42-EFT.WP.Heat.Decoherence v1.0 | Chapter 2: Terminology, Symbols & Metrology Baseline

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Chapter 2: Terminology, Symbols & Metrology Baseline

I. Scope & Dependency Anchors

• This chapter freezes the terminology, symbols, and metrology baseline of the volume: frequency/spectral-density conventions, Fourier transform conventions, units & dimensions, uncertainty & coverage factors, mandatory data-contract fields, and validation rules.
• All formulas/symbols/definitions are written in English and wrapped in backticks; cross-volume citations use the fixed style “volume + version + anchor (P/S/M/I)”.
• Dependency anchors: S20-1—S20-12 (this chapter). Cross-reference EFT.WP.Core.Metrology v1.0 Ch.1–3,5 and EFT.WP.Core.Equations v1.1 Ch.2 S20-*.
• Collision bans: T_fil (tension) ≠ T_trans (transmittance); n (number density) ≠ n_eff (effective refractive index / medium proxy).

II. Frequency, Spectral Density & Transform Conventions

1. S20-1 (Angular vs. linear frequency)
   Use ω = 2π f. Do not mix ω/f in a single expression unless explicitly stated.
2. S20-2 (Fourier dual)
   F(ω) = ∫ f(t) e^{-i ω t} dt; f(t) = (1/2π) ∫ F(ω) e^{i ω t} dω.
3. S20-3 (PSD convention)
   Explicitly declare one-sided or two-sided:
   • Two-sided: S_{xx}(ω), unit unit(x)^2/Hz.
   • One-sided: S_{xx}^{(1)}(f), same unit per Hz, domain f ≥ 0.
4. S20-4 (Cross-spectra & coherence)
   S_{xy}(ω) and coherence γ_{xy}^2(ω) = |S_{xy}(ω)|^2 / (S_{xx} S_{yy}). Default is two-sided unless declared.

III. Units, Dimensions & Uncertainty

1. S20-5 (Unit system)
   SI throughout: K, J, W, m, s, Hz, rad, and engineering dBm. Phase-PSD in rad^2/Hz.
2. S20-6 (Dimension checks)
   All expressions must pass check_dim; dataset tables must include unit columns. Typical verifications:
   • S_VV = 4 k_B T R ⇒ [V^2/Hz] = [J·K^-1][K][Ω].
   • φ_CP = (2π/λ)(ρ + c_ref b_t) ⇒ [rad].
   • f_D = - ( f_c / c_ref )( v_rel · n̂ ) ⇒ [Hz].
3. S20-7 (Uncertainty & coverage)
   Standard u(x), combined u_c, expanded U = k · u_c (coverage factor k); covariance Σ, observation covariance Σ_y.

IV. Canonical Noise & Response Conventions

1. S20-8 (Classical thermal noise)
   Johnson–Nyquist (classical limit) S_VV = 4 k_B T R.
2. S20-9 (Quantum FDT)
   S_{xx}(ω) = 2 coth( (ħω)/(2 k_B T) ) · Im χ_{xx}(ω). Each use of χ(ω) must specify response variable, excitation, and units.
3. S20-10 (Bath spectral density)
   If using a scaled family, default J(ω) = η ω (ω/ω_c)^{s-1} e^{-ω/ω_c}; record η, s, ω_c provenance in the dataset.
4. S20-11 (Heat transport & non-Fourier)
   Diffusion: ∂_t T = α ∇^2 T; Cattaneo–Vernotte: τ_q ∂_t q + q = -κ ∇T, with [α]=m^2·s^-1, [κ]=W·m^-1·K^-1.
5. S20-12 (Cross-disciplinary arrival-time fields)
   If the metrology chain reuses propagation/arrival time, the dataset must declare one convention:
   • T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ), or
   • T_arr = ( ∫ ( n_eff / c_ref ) d ell ),
     and explicitly provide gamma(ell) and d ell.

V. Symbol Table (Incremental)

VI. Data Contract (Required/Recommended Fields)

unit_system: "SI"

spectral_conventions:

fourier: {forward: "∫ f(t) e^{-iωt} dt", inverse: "(1/2π) ∫ F(ω) e^{iωt} dω"}

psd: {type: "one-sided|two-sided", freq: "ω|f"}

noise_models:

classical: "S_VV = 4 k_B T R"

quantum_fdt: "S_xx(ω) = 2 coth(ħω/2k_B T) · Im χ_xx(ω)"

bath_spectrum:

model: "ohmic|sub|super|custom"

params: {eta: "<…>", s: "<…>", omega_c: "<rad/s>"}

heat_transport:

model: "Fourier|Cattaneo|BTE"

params: {kappa: "<W/m/K>", alpha: "<m^2/s>", R_K: "<K m^2/W>"}

arrival_time: # if propagation chain is reused, mandatory

convention: "pulled_const|integrand"

delta_form: "c_ref^-1 * ∫ n_eff dℓ" # or "∫ (n_eff/c_ref) dℓ"

gamma: "piecewise: free|fixture|substrate|device|environment"

d_ell: "m"

uncertainty:

coverage_k: 2.0

Σ: "<block-diagonal or sparse>"

references: ["Heat.Decoherence v1.0:S20-*","Core.Metrology v1.0:Ch.1–3,5"]

VII. Quality Gates (This Chapter)

• Q1 Convention consistency: PSD one-/two-sided and ω/f choices are consistent; any change is versioned in the dataset.
• Q2 Units/dimensions: all formulas pass check_dim; unit columns for phase/frequency/power present.
• Q3 Identifiability: Fisher conditioning for J(ω) → {Γ_1, Γ_φ} is controlled; if cond(F) > κ_th, mark in the dataset and refer to Ch.10 strategies.
• Q4 Traceability: provenance of η, s, ω_c, χ(ω) (measurement/fit/literature) is recorded in references.
• Q5 Collision bans: never mix T_fil/T_trans or n/n_eff; distinguish rad from dimensionless clearly.

VIII. Cross-Volume References & Anchors (This Chapter)

• Cross-volume (fixed style): EFT.WP.Core.Metrology v1.0 Ch.1–3,5; EFT.WP.Core.Equations v1.1 Ch.2 S20-*.
• Anchors: S20-1—S20-12.