33-EFT.WP.Cosmo.EarlyObjects v1.0 | Appendix A — Symbols, Units, and Dimension Checks

Appendix A — Symbols, Units, and Dimension Checks

I. One-Sentence Goal

Provide a unified symbol set, SI units, and dimensional conventions for this Early Objects volume, together with an executable dimension-checking workflow and common-pitfall guards. The intent is to keep the dimensional algebra self-consistent across Catalog / Seeds / Trajectory, Phi_T / grad_Phi_T, L_nu / LC, n_eff, T_arr / Delta_T_arr, Delta_T_sigma, and the energy triplet—under both arrival-time forms and throughout data I/O.

II. Applicability & Non-Goals

• Covered: base dimension set, unified units, core symbols & dimensions, two-form arrival checks (continuous & discrete), spectroscopic and K-correction dimensionality, interfaces & energy triplet, an automated dimension-check sequence with a DimReport template, plus common errors & safeguards.
• Not covered: uncertainty propagation (Ch. 7) and numerical implementation details (Ch. 9); device-specific units for instrument response.

III. Base Dimensions & SI Units

• Base dimensions: [L] (length), [T] (time), [M] (mass). Angles are dimensionless (radian).
• SI units: length m; time s; speed m•s^-1; frequency Hz = s^-1; mass kg; power W.
• Line element: enforce dim(d ell) = [L]. Any coordinate-to-metric mapping must yield a physical arclength in consistent units.

IV. Fixed Symbols with Dimensions (EarlyObjects)

• Coordinates & metric
  eta (conformal time), dim = [T] when used as a time parameter; a(eta) (scale factor), dim = 1.
  chi (comoving radius) is mapped to physical arclength per metric_spec; post-mapping, dim(chi) ≡ [L] in engineering use.
• Object & state
  state = { M, R, J, a_bh, SFR, Z, … }
  dim(M)=[M]; dim(R)=[L]; dim(J)=[M•L^2•T^-1]; dim(a_bh)=1; dim(SFR)=[M•T^-1]; dim(Z)=1.
• Field & potential
  T_fil(x,t) (tension), dimension set by model/medium;
  Phi_T(x,t) (tension potential), recommended non-dimensionalization (declare Phi_ref in Contract) so dim(Phi_T)=1;
  grad_Phi_T(x,t), dim = dim(Phi_T)[L^-1].
• Propagation & path
  n_eff(x,t,f) (effective refractive index), dim = 1, with hard constraint n_eff ≥ 1;
  c_ref, dim = [L][T^-1]; c_loc = c_ref / n_eff, dim = [L][T^-1];
  gamma(ell) (arclength parameterization), dim(gamma)=[L]; d ell, dim=[L].
• Spectrum & light curve
  L_nu(f) (rest-frame spectral density), typical dim = [W•Hz^-1] (or declared photometric system in Contract);
  F_nu(f) (observed spectral density), dim = [W•m^-2•Hz^-1];
  LC(t) (light curve), unit depends on definition (luminosity/flux/count-rate) and must be declared in Contract.
• Arrival times & corrections
  T_arr, dim = [T]; Delta_T_arr, dim = [T];
  zero-thickness correction Delta_T_sigma, dim = [T].
• Interfaces & energy
  R_env, T_trans, A_sigma (reflection, transmission, absorption), dim = 1, with R_env + T_trans + A_sigma = 1.
  Jump terms: C_sigma = Phi_T^+ − Phi_T^-, dim = dim(Phi_T);
  J_sigma = dot( grad_Phi_T^+ − grad_Phi_T^- , n_vec ), dim = dim(grad_Phi_T).

V. Two-Form Arrival-Time Dimension Checks (continuous & discrete)

• Constant pull-out form
  Continuous: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) ⇒ ([L]/[L•T^-1]) = [T]
  Discrete: T_arr ≈ (1/c_ref) * ∑ n_eff[gamma[k]] • Δell[k] ⇒ [T]
• General form
  Continuous: T_arr = ( ∫ ( n_eff / c_ref ) d ell ) ⇒ ([T•L^-1]•[L]) = [T]
  Discrete: T_arr ≈ ∑ ( n_eff[gamma[k]] / c_ref[gamma[k]] ) • Δell[k] ⇒ [T]
• Lower-bound coherence: because n_eff ≥ 1, always T_arr ≥ L_path / c_ref (the general form encodes this inside the integrand).

VI. Spectroscopy & K-Correction Dimensionality

Observed spectrum:

F_nu(f_obs) = L_nu(f_em) / ( 4π D_L^2 ) • K(z_obs), with f_em = f_obs • (1+z_obs)

• dim(F_nu) = [W•m^-2•Hz^-1], dim(L_nu) = [W•Hz^-1], dim(D_L) = [L], dim(K) = 1.
• LC(t): if flux-based, dim(LC) = [W•m^-2]; if count-rate, declare unit in Contract and record in DimReport.
• Joint dimensionality: when building a joint likelihood over { T_arr, Delta_T_arr, F_nu, LC }, declare all units in the Contract and retain the stance for D_L in logs.

VII. Causation/Growth Dimensional Checks (examples)

• dM/dt = F_M( … ) ⇒ dim=[M•T^-1]
• dR/dt = F_R( … ) ⇒ dim=[L•T^-1]
• dJ/dt = F_J( … ) ⇒ dim=[M•L^2•T^-2]
• da_bh/dt = F_a( … ) ⇒ dim=[T^-1]
• dSFR/dt ⇒ dim=[M•T^-2]
• grad_Phi_T = g_T(T_fil) • grad(T_fil) ⇒ dim(g_T) = dim(Phi_T)/dim(T_fil); if Phi_T is non-dimensionalized, dim(g_T) = 1/dim(T_fil).

VIII. Layer/Interface Parameter Dimensions

• If a layer term uses Xi_k = | dW_k/dchi | (from LayeredSea), then dim(Xi_k) = [L^-1].
• If n_eff has a layer component u1_k • Xi_k, to remain dimensionless we need dim(u1_k) = [L].
• Event-type correction Delta_T_sigma ≈ k_sigma • H(crossing): dim(k_sigma) = [T], dim(H)=1.
• Directional term b1 • dot( grad_Phi_T , t_hat ): dim(b1) = 1 / dim(grad_Phi_T) = [L]/dim(Phi_T); if Phi_T is non-dimensionalized then dim(b1) = [L].

IX. Automated Dimension-Check Workflow (check_dimension sequence)

• Contract ingest: load units_spec, coords_spec, metric_spec, mode, tolerances:{eps_T,eta_T,eta_w,tau_switch}.
• Bind canonical dimensions: dim(c_ref)=[L][T^-1], dim(n_eff)=1, dim(d ell)=[L], dim(T_arr)=[T], dim(Delta_T_sigma)=[T]; if using layer terms, set dim(Xi_k)=[L^-1].
• Arrival checks: verify both two-form continuous and discrete equations reduce to [T]; emit ok(formula_const/general).
• Spectroscopic checks: verify dimensionality of F_nu = L_nu/(4π D_L^2)•K; confirm use sites of f_em = f_obs•(1+z).
• Coupling/layer checks: infer whether u1_k, b1, b1_n, k_sigma units match this Appendix; flag mismatches.
• Discrete coherence: validate unit alignment of Δell and c_ref; enforce explicit inclusion of { ell_i } endpoints; no cross-interface interpolation.
• Report: emit a DimReport and persist to logs (fields below).

X. Common Errors & Safeguards

• Unit mismatch: path in km while c_ref is m•s^-1 ⇒ T_arr off by 10^3. Guard: canonicalize units at ingress; record mapping.
• Missing parentheses: ∫ n_eff / c_ref d ell vs ∫ ( n_eff / c_ref ) d ell differ in meaning. Guard: always parenthesize the integrand.
• Angle units: degrees passed to trig expecting radians. Guard: declare radian convention and convert if needed.
• Layer coefficient mis-units: u1_k not cancelling with Xi_k to a dimensionless n_eff. Guard: infer & check at assembly time.
• Interface mis-spec: C_sigma / J_sigma logged dimensionless. Guard: correct to dim(Phi_T) / dim(grad_Phi_T) or reject.
• Naming confusion: using T_fil where T_trans is intended, or n for n_eff. Guard: block at the Contract layer.
• Missing metric: absent metric_spec leaves d ell ambiguous. Guard: reject Contract if missing.

XI. DimReport (EarlyObjects) — Minimal Fields

• Meta: units_spec, coords_spec, metric_spec, mode ∈ {constant, general}
• Arrival: ok(formula_const), ok(formula_general), ok(lower_bound), ok(discrete_forms)
• Dimension table:
  { c_ref:[L T^-1], n_eff:1, d_ell:[L], T_arr:[T], Delta_T_sigma:[T], Phi_T:?, grad_Phi_T:?•[L^-1], L_nu:[W Hz^-1], F_nu:[W m^-2 Hz^-1] }
• Parameter units: { u1_k:[L], b1:[L]/dim(Phi_T), b1_n:[L]/dim(Phi_T), k_sigma:[T] } (when layer/directional terms are enabled)
• Interface & energy: { C_sigma:dim(Phi_T), J_sigma:dim(grad_Phi_T), R_env:1, T_trans:1, A_sigma:1 }
• Differentials: { dim(Delta_T_arr):[T] }
• Notes: anomalies and auto-fix annotations
• Hashes: hash(Catalog/Seeds/Trajectory/SeaProfile/Phi_T/n_eff/gamma/code)

XII. Worked Checks (Four Illustrations)

• Ex.1 — Uniform medium: n_eff ≡ 1 ⇒ T_arr = L_path / c_ref, so [L]/([L][T^-1]) = [T]; lower bound attained.
• Ex.2 — Thin-layer zero-thickness correction: single crossing, Delta_T_sigma = k_sigma, dim(k_sigma) = [T]; T_arr_total = T_arr + Delta_T_sigma remains [T].
• Ex.3 — First-order band dispersion differential: n_path ≈ c_1(f−f0), dim(c_1)=[T] ⇒ dim(Delta_T_arr)=[T]; two-form coherence holds.
• Ex.4 — Directional term: n_eff ≈ a0 + b1 • dot(grad_Phi_T, t_hat); if Phi_T is non-dimensionalized, dim(b1)=[L], thus overall dimensionless.

XIII. Cross-References

• EFT.WP.Cosmo.EarlyObjects v1.0: Ch. 3 (minimal equations), Ch. 5 (coupling & growth), Ch. 6 (radiation & propagation), Ch. 9 (numerics), Ch. 12 (error budget).
• EFT.WP.Propagation.TensionPotential v1.0: two-form and path expressions.
• EFT.WP.Cosmo.LayeredSea v1.0: layer and interface dimensional alignment.
• EFT.WP.Core.Equations v1.1 / Metrology v1.0: notation and traceability.

XIV. Deliverables

• Dimension–unit binding sheet (drop-in to the Contract).
• check_dimension execution order & decision rules (incl. spectroscopy and layer terms).
• DimReport examples with log-field mapping (including eta_T / tau_switch and energy-triplet entries).