33-EFT.WP.Cosmo.EarlyObjects v1.0 | Chapter 1 — Terminology & Preliminaries

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Chapter 1 — Terminology & Preliminaries

I. One-Sentence Goal

Establish a unified lexicon—terms, coordinates and metrics, spectroscopic and arrival-time conventions, dimensions and units, path and segmentation notation—for “early objects” within the EFT framework. The goal is to provide consistent symbols and constraints for the modeling, metrology, and implementation of state / seed / trigger, L_nu / LC, n_eff, and the two arrival-time dialects T_arr / Delta_T_arr that will be used throughout this volume.

II. Scope & Non-Goals

• Covered: object taxonomy & data schema; redshift & time; metric and arc-length; spectra & light curves; propagation & the two arrival-time dialects; dimensions & units; naming isolation; avoidance of common ambiguities; typical dimensional checks; minimal data-contract fields.
• Not covered: detailed cosmological parameter fitting or observatory specifics; derivations of GR equations unrelated to this volume; any construct that violates n_eff ≥ 1.

III. Minimal Set of Terms & Symbols

1. Objects & taxonomy
   • O_i: an instance of an early object; type ∈ { PopIII, ProtoGalaxy, BHSeed, MiniQSO, ShockCloud }.
   • Key attributes: z_form (formation redshift), z_obs (observed redshift), M (mass), R (scale radius), J (angular momentum), a_bh (spin, if BHSeed), SFR (star-formation rate).
   • State vector: state = { M, R, J, a_bh, SFR, … }.
2. Coordinates & metric
   • t (coordinate time), eta (conformal time), chi (comoving radial coordinate), a(eta) (scale factor).
   • Line element (symbolic reference; its concrete form is persisted via metric_spec):
     ds^2 = a(eta)^2( - d eta^2 + d chi^2 + S_k(chi)^2 dΩ^2 ).
3. Fields & propagation
   • T_fil(x,t) (tension), Phi_T(x,t) (tensor potential), grad_Phi_T(x,t) (gradient), with Phi_T = G(T_fil), dG/dT_fil > 0.
   • n_eff(x,t,f) (effective refractive index, dimensionless, n_eff ≥ 1), c_ref (reference speed), c_loc = c_ref / n_eff.
4. Paths & segmentation
   • Path gamma(ell) (arc-length parameterization), ell ∈ [0, L], line element d ell; segmented paths gamma_i.
   • Environmental interfaces Sigma_env (if layered/interface coupling applies).
5. Spectra & light curves
   • L_nu(f) (intrinsic spectrum), F_nu(f) (observed spectrum with K-correction and propagation effects), LC(t) (light curve).
   • K-correction schematic: L_nu(f_em) ↔ F_nu(f_obs), f_em = f_obs • (1+z_obs).
6. Arrival times & differentials (two-form exemplar)
   • Constant-factored:
     T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • General form:
     T_arr = ( ∫ ( n_eff / c_ref ) d ell )
   • Band differential: Delta_T_arr(f1,f2) = T_arr(f1) − T_arr(f2) (same path and segmentation).
7. Energy closure at interfaces
   R_env (reflection), T_trans (transmission), A_sigma (absorption/loss), with R_env + T_trans + A_sigma = 1 (dimensionless).

IV. Inline Symbols & Units/Dimensions (SI)

• Base units: length m, time s, speed m•s^-1, frequency Hz = s^-1, mass kg.
• Dimension bindings: dim(T_arr) = [T], dim(c_ref) = [L][T^-1], dim(d ell) = [L], dim(n_eff) = 1, dim(L_nu) = [P•Hz^-1] (or according to the selected photometric system), dim(F_nu) = [P•m^-2•Hz^-1].
• Requirements: all inline symbols are wrapped in backticks; all divisions/integrals/composite operators must be parenthesized with explicit path and measure declarations (e.g., ∫_{gamma} (…) d ell).

V. Avoiding Mathematical Ambiguities

• Reparameterization: for sigma = h(ell), write the change of variables explicitly; T_arr integrals are invariant to reparameterization.
• Integrand parentheses: write ∫ ( n_eff / c_ref ) d ell, not ∫ n_eff / c_ref d ell.
• Metric → arc-length: declare chi → d ell in the contract’s metric_spec, ensuring dim(d ell) = [L].
• Redshift & frequency: distinguish f_em, f_obs, and z_obs; the spectral shape used for K-correction must state the source vs observer frame.
• Gauge indifference: if n_eff = F(grad_Phi_T, …) has no explicit dependence on the absolute Phi_T, the formulation is invariant under Phi_T → Phi_T + const.
• Energy closure & side limits: at every interface/layer event audit R_env + T_trans + A_sigma = 1 and n_eff^± ≥ 1.
• Naming isolation: T_fil (tension) vs T_trans (transmittance) must not be conflated; n (number density) vs n_eff (refractive index) must be treated distinctly.

VI. Typical Dimensional Check Examples

• Two-form arrival time
  Constant-factored: (1/c_ref)[L] → [T];
  General form: (n_eff/c_ref)[T•L^-1] × d ell[L] → [T].
• Spectral → observation: if F_nu = L_nu / (4π D_L^2) • (1+z_obs)^{-1}, the contract must declare the units and evaluation convention for luminosity distance D_L, preserving dimensional self-consistency.
• Differential: dim(Delta_T_arr) = [T].
• Environmental correction: Delta_T_sigma (zero-thickness correction) has units of s.

VII. Path, Arc-Length & Redshift–Frequency Mapping (Data & Implementation Constraints)

• Path discretization: { gamma[k], Δell[k] } (zero-based) with Δell[k] > 0; coordinates/metric/units consistent with the contract.
• Intersections & segmentation: interface indices { ell_i } satisfy gamma(ell_i) ∈ Sigma_env (if present); include segment endpoints explicitly in integrals—cross-interface interpolation is forbidden.
• Frequency grid: f_grid = { f_m } (observer frame), f_em = f_obs • (1+z_obs); compute differentials using the same path discretization and segmentation.
• Logging: minimal log includes coords_spec / units_spec / metric_spec, hash(gamma), interface_marks:{ ell_i }, and DimReport.

VIII. Acceptance Criteria & Falsification Lines (Chapter-Level)

1. Acceptance
   • Dimensional checks pass; two-form formulas and integrand parentheses are compliant;
   • Path/metric/units consistent within the contract; naming isolation respected;
   • n_eff ≥ 1 and T_arr ≥ L_path / c_ref are jointly satisfied.
2. Falsification
   • Any occurrence of n_eff < 1 or T_arr < L_path / c_ref;
   • Cross-interface interpolation or failure to include endpoints { ell_i } explicitly;
   • Misusing T_fil as T_trans, or conflating n with n_eff.

IX. Implementation Bindings & Data Objects (Aligned with Template Interfaces)

• Minimal data-contract fields: coords_spec, units_spec, metric_spec, mode ∈ { constant, general }, thresholds eps_T, eta_T, eta_w, tau_switch, hash(*), SolverCfg, seed.
• Template interface anchors: I.Path.Capture|Segment, I.Build.Phi|Neff, I.Arrival.Constant|General|Delta, I.RT.Estimate, I.Consistency.DualMode|ThinThick, I.Uncertainty.GUM|MC, I.Report.Log|Emit.
• Suggested mappings (examples within this volume):
  build_early_object_catalog (I70-1), seed_and_trigger (I70-2), evolve_object_state (I70-3), synthesize_spectrum (I70-4), predict_arrival_signature (I70-6).

X. Cross-References

• EFT.WP.Cosmo.EarlyObjects v1.0: Ch. 2 (postulates & applicability P70-*), Ch. 3 (object minimal equations S70-*), Ch. 6 (radiative & propagation signatures), Ch. 7 (metrology design).
• EFT.WP.Propagation.TensionPotential v1.0: two-form & path integrals.
• EFT.WP.Cosmo.LayeredSea v1.0: layered propagation & interface segmentation.
• EFT.WP.Core.Equations v1.1 / Metrology v1.0: notation & traceability conventions.

XI. Deliverables

• Terminology & symbol table (English • symbol • dimension), ready to copy into Appendix A’s dimension–unit register.
• Style card for expressions: path & measure notation, integrand parentheses, side-limits & energy-closure examples.
• Dimensional check templates for the two forms, Delta_T_sigma, and L_nu / F_nu.