30-EFT.WP.Propagation.TensionPotential v1.0 | Chapter 5 — Effective Refractive Index & Propagation Limit

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Chapter 5 — Effective Refractive Index & Propagation Limit

• I. One-Sentence Aim
  Within the established Phi_T(x,t) framework, construct an executable mapping and decomposition for the effective refractive index n_eff(x,t,f), define and measure the propagation limit c_loc(x,t,f) = c_ref / n_eff(x,t,f), and provide unified conventions for calibration, approximations, and limiting cases.
• II. Scope & Non-Goals
• Covered: functional dependence and bandwise decomposition of n_eff; definition and lower bounds of the propagation limit; applicability of approximations and series expansions; limits and boundary cases; metrology and calibration flows; interface specifications and verifiability criteria.
• Non-goals: no repetition of Chapter 4’s construction of Phi_T; no path integrations (delegated to Chapter 6); no device structures or electrical/mechanical parameters.
• III. Minimal Terms & Symbols
• Potential & gradient: Phi_T(x,t), grad_Phi_T(x,t).
• Effective index & speeds: n_eff(x,t,f), c_ref, c_loc(x,t,f) = c_ref / n_eff(x,t,f).
• Decomposition & bands: n_common(x,t) as the common (frequency-independent) term; n_path(x,t,f) as the path term.
• Auxiliaries: rho(x,t) (as needed); unit tangent t_hat(ell) along the path direction.
• IV. Dependencies & Invocations (from Chapter 2 P20-*, Chapter 3 S20-*)
• Invoke P20-1: n_eff ≥ 1; local propagation limit given by c_loc = c_ref / n_eff.
• Invoke P20-2, P20-3: Phi_T exists and the gauge is fixed; if observables depend only on grad_Phi_T, gauge shifts do not change them.
• Invoke P20-5: the bandwise decomposition n_eff = n_common + n_path must hold within preset residual thresholds.
• Invoke S20-4: the two arrival-time gauges are applied in Chapter 6; this chapter supplies n_eff.
• V. Definitions & Constructions (S20-25 to S20-31)
• S20-25 Base mapping
  n_eff(x,t,f) = F( Phi_T(x,t), grad_Phi_T(x,t), rho(x,t), f )
  Constraints: n_eff(x,t,f) ≥ 1, dim(n_eff) = 1; F is Lipschitz-continuous within the coherence window to ensure numerical stability.
• S20-26 Band decomposition
  n_eff(x,t,f) = n_common(x,t) + n_path(x,t,f)
  Decomposition residuals are included in the uncertainty budget.
• S20-27 Propagation-limit definition
  c_loc(x,t,f) = c_ref / n_eff(x,t,f)
  From n_eff ≥ 1, obtain the lower bound c_loc ≤ c_ref.
• S20-28 Small-gradient expansion (isotropic approximation)
  Near a reference state Phi_T = Phi_0, first–second order:
  n_eff ≈ a0 + a1 · ( Phi_T - Phi_0 ) + a2 · norm( grad_Phi_T )^2,
  with a0 ≥ 1; coefficients obtained by calibration.
• S20-29 Directional term (anisotropic extension)
  For directional response, add a first-order term along t_hat:
  n_eff ≈ a0 + a1 · ( Phi_T - Phi_0 ) + b1 · dot( grad_Phi_T , t_hat ) + a2 · norm( grad_Phi_T )^2.
• S20-30 Frequency-dependent structure
  On a band f ∈ [f0 − Δf, f0 + Δf], use a linear/low-order polynomial:
  n_path(x,t,f) ≈ ∑_{m=1}^M c_m(x,t) · ( f − f0 )^m, with M and residual thresholds determined during calibration.
• S20-31 Gauge-invariance criterion
  If F excludes the absolute Phi_T, then under Phi_T → Phi_T + const, n_eff is invariant.
• VI. Lower Bounds for the Propagation Limit & Equality Conditions (S20-32 to S20-34)
• S20-32 Path-level lower bound (prepares for Chapter 6)
  Under the constant-factored gauge: T_arr ≥ L_path / c_ref,
  with equality iff n_eff ≡ 1 along that path.
• S20-33 Monotonicity & feasibility
  Require ∂n_eff/∂Phi_T ≥ 0 and ∂n_eff/∂norm(grad_Phi_T) ≥ 0 to avoid infeasible n_eff < 1.
• S20-34 Regularization & clamping
  In implementation, clamp n_eff ∈ [1 , n_max]; choose n_max by physical priors and calibration.
• VII. Approximations, Limits & Boundary Cases
• Approximation hierarchy: Prefer the isotropic small-gradient expansion S20-28; enable the anisotropic extension S20-29 when pronounced directional effects or strong path shear are present.
• Low-frequency limit: n_path(x,t,f) → 0, hence n_eff → n_common(x,t); useful for estimating the background common term.
• High-frequency limit: where dispersion strengthens, limit the order M and increase out-of-band suppression; do not use extrapolations beyond the calibration band for formal evaluations.
• Boundaries & layers: on interfaces Sigma, apply Chapter 8 matching conditions to n_eff; if Phi_T jumps, capture it via S20-29 directional terms or explicit interface terms.
• VIII. Metrology & Calibration Flows (M20-8 to M20-12)
• M20-8 Reference-speed calibration
  Calibrate c_ref using a benchmark gamma_ref and reference T_arr_ref; record environment and uncertainties.
• M20-9 Background estimation
  Estimate n_common(x,t) in low-frequency bands; route residuals to u_sys.
• M20-10 Band-structure fitting
  Fit coefficients c_m(x,t) of n_path(x,t,f) on multi-band data; determine M and residual thresholds.
• M20-11 Anisotropy detection
  Use path ensembles with diverse orientations to test whether b1 is significant; enable the directional model as needed.
• M20-12 Consistency cross-check
  Cross-validate n_eff across paths and bands to ensure the two arrival-time gauges in Chapter 6 agree within tolerance.
• IX. Implementation Bindings & Interfaces (I20-8 to I20-12)
• I20-8 estimate_n_eff( Phi_T, grad_Phi_T, rho, f, params ) -> n_eff
  Implements the mapping and decomposition of S20-25 to S20-31.
• I20-9 decompose_n_eff( n_eff, f_grid ) -> n_common, n_path_params
  Outputs n_common(x,t) and parameter set c_m(x,t) for n_path.
• I20-10 local_speed( n_eff, c_ref ) -> c_loc
  Computes the propagation limit and enforces S20-34 clamping.
• I20-11 calibrate_c_ref( gamma_ref, T_arr_ref ) -> c_ref
  Calibrates c_ref; records environmental context and uncertainties.
• I20-12 check_monotonicity( params ) -> Report
  Audits the signs and magnitudes of ∂n_eff/∂Phi_T and ∂n_eff/∂norm(grad_Phi_T).
• X. Verification & Falsification Lines
• Verification:
• Across multiple bands, verify identifiability of n_path using differential formulas together with Chapter 6 arrival-time computations.
• Use contrasting path ensembles to test for b1 and directional effects.
• Check T_arr ≥ L_path / c_ref everywhere; closely review paths near the lower bound.
• Falsification:
• Existence of a path–band combination for which no n_common + n_path decomposition fits observations within the error budget.
• Observed n_eff < 1 persisting after excluding metrological and systematic errors.
• Gauge shifts change observables that should depend only on grad_Phi_T.
• XI. Systematic-Error Safeguards
• Band leakage: In M20-10, apply out-of-band suppression and leakage assessment; route residuals to u_sys.
• Directional confounding: Ensure path layouts span diverse orientations to avoid misattributing geometric differences to directional physics.
• Clamping & saturation: Log the n_eff clamping interval and trigger ratio to prevent masking systematic bias.
• Units & dimensions: At entries to I20-8 … I20-12, run dimensional checks to keep dim(c_loc) = [L][T^-1], dim(n_eff) = 1.
• XII. Cross-References
• EFT.WP.Core.Tension v1.0 S12-*
• EFT.WP.Core.Sea v1.0 S08-*
• EFT.WP.Core.Equations v1.1 S06-*
• EFT.WP.Core.Metrology v1.0 M05-, M10-
• EFT.WP.Core.Errors v1.0 M20-*
• EFT.WP.Metrology.PathCorrection v1.0 S03-*
• XIII. Deliverables
• Minimal equation cards: S20-25 to S20-34.
• Calibration & decomposition workflows: operating checklist and record templates for M20-8 … M20-12.
• Interface contracts & audit scripts: I/O, units, clamping, and logging standards for I20-8 … I20-12.