33-EFT.WP.Cosmo.EarlyObjects v1.0 | Chapter 3 — Object Taxonomy & Minimal Equations

Chapter 3 — Object Taxonomy & Minimal Equations

I. One-Sentence Goal

Provide a computable taxonomy and a minimal equation skeleton for early-Universe objects, unifying the dynamics of state / seed / trigger, the radiative connectors for L_nu / LC, and the propagation links to T_arr / Delta_T_arr. This forms the executable baseline for the origin modeling, metrology, and inversion developed in Chapters 4–7.

II. Scope & Non-Goals

• Covered: object taxonomy and data schema; state variables and evolution equations S70-*; origin initialization and triggers; spectral synthesis and observation connectors; coupling to Phi_T / SeaProfile; segmented connectors for the two T_arr dialects; parameter vectors and priors; sensitivity and numerical scaffolding.
• Not covered: detailed cosmological parameter fits or facility specifics; substitutes for the propagation numerics in Chapter 6 or solver implementations in Chapter 9; any construct violating n_eff ≥ 1.

III. Minimal Terms & Symbols

• Objects & types: O_i, type ∈ { PopIII, ProtoGalaxy, BHSeed, MiniQSO, ShockCloud }.
• State vector: state = { M, R, J, a_bh, SFR, Z, … } (mass, scale, angular momentum, BH spin, star-formation rate, metallicity, etc.).
• Origins & triggers: seed (initial conditions), trigger (events such as merge / collapse / cooling instability, etc.).
• Fields & propagation: T_fil(x,t), Phi_T(x,t), grad_Phi_T(x,t); n_eff(x,t,f) (dimensionless and ≥ 1), c_ref; path gamma(ell) with measure d ell.
• Radiation & observation: L_nu(f) (intrinsic spectrum), F_nu(f) (observed spectrum), LC(t) (light curve).
• Energy closure & interfaces: R_env, T_trans, A_sigma, with R_env + T_trans + A_sigma = 1.
• Two-form exemplars:
  T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
  T_arr = ( ∫ ( n_eff / c_ref ) d ell )

IV. Object Taxonomy & Data Schema

1. Classification schema (minimal fields)
   • Identity: id, type, z_form, z_obs.
   • Priors: seed (state_0 with uncertainty), trigger (event set and timestamps).
   • Environment: SeaProfile_id (if coupled to a layered sea), Sigma_env (interface type tags).
   • Observables: { T_arr(f_m, gamma_a), Delta_T_arr(f_{m1}, f_{m2}, gamma_a), F_nu(f_m), LC(t_n) } with uncertainties.
2. Type-specific fields (examples)
   • BHSeed: a_bh (spin), accretion rate dot_M, Eddington ratio λ_Edd.
   • PopIII: IMF parameters, SFR, and migration timescale tau_SF.
   • MiniQSO: radiative efficiency η, bolometric luminosity L_bol.

V. Minimal Equations (S70-1 … S70-5)

• S70-1 State evolution (skeleton)
• d state / dt = F_state( state, Φ_T, env, params )

Expanded examples:
dM/dt = F_M( state, Φ_T, env ), dJ/dt = F_J( state, Φ_T, env ),
da_bh/dt = F_a( state, Φ_T, env ), dSFR/dt = F_SFR( state, env ).

• S70-2 Origin initialization & triggers
• state(t_0) = seed ; state(t^+) = Trigger( state(t^-), event ), event ∈ { merge, collapse, inflow, … }.
• S70-3 Spectral synthesis & observation connector
• L_nu(f) = G_sed( state, params_sed )
• F_nu(f_obs) = L_nu(f_em) / ( 4π D_L^2 ) * K(z_obs), f_em = f_obs * (1+z_obs).
• S70-4 Field–object coupling term
• Φ_T = G( T_fil ), grad_Φ_T = ( dG/dT_fil ) * grad( T_fil )
• Couple( state ; Φ_T, grad_Φ_T, SeaProfile )

The coupling enters both F_state and G_sed.

• S70-5 Arrival-time and differential connectors
  The two-form segmented integrals (see Ch. 6) produce:
• T_arr(f, gamma)
• Delta_T_arr(f1,f2, gamma) = T_arr(f1, gamma) − T_arr(f2, gamma) (same path and segmentation).

VI. Constraints & Feasible Domain

• Feasibility: n_eff(x,t,f) ≥ 1; interface side limits n_eff^± ≥ 1; energy closure R_env + T_trans + A_sigma = 1.
• Lower bound: T_arr ≥ L_path / c_ref (the general dialect encodes the same via the integrand).
• Naming isolation: T_fil (tension) ≠ T_trans (transmittance); n (number density) ≠ n_eff (refractive index).

VII. Metrology & Observables (anchors M70-1 … M70-6)

• M70-1 Seed sampling & trigger logging: persist seed / trigger and RNG seeds; harden priors and hashes.
• M70-2 Reference-speed calibration: calibrate c_ref from gamma_ref, T_arr_ref; output u_stat, u_sys.
• M70-3 Two-form consistency: compute T_arr^{const}, T_arr^{gen} in parallel; output eta_T.
• M70-4 Band differential isolation: on the same path, estimate n_path parameters and log out-of-band leakage ratio.
• M70-5 Energy closure & interface audit: enforce R_env + T_trans + A_sigma = 1 and n_eff^± ≥ 1.
• M70-6 Archival & replay: persist Contract/Log, hash(*), SolverCfg, seed, and falsification samples.

VIII. Parameter Vector, Priors & Identifiability

• Parameter vector: θ = { θ_state, θ_sed, θ_couple, θ_path }, e.g., coefficients/scales for { F_M, F_J, F_a, F_SFR }, params_sed, coupling coefficients, and the polynomial coefficients c_m of n_path.
• Priors: physical bounds (non-negativity, upper limits, sparsity penalties), layered-sea gate eta_w, and thin↔thick consistency gate tau_switch.
• Identifiability: leverage multi-band differentials on the same path and multi-angle multi-path designs to reduce degeneracies between θ_state and θ_path.

IX. Data Objects & I/O (minimal set)

• Catalog: object list with type labels.
• Seeds / Triggers: origin records.
• Trajectory: state(t) series.
• SED: L_nu(f).
• Observations: T_arr / Delta_T_arr / F_nu / LC with uncertainties.
• Env: SeaProfile and Sigma_env metadata.
  All objects must carry coords_spec / units_spec / metric_spec and hash(*).

X. First-Order Variations & Sensitivities (for fitting & uncertainty propagation)

• Constant-factored dialect
• delta T_arr = (1/c_ref) * ∫ [ (∂n_eff/∂Φ_T)•delta Φ_T
• + (∂n_eff/∂grad_Φ_T)•grad(delta Φ_T)
• + (∂n_eff/∂θ)•delta θ ] d ell
• + ∑ crossings delta k_sigma .
• General dialect
• ∂T_arr/∂θ = ∫ ( ∂n_eff/∂θ ) / c_ref d ell .
• Use: least-squares / MAP fitting for θ, and GUM/MC uncertainty propagation (see Chs. 7 & 12).

XI. Numerical Skeleton & Pseudoflow (bridges to Chapter 9)

• Ingest Catalog / Seeds / Env → initialize state(t_0).
• Time-march: state(t+Δt) = state(t) + F_state • Δt (or solver stepping).
• At each output epoch: synthesize L_nu → evaluate n_eff (with coupling/layer corrections) → segmented integration for T_arr; form Delta_T_arr by differencing.
• Record: mode, eta_T, tau_switch, hash(*), DimReport; produce audit & replay bundles.

XII. Acceptance Criteria & Falsification Lines

• Accept: n_eff ≥ 1; T_arr ≥ L_path / c_ref; eta_T and tau_switch within thresholds; energy closure R_env + T_trans + A_sigma = 1; differential linear-regime / specified order satisfied; dimensional checks pass.
• Falsify: any failure of the above after excluding implementation error; missing segmentation or cross-interface interpolation; naming misuse.

XIII. Implementation Bindings & Interfaces (suggested I70-* mapping)

• I70-1 build_early_object_catalog(params) -> Catalog
• I70-2 seed_and_trigger(Phi_T, env, priors) -> Seeds
• I70-3 evolve_object_state(O, env, tgrid) -> Trajectory
• I70-4 synthesize_spectrum(O, state, fgrid) -> L_nu
• I70-6 predict_arrival_signature(n_eff, gamma, mode, c_ref) -> { T_arr, Delta_T_arr }
• I70-7 estimate_energy_triplet(data, Sigma_env) -> { R_env, T_trans, A_sigma }
  Constraints: enforce dimension checks, lower bounds, and energy closure at entry; naming isolation is mandatory; logs include hash(Catalog/Seeds/Trajectory) and threshold fields.

XIV. Cross-References

• EFT.WP.Cosmo.EarlyObjects v1.0: Ch. 4 (origins & triggers), Ch. 5 (tensor-potential coupling), Ch. 6 (radiative & propagation signatures), Ch. 7 (metrology), Ch. 9 (numerical implementation).
• EFT.WP.Propagation.TensionPotential v1.0: two forms & path integrals.
• EFT.WP.Cosmo.LayeredSea v1.0: layer coupling & segmented corrections.
• EFT.WP.Core.Tension v1.0 / Equations v1.1 / Metrology v1.0: potential mappings, notation, and traceability.

XV. Deliverables

• Equation cards S70-1 … S70-5 plus the variable/unit register.
• Parameter & prior templates: bounds for θ, regularization, and identifiability guidance.
• Audit checklists: eta_T, tau_switch, lower-bound margins, energy-closure margins, and falsification-sample templates.