) / Σ_eff | effective conductivity (iso/anisotropic) | scalar/tensor | S·m⁻¹ | σ_eff ≥ 0; tensor as Σ_eff |
Z_ref(omega) | reference impedance | complex scalar | Ω | no-tension/radiation limit or calibrated baseline |
ΔZ_T(omega) | tension correction | complex scalar | Ω | from kernels + path weights |
ΔZ_rad(omega) | radiation positive-real correction | complex scalar | Ω | Re{ΔZ_rad} ≥ 0, K–K consistent |
Z_c(omega) | characteristic impedance | complex scalar | Ω | high-speed/modal analysis |
VI. High-Speed & Dispersion
Symbol | Name | Type/Domain | SI Unit | Definition/Notes |
|---|---|---|---|---|
alpha(omega) | attenuation constant | scalar | Np·m⁻¹ | material/radiation/conductor loss |
beta(omega) | phase constant | scalar | rad·m⁻¹ | beta·L ≈ omega·T_arr + φ_T |
k(omega) | complex propagation constant | scalar | Np·m⁻¹ + i·rad·m⁻¹ | k = alpha + i·beta |
E_phase | phase linearity error | scalar | rad | design/acceptance KPI |
GDR | group-delay ripple | scalar | s | design/acceptance KPI |
VII. EMI/EMC (Radiation & Immunity)
Symbol | Name | Type/Domain | SI Unit | Definition/Notes |
|---|---|---|---|---|
E_rad(omega,r) | radiated E-field | vector | V·m⁻¹ | near/far-field scans |
H_rad(omega,r) | radiated H-field | vector | A·m⁻¹ | near/far-field scans |
I_CM(omega) | common-mode current | scalar | A | EMI key observable |
sigma_seam | seam effective conductance | scalar | S | shielding continuity metric |
VIII. Metrology & Alignment
Symbol | Name | Type/Domain | SI Unit | Definition/Notes |
|---|---|---|---|---|
Δt_sync | timebase mis-alignment | scalar | s | phase correction Δφ = omega·Δt_sync |
baseline_id | baseline identifier | string | — | for de-embedding/alignment |
binding_ref | binding record ID | string | — | layout ↔ gamma(ell) |
check_dim | dimensional check | flag | — | key equalities must pass |
KK_consistency | K–K consistency | flag | — | frequency-domain causality |
passivity | passivity gate | flag | — | min Re{Z_eft} ≥ 0 |
IX. Implementation APIs & Data-Card Keys
Field/Symbol | Name | Type | Unit/Range | Notes |
|---|---|---|---|---|
compile/predict/jacobian/invert/ppc/export | binding & inversion APIs | functions | — | I40-* / Mx-* |
dataset_card | dataset card | struct | — | minimal dataset fields |
pipeline_card | pipeline card | struct | — | stages/params/version lock |
env_lock | environment lock | struct | — | interpreter/deps/container |
qa_gates | QA gates | list | — | check_dim / passivity / KK |
X. Canonical Relations (dialect-aligned, quick reference)
- Continuity: ( ∂rho / ∂t ) + ( ∇ · J ) = 0
- Effective conduction (minimal): J = ( σ_eff * ( -∇phi ) ) + ( chi_T * ∇T_fil ) + ( psi_T * ( ∂T_fil / ∂t ) )
- Tension kernel response: J_T = ( K_s ⊛ ∇T_fil ) + ( K_t ⊛ ( ∂T_fil / ∂t ) )
- Impedance mapping: Z_eft(omega) = Z_ref(omega) + ΔZ_T(omega) [+ ΔZ_rad(omega)]
- Group delay: T_group(omega) = d/domega( arg Z_eft(omega) )
- Phase (coherence window): arg Z_eft(omega) ≈ ( omega · T_arr ) + φ_T(omega)
XI. Forbidden Aliases & Writing Examples (Compliance)
- Forbidden: bare c, T, n; use c_ref, T_fil/T_trans, n_eff.
- Arrival record (example):
- arrival:
- form: "n_over_c" # or "one_over_c_times_n"
- gamma: "explicit"
- measure: "d_ell"
- c_ref: 299792458.0
- Tarr_s: 1.234e-09
- u_Tarr_s: 6.0e-12
- delta_form: "n_over_c"
- Phase correction (metrology): arg Z_corr(omega) = arg Z_raw(omega) - ( omega · Δt_sync )
XII. SI Quick Reference
Current A; voltage V; impedance Ω; phase rad; time s; length m; power density W·m⁻³; energy density J·m⁻³; energy flux W·m⁻²; conductivity S·m⁻¹; attenuation Np·m⁻¹; phase constant rad·m⁻¹.
XIII. Consistency & Gates (Release Hard Gates)
check_dim = pass; Re{Z_eft} ≥ 0; KK_consistency = pass; two-dialect T_arr agreement; E_phase and GDR meet in-window gates; if ΔZ_rad is enabled then Re{ΔZ_rad} ≥ 0.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
License link：https://creativecommons.org/licenses/by/4.0/