33-EFT.WP.Cosmo.EarlyObjects v1.0 | Chapter 2 — Postulates & Applicability

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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.