Chapter 2 — Postulates & Applicability
I. One-Sentence Goal
State the minimal postulates P70-* and applicability limits for “early objects” within the EFT framework, fix hard constraints for the two arrival-time dialects, energy closure, and feasible domains, and provide an auditable, implementation-ready metrological convention.
II. Scope & Non-Goals
- Covered: object origins and triggers; spectral and arrival-time observables; coupling to the layered sea / interfaces; two-form consistency and energy closure; thin↔thick layer switching; tolerances and logging requirements; minimal anchors for metrology and implementation.
- Not covered: explicit derivations of cosmological metrics/parameters; device-level observation pipelines; any constructs that violate n_eff ≥ 1 or introduce empirical “black-box refractivity.”
III. Minimal Terms & Symbols
- Objects & state: O_i; type ∈ { PopIII, ProtoGalaxy, BHSeed, MiniQSO, ShockCloud };
state = { M, R, J, a_bh, SFR, … }. - Coordinates & path: eta (conformal time), chi (comoving radial), a(eta) (scale factor); path gamma(ell) with line element d ell.
- Fields & propagation: T_fil(x,t), Phi_T(x,t), grad_Phi_T(x,t); n_eff(x,t,f) (dimensionless, n_eff ≥ 1), c_ref, c_loc = c_ref / n_eff.
- Spectra & light curves: L_nu(f) (intrinsic), F_nu(f) (observed), LC(t) (light curve).
- Events & energy: R_env, T_trans, A_sigma, with R_env + T_trans + A_sigma = 1.
- Two arrival-time dialects:
Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
General form: T_arr = ( ∫ ( n_eff / c_ref ) d ell ) - Naming isolation: T_fil ≠ T_trans; n (number density) ≠ n_eff (effective refractive index).
IV. Postulates & Constraints P70-*
- P70-1 Finite families of origins.
The initial seed/trigger of early objects reduces to a finite family (e.g., gravitational collapse, mergers, external inflow / cooling instabilities). Trigger rates are parametrizable and metrically traceable. - P70-2 Finite-parameter spectral approximations.
Within the target band, the intrinsic spectrum L_nu(f) admits a finite-parameter family (e.g., broken power law / thermal + non-thermal tail) that is invertible with the observed spectrum F_nu(f) via K-correction to a stated tolerance. - P70-3 Propagation feasibility & lower bound.
Media satisfy n_eff(x,t,f) ≥ 1; therefore T_arr ≥ L_path / c_ref under both dialects (the general form encodes the same via the integrand). - P70-4 Energy closure & side limits.
At any environment/layer event, R_env + T_trans + A_sigma = 1 and side limits obey n_eff^± ≥ 1. - P70-5 Coupling to the layered sea & switching.
When near-field coupling to SeaProfile applies and Delta_k / L_char ≤ eta_w, a zero-thickness correction Delta_T_sigma may represent the layer; otherwise use thick-layer volume integrals and audit thin↔thick consistency via tau_switch. - P70-6 Two-form consistency & band decomposition.
Within the target band, admit n_eff = n_common(x,t) + n_path(x,t,f). The two-form consistency metric
eta_T = | T_arr^{const} − T_arr^{gen} | must remain below threshold; violations require revisiting c_ref calibration and the decomposition residuals. - P70-7 Measurable set & traceability.
The minimal observable set is { T_arr(f, gamma), Delta_T_arr(f1,f2, gamma), F_nu(f), LC(t) }. All measurements/inferences must trace back to Contract/Log units, coordinates, and thresholds.
V. Default Boundary Conditions & Consistency
- Far-field boundary (choose & record):
Dirichlet: Phi_T → 0; Neumann: dot( grad_Phi_T , n_vec ) = 0; Robin: alpha • Phi_T + beta • dot( grad_Phi_T , n_vec ) = g(x,t). - Two-form segmented controls:
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) - Hard constraints: n_eff ≥ 1, T_arr ≥ L_path / c_ref, R_env + T_trans + A_sigma = 1, naming isolation enforced.
VI. Applicable Scales & Neglected Terms
- Coherence windows: assume quasi-stationarity and piecewise smoothness within spatial ell_coh and temporal tau_coh; faster phenomena than tau_coh are logged as impulse corrections.
- Geometry & metric: declare metric and the mapping chi → d ell in the Contract to guarantee dim(d ell) = [L]; treat small curvature via piecewise smoothing.
- Thin/thick gate: if Delta_k / L_char ≤ eta_w, enable zero-thickness corrections; within the switching neighborhood, run dual solves and report tau_switch.
- Out-of-band leakage: fold band-edge energy into u_sys and record the leakage ratio and alignment strategy.
VII. Minimal Equations & Calls S70-*
- S70-1 object state vector state = { M, R, J, a_bh, SFR, … } and evolution skeleton.
- S70-2 parameterized seed/trigger-rate models and priors (Ch. 4).
- S70-3 connector between L_nu = G(state, params) and F_nu with K-correction.
- S70-4 propagation link: two-form arrival times, segmented integrals, and Delta_T_arr isolation (Ch. 6).
- S70-5 coupling to Phi_T / grad_Phi_T and SeaProfile (Ch. 5 and LayeredSea Chs. 6–8).
This chapter provides call anchors; complete expressions and derivations are given in Chs. 3/4/5/6.
VIII. Metrology & Calibration Workflow M70-*
- M70-1 Seed sampling & trigger logging: choose priors, generate Seeds, persist; record RNG seeds and parameter hashes.
- M70-2 Reference-speed calibration: calibrate c_ref using gamma_ref, T_arr_ref; output u_stat, u_sys.
- M70-3 Two-form audit: compute both dialects in parallel, produce eta_T; if exceeded, back-trace c_ref and the n_eff decomposition.
- M70-4 Band differential isolation: evaluate Delta_T_arr on multiple frequencies along the same path; estimate n_path parameters; record out-of-band leakage ratio.
- M70-5 Energy closure & layer coupling: at interface/layer events audit R_env + T_trans + A_sigma = 1; within eta_w run thin/thick dual solves and output tau_switch.
- M70-6 Archival & replay: harden Contract/Log, hash(*), and SolverCfg; register falsification samples and replay handles.
IX. Implementation Bindings & Interfaces I70-*
- build_early_object_catalog( params ) -> Catalog
- seed_and_trigger( Phi_T, env, priors ) -> Seeds
- evolve_object_state( O, env, tgrid ) -> Trajectory
- synthesize_spectrum( O, state, fgrid ) -> L_nu
- 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 }
Constraints: enforce dimension checks and lower bounds at entry; energy closure and naming isolation are mandatory; logs include hash(Catalog/Seeds/Trajectory), mode, eta_T, tau_switch.
X. Cross-References
- EFT.WP.Cosmo.EarlyObjects v1.0: Ch. 3 (object minimal equations), Ch. 4 (origins & triggers), Ch. 5 (tensor-potential coupling), Ch. 6 (radiative & propagation signatures), Ch. 7 (metrology workflows).
- EFT.WP.Propagation.TensionPotential v1.0: two-form and path segmentation.
- EFT.WP.Cosmo.LayeredSea v1.0: layer coupling, matching, and thin/thick consistency.
- EFT.WP.Core.Metrology v1.0: traceability and reporting.
XI. Verification & Falsification Lines
- Verify: T_arr ≥ L_path / c_ref; eta_T within gate; R_env + T_trans + A_sigma = 1 within residual thresholds; tau_switch within bounds; Delta_T_arr linear-regime / specified-order agreement.
- Falsify: persistent n_eff < 1; sustained two-form inconsistency with no successful back-trace; energy-closure violation; tau_switch beyond threshold; missing segmentation or cross-interface interpolation.
XII. Systematic-Error Guarding
- Path & segmentation: include endpoints { ell_i } explicitly in integrals; forbid cross-interface interpolation; enforce step refinement in high-Xi_k(chi) regions.
- Band leakage: reuse the same { gamma[k], Δell[k] } and correction configuration across frequencies; fold out-of-band residuals into u_sys.
- Thin/thick switching: run dual solves in the eta_w neighborhood and record tau_switch; if exceeded, fix to thick-layer volume integrals.
- Names & units: unify coords_spec / units_spec / metric_spec at entry; execute check_dimension and persist a DimReport.
XIII. Deliverables
- Postulate cards P70-1 … P70-7 with invocation guidance.
- Audit checklists for the two forms, energy closure, and thin/thick consistency (with threshold fields).
- Segmentation & interface log templates: { ell_i }, type tags, tau_switch, eta_T, energy residuals, and the falsification sample register.