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

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Appendix C — Derivation Details & Proof Sketches

• I. Preliminaries & Notation
• Path & measure: gamma(ell) is a piecewise-C^1 path parameterized by arclength, ell ∈ [0, L]. The line element is d ell, and the path length is L_path = ∫_0^L d ell.
• Coordinates & metric: conformal time eta, comoving radius chi, scale factor a(eta). The metric_spec is declared in the Contract so that dim(d ell) = [L] is guaranteed.
• Layers & profiles: layer index k = 1…K, profiles W_k(chi), strengths Xi_k(chi) = | dW_k/dchi |, interface set Sigma_sea.
• Fields & propagation: Phi_T(x,t), grad_Phi_T(x,t), n_eff(x,t,f), c_ref, with local speed c_loc = c_ref / n_eff.
• Energy consistency: R_sea + T_trans + A_sigma = 1.
• Two formulations:
• Constant factored form: T_arr = (1/c_ref) * ∫_gamma n_eff d ell.
• General form: T_arr = ∫_gamma ( n_eff / c_ref ) d ell.
• Regularity assumptions: within a coherence window, F 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 (corresponds to S60-5)
  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. Apply the chain rule with the change of variables; monotonic invertibility ensures a valid measure transform.
• Lemma 2 (Additivity under concatenation).
  If gamma = gamma_1 ∘ gamma_2 with a continuous junction at the endpoint, then
  ∫_gamma g d ell = ∫_{gamma_1} g d ell + ∫_{gamma_2} g d ell.
  Corollary. Under both formulations, T_arr[gamma] = T_arr[gamma_1] + T_arr[gamma_2].
• III. Two-Formulation Agreement & Lower Bound (corresponds to S60-5)
  Proposition 1 (Sufficient conditions for agreement).
  When c_ref is constant—or its variability can be absorbed into the unified integrand n_eff/c_ref with residuals below metrological thresholds—the difference | T_arr^{const} − T_arr^{gen} | = eta_T stays within the specified bound.
  Sketch. For the constant case, (1/c_ref) factors out. When c_ref varies piecewise, treat (n_eff/c_ref) as the single integrand; differences reduce to higher-order remainders under the adopted thresholds.
• Theorem 1 (Arrival-time lower bound).
  If n_eff ≥ 1, then
• Constant factored: T_arr ≥ L_path / c_ref.
• General: T_arr ≥ ∫_gamma (1/c_ref) d ell.
  Sketch. Take the pointwise lower bound on the integrand and use the order-preserving property of integration.
• Proposition 2 (Tightness).
  The bound is attained iff n_eff ≡ 1 along the path.
• IV. Layer Profiles & Strengths (corresponds to S60-1)
  Proposition 3 (Monotone profile implies unique crossing).
  If W_k(chi) is monotone on (chi_k − 0.5·Delta_k, chi_k + 0.5·Delta_k), then the discriminator F_k(chi) = W_k(chi) − 0.5 has at most one zero in that interval.
  Sketch. A monotone function has at most one root; separated layers imply a finite intersection sequence { ell_i }.
• Proposition 4 (Strength peaks guide refinement).
  Xi_k(chi) = | dW_k/dchi | peaks near interfaces and is effective as an adaptive step-size trigger, enforcing symmetric sampling and controlling local error around endpoints.
  Use. Provides an actionable threshold tau_sea for the step strategy in Chapter 9.
• V. Potential Mapping & Gradient Chain Rule (corresponds to S60-2)
  Proposition 5 (Order preservation & chain rule).
  With Phi_T = G(T_fil) and g_T = dG/dT_fil > 0,
  grad_Phi_T = g_T(T_fil) · grad(T_fil).
  Significance. Enables gauge-invariant constructions that depend solely on grad_Phi_T, and supplies analytic sensitivity channels.
• VI. Layered Coupling & Expansions for n_eff (corresponds to S60-3, S60-4)
  Proposition 6 (Minimal form).
  n_eff = F( Phi_T, grad_Phi_T, rho, f ) + H_sea( { W_k, Xi_k }, f ), with n_eff ≥ 1.
• Proposition 7 (Isotropic small-gradient expansion).
  Near a reference state Phi_0:
  n_eff ≈ a0 + a1·(Phi_T − Phi_0) + a2·norm(grad_Phi_T)^2 + ∑_k ( u0_k·W_k + u1_k·Xi_k ).
• Proposition 8 (Leading directional terms, if enabled).
  With a preferred direction: + b1·dot(grad_Phi_T, t_hat) + b1_n·dot(grad_Phi_T, n_vec).
  Dimensionality. See Appendix A; u1_k·Xi_k, b1·dot(…), etc., must be dimensionless.
• VII. In-Band Differencing Cancels the Common Term (corresponds to S60-4)
  Theorem 2 (Differencing eliminates n_common on the same path).
• Constant factored: Delta_T_arr(f1,f2) = (1/c_ref) * ∫ ( n_path(f1) − n_path(f2) ) d ell.
• General: Delta_T_arr(f1,f2) = ∫ ( ( n_path(f1) − n_path(f2) ) / c_ref ) d ell.
  Sketch. Substitute n_eff = n_common + n_path and subtract; the n_common contribution cancels exactly.
• VIII. Segment Endpoints & Zero-Thickness Correction (corresponds to S60-5)
  Proposition 9 (Explicit endpoints restore piecewise smoothness).
  Interpolating across interfaces without explicitly inserting { ell_i } degrades convergence because of integrand jumps. Explicit endpoints restore piecewise smooth integrands and enable high-order quadrature.
• Proposition 10 (Zero-thickness equivalence).
  When Delta_k/L_char ≤ eta_w, the layer-band contribution to T_arr admits
  Delta_T_sigma ≈ k_sigma · H(crossing).
  Quantifying consistency. Define tau_switch = | T_arr^{thick} − (T_arr^{thin} + Delta_T_sigma) |; if within threshold, treat as equivalent (numerical validation in Chapter 9).
• IX. First Variations & Parameter Sensitivities (backbone for metrology & inversion)
  Theorem 3 (First variation under constant factored form).
  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 ] d ell + ∑ crossings delta k_sigma.
• Theorem 4 (Parameter gradients under the general form).
  ∂T_arr/∂theta = ∫ ( ∂n_eff/∂theta ) / c_ref d ell, with theta ∈ { a1,a2,u0_k,u1_k,b1,b1_n,c_m,k_sigma,… }.
  Use. Supports least-squares/MAP inversion for SeaProfile and n_path, and GUM propagation (Chs. 7 & 12).
• X. Energy Consistency & One-Sided Feasible Set (corresponds to S60-10 … S60-12)
  Proposition 11 (One-sided lower bounds).
  After interface matching, both sides must satisfy n_eff^± ≥ 1.
• Proposition 12 (Energy balance).
  Every interface event obeys R_sea + T_trans + A_sigma = 1. This constraint is independent of the two arrival-time formulations but informs path weighting and feasibility.
  Role. Provides testable conditions for coupling (Ch. 8) and for audits (Ch. 11).
• XI. Error Order for Thin/Thick Consistency (corresponds to S60-5, Chapter 9)
  Proposition 13 (Convergence to event correction).
  As Delta_k/L_char → 0, the thick-layer volumetric contribution T_arr^{layer_k} converges to Delta_T_sigma, hence tau_switch → 0. The error order is controlled by sup | d(n_eff/c_ref)/d ell | and Delta_k.
  Sketch. View the band as a narrow-support region; expand the integrand about the center (first- or second-order), then apply a mean-value estimate.
• XII. Dimensional Self-Consistency of the Metric→Arclength Mapping
  Proposition 14 (Consistency of chi → d ell).
  If the Contract declares d ell = a(eta) · norm( d x_comov ) (with pure radial d ell = a(eta) · d chi), then dim(d ell) = [L] holds identically.
  Significance. Ensures dimensional closure of both formulations and of differencing, preventing coordinate-mixing bias (cf. Chapter 4 and Appendix A).
• XIII. Multi-Path “Echo” Approximation (corresponds to S60-14)
  Proposition 15 (Approximate delay for round-trip echoes).
  If a near-layer round-trip of length L_loop exists, the k-th echo delay satisfies
  Delta_T_echo(k) ≈ k · ∫_{loop} ( n_eff / c_ref ) d ell (general form).
  Sketch. Treat each round trip as the time increment of a closed segment, then sum linearly; weights arise from the energy triplet and path geometry.
• XIV. Numerical Criteria for Convergence & Consistency (echoing Chapter 9)
• Refinement convergence: | T_arr^{(fine)} − T_arr^{(coarse)} | ≤ eps_T.
• Two-formulation agreement: eta_T = | T_arr^{const} − T_arr^{gen} | within threshold.
• Thin/thick agreement: tau_switch within limit.
• Differencing linear region: correlation and slope for Delta_T_arr meet targets; out-of-band residuals folded into u_sys.
• Explicit endpoints: { ell_i } must be included; cross-interface interpolation is forbidden.
• XV. Cross-References
• EFT.WP.Cosmo.LayeredSea v1.0: Ch. 3 (Minimal Equations & Layered Representation), Ch. 4 (Geometry & Coordinate Choice), Ch. 6 (Propagation & Arrival Time), Ch. 8 (Interface Matching), Ch. 9 (Modeling Methods & Numerical Realization), Ch. 11 (Validation), Ch. 12 (Error Budget).
• EFT.WP.Propagation.TensionPotential v1.0: dual formulations, differencing, and data conventions.
• EFT.WP.Core.Equations v1.1 / Metrology v1.0 / Errors v1.0: baseline for notation, metrology, and uncertainty propagation.