33-EFT.WP.Cosmo.EarlyObjects v1.0 | Chapter 4 — Initial Conditions & Origins (Seeds and Triggers)

Chapter 4 — Initial Conditions & Origins (Seeds and Triggers)

I. One-Sentence Goal

Define the seed and trigger mechanisms for early-Universe objects, specifying priors, trigger rates, and state initialization in minimal, computable models. Tie these causative pieces traceably into the tensor potential Phi_T, the layered medium SeaProfile, and the two arrival-time dialects T_arr / Delta_T_arr for end-to-end metrology and implementation.

II. Scope & Non-Goals

• Covered: seed priors and sampling; trigger event families and trigger rates; state initialization and event updates; causal coupling to Phi_T / SeaProfile; metrological logging and uncertainty propagation; interface and data-object constraints.
• Not covered: instrument-specific observing workflows or full cosmological parameter fitting; any propagation construct that violates n_eff ≥ 1; any empirical “black-box refractivity” in place of interface side-limits.

III. Minimal Terms & Symbols

• Object & state: O_i; type ∈ { PopIII, ProtoGalaxy, BHSeed, MiniQSO, ShockCloud };
  state = { M, R, J, a_bh, SFR, Z, … }.
• Origins & triggers: seed (initial conditions), trigger with event set
  event ∈ { merge, collapse, inflow, shock, cooling-instability }.
• Tensor potential & environment: T_fil(x,t), Phi_T(x,t) = G(T_fil), grad_Phi_T(x,t); layered sea SeaProfile and interface set Sigma_env.
• Propagation variables: n_eff(x,t,f) (dimensionless, n_eff ≥ 1), c_ref, path gamma(ell), measure d ell.
• Observables: T_arr(f, gamma), Delta_T_arr(f1,f2, gamma), F_nu(f), LC(t).
• Energy closure: R_env + T_trans + A_sigma = 1 (dimensionless).
• Naming isolation: T_fil (tension) ≠ T_trans (transmittance); n (number density) ≠ n_eff (effective refractive index).

IV. Postulates & Assumptions (P70-7 … P70-10)

• P70-7 Parametrizable seed priors.
  Seed distributions are representable by a finite-parameter prior, e.g. p(seed | type, z_form, Φ_T). Prior parameters are metrically estimable and must be persisted in the Contract.
• P70-8 Finite, measurable trigger families.
  Early-object triggers belong to a finite set event ∈ { merge, collapse, inflow, shock, cooling-instability }. Their rates λ_event(z, Φ_T, env) admit parametric forms calibrated to observations or simulations.
• P70-9 Order-preserving causation.
  Formation and early growth respond to Phi_T monotonically (dG/dT_fil > 0), ensuring causal updates do not violate the propagation feasibility n_eff ≥ 1.
• P70-10 Metrological traceability.
  Every seed/trigger must record samples, timestamps, RNG seeds, parameter hashes, and environment snippets into the Contract/Log to enable replay and falsification audits.

V. Minimal Models (S70-6 … S70-9)

1. S70-6 Seed priors & sampling
   • Form (example):
     seed ~ p(M_0, R_0, J_0, a_bh0, Z_0 | type, z_form, Φ_T).
   • Sampling: { seed_j } are drawn from the priors and the environment slice env(z_form); record seed_rng.
2. S70-7 Trigger rates & arrival process
   • Intensity:
     λ_event(z) = Λ_event( Φ_T, grad_Φ_T, SeaProfile, … ).
   • Event times:
     t_event ~ InhomogeneousPoisson( λ_event(z(t)) ).
   • Event update:
     state(t^+) = Trigger( state(t^-), event ).
3. S70-8 Order-preserving causal coupling
   • Φ_T = G(T_fil), grad_Φ_T = ( dG/dT_fil ) • grad(T_fil).
   • Coupling impact:
     Couple_seed( seed ; Φ_T, grad_Φ_T, SeaProfile ) and
     Couple_trigger( event ; Φ_T, env ) inform state initialization and event updates.
4. S70-9 Initialization → propagation linkage
   • Initialize: state(t_0) = seed.
   • Propagation signatures: via Ch. 6 segmented two-form integrals to obtain T_arr(f, gamma) and
     Delta_T_arr(f1,f2, gamma) = T_arr(f1, gamma) − T_arr(f2, gamma) (same path and segmentation).
   • Hard constraints: n_eff ≥ 1, lower bound T_arr ≥ L_path / c_ref, and energy closure at interfaces.

VI. Metrology & Observables (M70-1 … M70-4)

• M70-1 Seed sampling & recording
  Inputs: priors, environment slice (Φ_T / SeaProfile), type.
  Outputs: Seeds (including seed_rng, prior parameters & hashes).
  Traceability: write coords_spec / units_spec / metric_spec and hash(Seeds) into the Contract.
• M70-2 Trigger-rate calibration
  Estimate parameters of Λ_event via joint likelihoods over T_arr / Delta_T_arr / F_nu / LC (or benchmarks).
  Deliverables: theta_hat, covariance, and residual diagnostics.
• M70-3 Two-form and energy-closure audits
  Compute T_arr^{const} and T_arr^{gen} in parallel, report eta_T;
  for each interface event, enforce R_env + T_trans + A_sigma = 1 and n_eff^± ≥ 1.
• M70-4 Thin/thick consistency
  If coupled to SeaProfile, run dual solves in the eta_w neighborhood:
  T_arr^{thin} + Delta_T_sigma vs T_arr^{thick}; report tau_switch; if exceeded, fix to thick-layer chain.

VII. Implementation Bindings & Prototypes (I70-1 … I70-4)

• I70-1 build_early_object_catalog(params) -> Catalog
  Produce the object list with type labels, z_form / z_obs, environment references & hashes.
• I70-2 seed_and_trigger(Phi_T, env, priors) -> Seeds
  Sample seeds given priors and Φ_T / SeaProfile; derive an event-process handle (supports inhomogeneous Poisson).
• I70-3 evolve_object_state(O, env, tgrid) -> Trajectory
  Apply Trigger at event times; output the state(t) series and an event log.
• I70-4 synthesize_spectrum(O, state, fgrid) -> L_nu
  Generate the intrinsic spectrum from state and params_sed (for Ch. 6 propagation & metrology).

Unified constraints: enforce check_dimension at entry; never conflate T_fil with T_trans; if the two arrival-time dialects are used downstream, record mode and eta_T.

VIII. Acceptance Criteria & Falsification Lines

1. Accept if:
   • Seeds/triggers carry samples, timestamps, RNG seeds, hashes;
   • Trigger-rate calibration residuals pass gates;
   • eta_T ≤ threshold, tau_switch ≤ gate;
   • All linked propagation chains satisfy n_eff ≥ 1, T_arr ≥ L_path / c_ref, and energy closure.
2. Falsify if:
   • Any propagation chain shows stable n_eff < 1 or T_arr < L_path / c_ref;
   • Persistent two-form inconsistency with no successful back-trace;
   • tau_switch routinely exceeds gate, or segmentation endpoints { ell_i } are missing;
   • Missing seed_rng / hashes / contract fields renders the run unreproducible.

IX. Cross-References

• EFT.WP.Cosmo.EarlyObjects v1.0: Ch. 3 (object minis), Ch. 5 (tensor-potential coupling), Ch. 6 (radiation & propagation signatures), Ch. 7 (metrology), Ch. 9 (numerical implementation).
• 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 / Core.Equations v1.1: traceability and notation.

X. Deliverables

• Seed/trigger prior & event cards (drop-in for Appendix B data schema).
• Trigger-rate calibration template (with theta_hat, covariance, residual panels).
• Metrology-to-implementation handoff checklist (Contract/Log fields, hash(*), mode, eta_T / tau_switch, interface_marks:{ ell_i }).