33-EFT.WP.Cosmo.EarlyObjects v1.0 | Chapter 5 — Tensor-Potential Coupling & Growth Laws

Chapter 5 — Tensor-Potential Coupling & Growth Laws

I. One-Sentence Goal

Unify the tensor potential Phi_T = G(T_fil) with the growth laws for mass, size, angular momentum, spin, and star formation of early objects into computable minimal equations S70-*. Specify the coupling to the layered environment (SeaProfile) and the thin/thick switching convention, yielding a measurable, auditable parametrization with concrete implementation bindings.

II. Scope & Non-Goals

III. Minimal Terms & Symbols

IV. Postulates & Assumptions (P70-11 … P70-14)

V. Minimal Equations & Coupled Growth Laws (S70-7 … S70-12)

S70-7 Mass growth

dM/dt = F_M( state, Phi_T, grad_Phi_T, env ) − \dot{M}_out( state, env )

\dot{M}_out is the feedback-driven outflow (see S70-11).

S70-8 Size evolution

dR/dt = F_R( state, Phi_T, env )

(encodes self-gravitational contraction / powered expansion, tidal terms, and effective pressure support).

S70-9 Angular momentum & spin

dJ/dt = F_J( state, Phi_T, grad_Phi_T, env )

da_bh/dt = F_a( state, Phi_T, env )

(accretion, mergers, and magnetic/tidal coupling).

S70-10 Star formation / cooling flux

dSFR/dt = F_SFR( state, env ) − Q_feedback( state, env )

(cooling–heating balance).

S70-11 Feedback & energy-closure constraint
Let L_rad(state) be radiative power and \dot{E}_kin the kinetic outflow power. Then

L_rad + \dot{E}_kin + \dot{E}_therm = \dot{E}_acc + \dot{E}_grav

and at interfaces R_env + T_trans + A_sigma = 1 holds.

S70-12 Field–object chaining

grad_Phi_T = g_T(T_fil) • grad(T_fil)

Only dimensionless combinations of Phi_T, grad_Phi_T may enter F_* (declare scaling factors in the Contract).

VI. Near-Field Propagation–Coupling with the Layered Sea

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

is reported for audit/falsification (see Chs. 6 & 12).

VII. Metrology & Observables (M70-2 extended; M70-7 … M70-10)

  1. M70-7 Parameterization & priors. Persist physical bounds and priors for
    θ_growth = { θ_M, θ_R, θ_J, θ_a, θ_SFR } in the Contract.
  2. M70-8 Joint fit (observations ↔ model). Fit θ_growth and env via a joint likelihood over { T_arr, Delta_T_arr, F_nu, LC }; deliver theta_hat, Cov.
  3. M70-9 Uncertainty propagation (GUM/MC).
    • GUM: synthesize u_c via first-order sensitivities ∂T_arr/∂θ, ∂F_nu/∂θ.
    • MC: sample { θ_growth, env }, report quantiles and tail risk.
  4. M70-10 Consistency & guarding. Audit eta_T (two-form) and tau_switch (thin/thick); log energy-closure residuals and falsification samples.

VIII. Implementation Bindings & Prototypes (suggested I70-10 … I70-14)

Unified constraints: at entry, run check_dimension; enforce n_eff ≥ 1, T_arr ≥ L_path/c_ref, and R_env + T_trans + A_sigma = 1; record hash(Trajectory/env), mode, eta_T, tau_switch.

IX. Acceptance Criteria & Falsification Lines

  1. Accept if:
    • Growth-law equations are dimensionally self-consistent; θ_growth respects priors/physical bounds;
    • Two-form consistency eta_T ≤ gate; thin/thick consistency tau_switch ≤ gate;
    • Energy-closure residuals within thresholds; propagation lower bound T_arr ≥ L_path/c_ref;
    • GUM/MC reports are complete; logs contain hash(*) / SolverCfg / metric_spec.
  2. Falsify if:
    • Any path/band shows stable n_eff < 1 or T_arr < L_path/c_ref;
    • eta_T or tau_switch persistently exceeds gates;
    • Energy closure is violated or feedback conservation fails;
    • Missing segmentation / cross-interface interpolation degrades convergence;
    • Naming misuse (T_fil ↔ T_trans, n ↔ n_eff).

X. Cross-References

XI. Deliverables