33-EFT.WP.Cosmo.EarlyObjects v1.0 | Appendix C — Derivation Details & Proof Sketches

Appendix C — Derivation Details & Proof Sketches

I. Preliminaries & Notation

  1. Path & measure: gamma(ell) is a piecewise-C^1 arc-length parameterized path with ell ∈ [0,L], line element d ell, and total length L_path = ( ∫_0^L d ell ).
  2. Coordinates & metric: conformal time eta, comoving radius chi, scale factor a(eta); the concrete metric_spec is recorded in the Contract to ensure dim(d ell) = [L].
  3. Objects & state: O_i, type ∈ {PopIII, ProtoGalaxy, BHSeed, MiniQSO, ShockCloud}, state = { M, R, J, a_bh, SFR, Z, … }.
  4. Field & propagation: Phi_T = G(T_fil), grad_Phi_T = ( dG/dT_fil ) • grad(T_fil); n_eff(x,t,f) ≥ 1, c_ref, c_loc = c_ref / n_eff.
  5. Spectrum & observation: L_nu(f) (intrinsic), F_nu(f) (observed), LC(t); f_em = f_obs • (1+z_obs).
  6. Two arrival-time forms:
    • Constant pull-out: T_arr = (1/c_ref) * ( ∫_gamma n_eff d ell )
    • General form: T_arr = ( ∫_gamma ( n_eff / c_ref ) d ell )
  7. Regularity: within a coherence window, F, G, and H_sea are Lipschitz in their arguments; n_eff(•) is piecewise continuous and bounded; c_ref is constant or piecewise continuous.

II. Basic Properties of Path Integrals (cf. §6 S70-11)

Lemma 1 (Reparameterization invariance).
If sigma = h(ell) is strictly monotone and differentiable, then
( ∫_0^L g( gamma(ell) ) d ell ) = ( ∫_{h(0)}^{h(L)} g( gamma( h^{-1}(sigma) ) ) d sigma ).
Sketch: chain rule and change of variables; monotone invertibility legitimizes the measure transform.

Lemma 2 (Additivity under concatenation).
If gamma = gamma_1 ∘ gamma_2 with continuous endpoint matching, then
( ∫_gamma g d ell ) = ( ∫_{gamma_1} g d ell ) + ( ∫_{gamma_2} g d ell ).
Corollary: for both forms, T_arr[gamma] = T_arr[gamma_1] + T_arr[gamma_2].

III. Two-Form Consistency & Lower Bounds (cf. §2/§6)

Proposition 1 (Sufficient condition for two-form agreement).
When c_ref is constant, or its variability can be absorbed into a unified integrand n_eff/c_ref within metrological tolerance, eta_T = | T_arr^{const} − T_arr^{gen} | stays below threshold.
Sketch: constant case pulls out (1/c_ref); otherwise treat n_eff/c_ref as a single integrand—differences are higher-order smalls.

Theorem 1 (Arrival-time lower bound).
If n_eff ≥ 1, then
Constant pull-out: T_arr ≥ L_path / c_ref;
General form: T_arr ≥ ( ∫_gamma (1/c_ref) d ell ).
Sketch: take the pointwise lower bound of the integrand and apply order-preserving integration.

Proposition 2 (Equality condition).
The bound is tight iff n_eff ≡ 1 along the path.

IV. Causation Process & Trigger Times (cf. §4 S70-7)

Proposition 3 (Inhomogeneous Poisson arrivals).
With intensity λ_event(eta) = Λ_event( Phi_T(eta), grad_Phi_T(eta), env(eta) ), the survival function for the first trigger time eta_* is
S(eta) = exp( − ( ∫_{eta_0}^{eta} λ_event(s) ds ) ), with density p(eta_*) = λ_event(eta_*) • S(eta_*).
Sketch: standard inhomogeneous Poisson; dependence on Phi_T/SeaProfile enters via Λ_event.

Proposition 4 (Measurable event updates).
If Trigger(state, event) is Lipschitz, then state(t^+) = Trigger( state(t^-), event ) preserves measurability and boundedness, ensuring subsequent spectra and propagation remain integrable.

V. Chain Rule & Order Preservation for the Potential Map (cf. §5 S70-12)

Proposition 5 (Chain relation).
With Phi_T = G(T_fil) and g_T = dG/dT_fil > 0,
grad_Phi_T = g_T(T_fil) • grad(T_fil).
Implication: couplings written in Phi_T, grad_Phi_T respond monotonically to T_fil, and formulations depending only on grad_Phi_T enjoy gauge-shift invariance in Phi_T.

VI. Spectrum–Observation Link & K-Correction (cf. §6 S70-10)

Proposition 6 (Spectral dimensional closure).
Given F_nu(f_obs) = L_nu(f_em) / ( 4π D_L^2 ) • K(z_obs), f_em = f_obs • (1+z_obs), with dim(L_nu)=[W•Hz^-1], dim(D_L)=[L], dim(K)=1, then dim(F_nu)=[W•m^-2•Hz^-1].
Sketch: direct substitution; K(z) definition is fixed in the Contract.

VII. Thin/Thick Layer Equivalence (cf. LayeredSea & §8)

Theorem 2 (Zero-thickness correction as a limit).
If a layer has width Delta_k with Delta_k / L_char → 0, then its contribution
T_arr^{layer_k} = ( ∫_{layer_k} ( n_eff / c_ref ) d ell ) → Delta_T_sigma = k_sigma • H(crossing), and
tau_switch = | T_arr^{thick} − ( T_arr^{thin} + Delta_T_sigma ) | → 0.
Sketch: treat the layer as a narrow-support contribution and Taylor expand around the crossing; the error is governed by sup | d(n_eff/c_ref)/d ell | and Delta_k.

VIII. Differential Cancellation of Common Terms (cf. §6 S70-13)

Theorem 3 (n_common cancellation).
On the same path,
Constant pull-out: 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 ).
Sketch: substitute n_eff = n_common + n_path into both frequencies and subtract; the shared n_common cancels.

IX. Multi-Path “Echo” Times (cf. §6/§8)

Proposition 7 (Echo order approximation).
If a near-layer closed round-trip of length L_loop exists, the k-th echo obeys
Delta_T_echo(k) ≈ k • ( ∫_{loop} ( n_eff / c_ref ) d ell ) (general form).
Sketch: treat each round trip as an additive closed-segment delay; weights are set by {R_env,T_trans,A_sigma} and geometry.

X. First-Order Variations & Parameter Sensitivities (cf. §3/§5/§6/§7)

Theorem 4 (First variation, constant pull-out).
delta T_arr = (1/c_ref) * ( ∫ [ (∂n_eff/∂Phi_T) • delta Phi_T + (∂n_eff/∂grad_Phi_T) • grad(delta Phi_T) + (∂n_eff/∂rho) • delta rho + (∂n_eff/∂θ) • delta θ ] d ell ) + ∑ crossings delta k_sigma.
Here θ aggregates θ_state, θ_sed, θ_path. Boundary terms are handled per matching conditions.

Theorem 5 (Parameter gradient, general form).
∂T_arr/∂θ = ( ∫ ( ∂n_eff/∂θ ) / c_ref d ell ).
Use: least-squares/MAP estimation for θ and GUM propagation; cross-check against MC.

XI. Energy Consistency & Side Limits (cf. §8)

Proposition 8 (Side-limit lower bound).
For any matched interface event, both sides satisfy n_eff^± ≥ 1.

Proposition 9 (Energy conservation).
Every interface event obeys R_env + T_trans + A_sigma = 1.
Meaning: causal/energetic constraints for propagation; violations are registered as falsifications.

XII. Numerical Segmentation & Convergence (cf. §9)

Lemma 3 (Explicit endpoints & uniform convergence).
Omitting explicit { ell_i } and interpolating across interfaces produces jump errors and breaks uniform convergence of T_arr; explicitly integrating endpoints restores piecewise smoothness and enables higher-order quadrature.

Lemma 4 (Monotone refinement).
Under the three-threshold controls (geometry curvature / medium variation / layer strength), refinement yields monotone decrease of | T_arr^{(fine)} − T_arr^{(coarse)} | until eps_T; non-monotonic behavior flags endpoint tolerances or step strategy issues.

XIII. Joint Spectrum–Arrival Consistency Weights (cf. §11 M7)

Proposition 10 (Joint likelihood weighting).
Let y = [ T_arr, Delta_T_arr, F_nu, LC ] with covariance Σ_y. The joint likelihood
L(θ) ∝ exp( − 0.5 • ( y_obs − y_mod(θ) )^T Σ_y^{-1} ( y_obs − y_mod(θ) ) ).
Note: cross-covariances in Σ_y arise from shared parameters θ_state/θ_sed/θ_path and system terms (c_ref, path geometry). Mis-weighting induces bias.

XIV. Identifiability & Degeneracies (abridged guidance)

Proposition 11 (Differentials break path/physics degeneracy).
On a fixed path, multi-frequency Delta_T_arr can separate linear dependencies between θ_state and θ_path, provided the frequency set covers the target band and controls out-of-band leakage.

Proposition 12 (Multi-angle paths break geometric degeneracy).
Cross-angle path families reduce coupling between object geometry and layered structure, improving identifiability of directional terms.

XV. Worked Dimensional Checks for Three Chains

XVI. Cross-References

XVII. Deliverables