33-EFT.WP.Cosmo.EarlyObjects v1.0 | Chapter 6 — Radiation & Propagation Signatures

Home ／ Docs-Technical WhitePaper ／ 33-EFT.WP.Cosmo.EarlyObjects v1.0

Chapter 6 — Radiation & Propagation Signatures

I. One-Sentence Goal

Unify an early object’s intrinsic radiation L_nu(f) with observed spectra/light curves F_nu(f), LC(t), and with its propagation signatures T_arr(f, gamma) and Delta_T_arr(f1,f2, gamma) in a two-form + segmented computable convention. Provide consistent thin/thick corrections against the layered sea (SeaProfile) and interface set Sigma_env, together with auditable rules.

II. Scope & Non-Goals

• Covered: spectral synthesis and K–correction; two-form arrival times and segmented integrals; zero-thickness correction Delta_T_sigma vs thick–layer volume integral consistency tau_switch; band-differential isolation of the path term; multi-path / “echo” approximations; metrological workflow and implementation bindings; logging and falsification lines.
• Not covered: instrument response and observatory specifics; substitutes for the full solver (Ch. 9) or the uncertainty propagation corpus (Ch. 12); any propagation construct violating n_eff ≥ 1.

III. Minimal Terms & Symbols

• Radiation: L_nu(f) (intrinsic spectrum), F_nu(f) (observed spectrum), LC(t) (light curve).
• Coordinates & frequency: f_obs (observer frame), f_em (source frame), f_em = f_obs • (1+z_obs).
• Propagation & path: n_eff(x,t,f) (dimensionless, n_eff ≥ 1), c_ref, path gamma(ell) with measure d ell.
• Environment & layers: SeaProfile (layered sea), interface set Sigma_env; zero-thickness correction Delta_T_sigma; consistency metric tau_switch.
• Energy closure: R_env + T_trans + A_sigma = 1 (dimensionless).
• Two-form exemplars:
  Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
  General form: T_arr = ( ∫ ( n_eff / c_ref ) d ell )

IV. Postulates & Assumptions (P70-15 … P70-17)

• P70-15 Spectral parametrizability. Within a target band, L_nu(f) admits a finite-parameter family (e.g., broken power laws / thermal + non-thermal) to metrological tolerance.
• P70-16 Feasible propagation & energy closure. Media obey n_eff ≥ 1; interface events obey R_env + T_trans + A_sigma = 1.
• P70-17 Segmented consistency. Paths are segmented at Sigma_env / SeaProfile; segment endpoints are integrated explicitly; cross-interface interpolation is forbidden. Thin/thick dual chains are audited via tau_switch.

V. Minimal Equations & Models (S70-10 … S70-13)

S70-10 Spectral synthesis & K–correction

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).

S70-11 Two-form arrival time (segmented)
With segment boundaries { ell_i }:

T_arr = ( 1 / c_ref ) * ( ∑_i ∫_{gamma_i} n_eff d ell ) (constant-factored)

T_arr = ∑_i ∫_{gamma_i} ( n_eff / c_ref ) d ell (general form)

Lower bound: T_arr ≥ L_path / c_ref.

S70-12 Thin/thick consistency

Thin: Delta_T_sigma ≈ k_sigma • H(crossing)

Thick: T_arr^{layer} = ∫_{layer} ( n_eff / c_ref ) d ell

Consistency: tau_switch = | T_arr^{thick} − ( T_arr^{thin} + Delta_T_sigma ) |

S70-13 Band-differential isolation of the path term

Constant-factored:

Delta_T_arr(f1,f2) = ( 1 / c_ref ) * ∫ ( n_path(f1) − n_path(f2) ) d ell

General form:

Delta_T_arr(f1,f2) = ∫ ( ( n_path(f1) − n_path(f2) ) / c_ref ) d ell

Requirement: reuse the same { gamma[k], Δell[k] } and the same segmentation/correction configuration.

VI. Metrology & Observables (M70-3 Expanded)

• Joint sampling (spectrum + arrival time): on f_grid, acquire F_nu(f_obs) and T_arr(f_obs, gamma); persist coords_spec / units_spec / metric_spec.
• Two-form consistency: compute T_arr^{const} and T_arr^{gen} in parallel, produce eta_T; if exceeded, revisit c_ref calibration and the n_eff decomposition.
• Differential isolation & OOB evaluation: analyze correlation/slope of Delta_T_arr_obs / Delta_T_arr_mod; fold out-of-band residuals into u_sys.
• Thin/thick dual run: in the eta_w neighborhood, compute tau_switch; if exceeded, lock to thick layer and revisit SeaProfile and endpoint tolerances.
• Energy-closure audit: at each interface, report residual curves for R_env / T_trans / A_sigma.
• Minimal log set: hash(state / L_nu / n_eff / gamma), interface_marks:{ ell_i }, eta_T, tau_switch, DimReport.

VII. Multi-Path & “Echo” Approximation (Observation-Friendly)

• Weighted synthesis:
  T_arr_total = ∑_m w_m • T_arr[ gamma_m ], with ∑_m w_m = 1 (or amplitude-normalized). Weights are set by geometry and R_env, T_trans, A_sigma.
• Echo order: if a closed loop of length L_loop exists, the k-th echo delay
• Delta_T_echo(k) ≈ k • ∫_{loop} ( n_eff / c_ref ) d ell (general form)

serves as a template component for LC(t) pulse fitting and decomposition.

VIII. Implementation Bindings & Interfaces (aligned with I70-*)

• 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 }
• estimate_energy_triplet( data, Sigma_env ) -> { R_env, T_trans, A_sigma }
• log_radiative_transport( meta, hashes, metrics ) -> Log
  Constraints: enforce entry dimension checks; audit T_arr ≥ L_path / c_ref and R_env + T_trans + A_sigma = 1; record mode, eta_T, tau_switch.

IX. Acceptance Criteria & Falsification Lines

• Accept if: spectrum–propagation linkage is dimensionally self-consistent; eta_T and tau_switch within thresholds; R_env + T_trans + A_sigma = 1; Delta_T_arr meets linear-regime / specified-order criteria; segment endpoints { ell_i } are explicitly integrated.
• Falsify if: stable n_eff < 1; persistent two-form inconsistency; tau_switch beyond gate; cross-interface interpolation; failure of differential linear regime not explained by out-of-band leakage.

X. Cross-References

• EFT.WP.Cosmo.EarlyObjects v1.0: Ch. 3 (minimal equations), Ch. 4 (seeds & triggers), Ch. 5 (coupling & growth laws), Ch. 7 (metrology), Ch. 9 (numerics).
• EFT.WP.Cosmo.LayeredSea v1.0: Chs. 6–8 (segmented conventions, interface matching, consistency).
• EFT.WP.Propagation.TensionPotential v1.0: two forms, differentials, and path conventions.

XI. Deliverables

• Spectrum–propagation connector cards: S70-10 … S70-13 with unit/dimension registers.
• Metrology script anchors: two-form eta_T, thin/thick tau_switch, differential linear-regime panels.
• Audit log templates: hash(*), interface_marks, DimReport, R/T/A residuals, and falsification-sample registry.