31-EFT.WP.BH.TensionWall v1.0 | Appendix C — Derivation Details & Proof Sketches (TW Edition)

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Appendix C — Derivation Details & Proof Sketches (TW Edition)

• I. Preliminaries & Notation
• Path & measure: gamma(ell) is a piecewise-C^1 path with ell ∈ [0,L]; the line element is d ell, and the path length is L_path = ∫_0^L d ell.
• Fields & quantities: Phi_T(x,t), grad_Phi_T(x,t), n_eff(x,t,f), c_ref, and local speed c_loc = c_ref / n_eff.
• Wall & profile: Sigma_TW, r_H, Delta_w, W(r), Xi_TW(r) = | dW/dr |.
• Energy consistency: R_TW + T_trans + A_sigma = 1.
• Two arrival-time gauges
• Constant-factored: T_arr = (1/c_ref) * ∫_gamma n_eff d ell.
• General: T_arr = ∫_gamma ( n_eff / c_ref ) d ell.
• Regularity assumptions: within the coherence window, F and H_TW are Lipschitz-continuous 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 S40-10)
  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.
  Proof sketch: chain rule and monotone-invertible change of variables.
• Lemma 2 (Additivity under splicing).
  Let gamma = gamma_1 ∘ gamma_2 with a continuous junction; then
  ∫_gamma g d ell = ∫_{gamma_1} g d ell + ∫_{gamma_2} g d ell.
  Corollary. Under both gauges, T_arr[gamma] = T_arr[gamma_1] + T_arr[gamma_2].
• III. Agreement of the Two Gauges & Lower Bounds (corresponds to S40-10, S40-16)
  Proposition 1 (Sufficient conditions for gauge agreement).
  When c_ref is constant—or when its variability can be absorbed inside the integrand as n_eff/c_ref, with fluctuations below metrological thresholds—the two gauges agree within tolerance.
  Proof sketch: pull out the constant factor when possible; otherwise treat n_eff/c_ref as a single integrand, with differences controlled by higher-order small terms.
• Theorem 1 (Arrival-time lower bounds).
  If n_eff ≥ 1, then
• Constant-factored: T_arr ≥ L_path / c_ref;
• General: T_arr ≥ ∫_gamma (1/c_ref) d ell.
  Proof sketch: substitute the lower bound into the integrand and use the monotonicity of the integral.
• Proposition 2 (Equality conditions).
  The lower bound is tight iff n_eff ≡ 1 along the path.
• IV. Gauge Invariance & Equivalence Classes (corresponds to S40-8)
  Proposition 3 (Invariance under gauge shifts).
  If n_eff = F( grad_Phi_T, rho, f ) + H_TW(·) contains no absolute Phi_T, then Phi_T → Phi_T + const leaves T_arr unchanged.
  Proof sketch: grad( Phi_T + const ) = grad_Phi_T; therefore the integrand is unchanged.
• Proposition 4 (Gauge fixing when absolute potential appears).
  If F or H_TW explicitly depends on absolute Phi_T, fix Phi_T(x_ref,t_ref) = 0; differences across gauges are absorbed by calibration constants.
• V. Wall Profile & Potential Mapping (corresponds to S40-1, S40-2, S40-3)
  Proposition 5 (Monotonic profile & interface strength).
  W(r) is monotone in the transition zone and near-constant on both sides; define Xi_TW = | dW/dr | as the interface-strength measure.
  Proof sketch: differentiate tanh/logistic families to show monotonicity and boundary limits; for splines impose monotonic constraints with nonnegative first differences.
• Proposition 6 (Chain relation).
  With Phi_T = G(T_fil) and g_T = dG/dT_fil > 0, we have grad_Phi_T = g_T(T_fil) * grad(T_fil).
  Significance: order preservation and comparability; facilitates writing wall terms in W(r) and grad_Phi_T.
• VI. Wall-Term Construction for the Effective Index (corresponds to S40-4 … S40-7)
  Proposition 7 (Minimal form).
  n_eff = F( Phi_T, grad_Phi_T, rho, f ) + H_TW( W, Xi_TW, f ), with n_eff ≥ 1.
  Proof sketch: collect wall-induced responses into H_TW, conventional terms into F, and clamp to enforce the lower bound.
• Proposition 8 (Isotropic expansion).
  Near a reference state Phi_0,
  n_eff ≈ a0 + a1*(Phi_T − Phi_0) + a2*norm(grad_Phi_T)^2 + u0*W + u1*Xi_TW.
  Sketch: symmetry excludes odd terms; quadratic form uses norm(grad_Phi_T)^2.
• Proposition 9 (Leading directional terms).
  With a preferred direction present, allow
  b1*dot(grad_Phi_T,t_hat) + b1_sigma*dot(grad_Phi_T,n_vec).
  Sketch: broken isotropy permits first-order scalars from inner products with unit vectors.
• Proposition 10 (Band polynomial).
  In-band expansion n_path(f) ≈ ∑_{m=1}^M c_m (f − f0)^m; absorb band-independent parts into n_common.
  Sketch: polynomial approximation on compact intervals, with order set by residual thresholds.
• VII. Interface Matching & Zero-Thickness Corrections (corresponds to S40-9 … S40-12)
  Theorem 2 (Three matching types & side limits).
• Continuous: Phi_T^+ = Phi_T^-, J_sigma = 0; if F is continuous, then n_eff^+ = n_eff^-.
• Potential jump: C_sigma ≠ 0, J_sigma = 0; side limits may differ.
• Flux jump: C_sigma = 0, J_sigma ≠ 0; include wall-normal response in n_eff.
  Sketch: derive side-limit relations from one-sided limits and continuity.
• Proposition 11 (Zero-thickness correction equivalence).
  When Delta_w / r_H ≤ eta_w,
  Delta_T_sigma ≈ k_sigma * H(crossing), where H counts crossings.
  Sketch: approximate the wall layer as a step plus a thin-layer weight; path crossings add linearly.
• VIII. Multi-Path Composition & Echo Delays (corresponds to S40-13 … S40-15)
  Proposition 12 (Multi-path weighted sum).
  T_arr_total = ∑_m w_m * T_arr[gamma_m], with ∑_m w_m = 1 or amplitude normalization.
  Sketch: linear additivity with energy-allocation weights.
• Proposition 13 (Echo-order delays).
  If a near-wall loop of length L_loop exists, the k-th echo satisfies
  Delta_T_echo(k) ≈ k * ∫_{loop} ( n_eff / c_ref ) d ell.
  Sketch: treat each round-trip as an additive delay over the closed loop.
• IX. Band Differencing Isolates the Path Term (corresponds to S40-15)
  Theorem 3 (Common-term cancellation).
  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; n_common cancels.
• X. First Variations & Parameter Sensitivities (corresponds to S40-20, S40-21)
  Theorem 4 (Variation under constant-factored gauge).
  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.
  Sketch: Gâteaux variation; swap variation and integration; treat boundary terms via matching conditions.
• Theorem 5 (Parameter gradient under general gauge).
  ∂T_arr/∂theta = ∫ ( ∂n_eff/∂theta ) / c_ref d ell.
  Sketch: differentiate the integrand and integrate.
• XI. Thin vs Thick-Wall Consistency & Error Orders (corresponds to S40-23, S40-24)
  Proposition 14 (Convergent consistency).
  As Delta_w / r_H → 0, the wall-layer contribution from thick-wall volume integration T_arr_layer converges to Delta_T_sigma; the difference tau_switch → 0, with an error order proportional to a power of Delta_w.
  Sketch: treat the wall as a narrow-support integrand; use mean-value estimates; error controlled by derivative bounds and layer width.
• XII. Symmetry Arguments for Directional Terms (corresponds to S40-6)
  Proposition 15 (Isotropy excludes first-order terms).
  Under full isotropy, all first-order terms beyond the constant vanish; the minimal nonzero invariant is norm(grad_Phi_T)^2.
  Sketch: invariance under all rotations leaves only quadratic scalars.
• Proposition 16 (First allowed terms under broken symmetry).
  Given a unit vector t_hat or n_vec, the scalars dot(grad_Phi_T,t_hat) or dot(grad_Phi_T,n_vec) are permitted first-order terms.
  Sketch: once the rotation group is reduced, new invariants arise from inner products.
• XIII. Segment Endpoints & Numerical Convergence (echoing Chapter 9)
  Lemma 3 (Explicit endpoints for stability).
  Failing to include { ell_i } explicitly and interpolating across interfaces introduces step errors that spoil consistent convergence of T_arr.
  Sketch: interpolation error at a jump does not converge uniformly with step-size; explicit segmentation restores piecewise smoothness and high-order quadrature.
• Lemma 4 (Monotone refinement).
  With dual thresholds on geometric curvature and medium variation, refinement makes discrete errors decrease monotonically, targeting | T_arr^{(fine)} − T_arr^{(coarse)} | ≤ eps_T.
  Sketch: after segmentation, the integrand is piecewise smooth, enabling standard quadrature bounds.
• XIV. Proof-Sketch Summary & Usage Notes
• The two gauges are equivalent under constant or weakly varying c_ref; in practice, audit via the consistency metric eta_T.
• Wall effects separate cleanly into geometric profile W and strength Xi_TW; use zero-thickness corrections for thin walls and volume integration for thick walls.
• Band differencing is essential to isolate the path term—reuse identical path discretization and correction settings.
• Variational formulas supply sensitivity sources for inversion and uncertainty propagation; interface crossings add via the k_sigma variation term.
• Thin/thick consistency is audited via tau_switch; if the threshold is not met, switch to the thick-wall route.
• Directional terms must be supported by data and significance tests to avoid overfitting and spurious directionality.