28-EFT.WP.Propagation.PathRedshift v1.0 | Chapter 8 — Worldlines, Rays, and Path Integrals

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Chapter 8 — Worldlines, Rays, and Path Integrals

One-sentence goal: Under weak-field, engineering assumptions, provide a unified method that goes from worldline/ray solving to path integrals (phase/group conventions): construct gamma(ell), solve the ray equation, stitch interfaces/boundaries, and manifest the implementation, errors, and evidence for integrating z and T_arr^{form1/form2}.

I. Scope & Objects

1. Inputs
   • Media / potentials: n_phi(f,x,t), n_g(f,x,t) or beta(ω,x); (optional) weak-field phi_grav(x) (used only to weigh path terms; redshift components remain in Chapter 4).
   • Boundaries & geometry: TX/RX positions and attitudes { r_emit, r_obs, att }, interface set { Σ_j } (core / air / glass, etc.), constraints and tolerances bc.
   • Initial guess / prior geometry: initial ray gamma0(ell) (straight / piecewise straight / geodesic); band/bandwidth B and operating frequency f_0.
   • References: RefCond = { timebase.hash, map/hash, met.hash, iono.hash, gravity.hash, tz, … }.
2. Outputs
   • Ray path gamma(ell) and unit tangent t_hat(ell); interface normals / refraction satisfaction;
   • Path integrals:
     T_phi = ( ∫ n_phi/c_ref d ell ), T_g = ( ∫ n_g/c_ref d ell ), the two-form T_arr^{form1/form2} and delta_form;
   • Manifest: manifest.redshift.ray.* with uncertainties u/U.
3. Boundary
   Default isotropic / weak dispersion / weak field engineering convention (Hamilton–Fermat); strong anisotropy / vectorial effects / strong-field use extensions that record the model and order.

II. Terms & Variables

• Ray & path: gamma: [0,L_gamma]→R^3, t_hat = d gamma / d ell / | d gamma / d ell |, L_gamma = ( ∫_gamma 1 d ell ).
• Phase function & eikonal: S(x), k = ∇S, |k| = k0 n_phi, k0 = 2π/λ_0 = ω_0/c_ref.
• Hamiltonian optics: H(x,k) = (1/2)(|k|^2 - k0^2 n_phi^2(x)) = 0.
• Refraction / interfaces: interface normal n_j, Snell: n_1 sin θ_1 = n_2 sin θ_2.
• Dimensions: unit(n_*)=1, unit(L_gamma)=[L], unit(T_*)=[T], unit(delta_form)=[T].

III. Postulates P65-8x

• P65-801 (Explicit, monotone paths): gamma(ell) must be monotone (no backtracking) and piecewise explicit; interface/stitch points must be recorded.
• P65-802 (Two-form pairing): For each gamma, integrate T_arr^{form1/form2} in parallel and record delta_form ≤ tol_Tarr.
• P65-803 (Explicit domains & measures): any integral/average declares ( ∫_{gamma(ell)} • d ell ), ( ∫_{t∈W} • dt ), ( ∫_{f∈B} • df ); interfaces use explicit surface elements/normals.
• P65-804 (Dimensions / timebase): pass check_dim( y − f(x) ); compute on tau_mono, publish at ts; record log↔linear conversions in scale.note.
• P65-805 (RefCond traceability): sources and interpolation policies for n_* / maps / ephemerides / meteorology are persisted in RefCond with hash / validity.

IV. Minimal Equations S65-8x

1. Eikonal / Hamilton–Fermat (isotropic, weak field)

• S65-801 (Eikonal): |∇S(x)| = k0 n_phi(x), with phase fronts orthogonal to rays.
• S65-802 (Hamilton’s equations):
• d x / d s = ∂H/∂k = k, d k / d s = -∂H/∂x = (k0^2/2) ∇(n_phi^2),

choose arc-length parametrization and normalize to obtain:

d ( n_phi t_hat ) / d ell = ∇ n_phi (Fermat form).

2. Group-convention correction (weak dispersion, narrowband)

• S65-803: for group-velocity convention, substitute n_phi → n_g = n_phi − f ( d n_phi / d f ); the ray geometry is the same for phase/group; the difference appears only in the T_* integrand.

3. Interface / stitching conditions

• S65-804 (Snell / normals): at interface Σ_j, n_1 sin θ_1 = n_2 sin θ_2, tangential k_t is continuous; reflection events must be tagged and branched (multipath belongs to Chapter 12).

4. Weak-field path term (optional)

• S65-805: if including a gravitational path-term approximation, apply an effective correction n_eff ≈ n_phi ( 1 − 2φ_grav/c_ref^2 ); the corresponding Shapiro increment enters ΔT_grav (Chapter 4); this chapter only affects ray geometry.

5. Path integrals (two forms)

• S65-806:
• T_phi = ( ∫_{gamma(ell)} n_phi / c_ref d ell ), T_g = ( ∫_{gamma(ell)} n_g / c_ref d ell );

for Chapter 2:
T_arr^{form1} = (1/c_ref)( ∫ n_eff d ell ), T_arr^{form2} = ( ∫ ( n_eff / c_ref ) d ell ).

6. Sensitivities & Jacobians (GUM)

• S65-807 (Arrival-time sensitivities):
• ∂T/∂n ≈ (1/c_ref) ∫ d ell, ∂T/∂x ≈ (1/c_ref) ∫ (∇ n • δx) d ell,

and under discrete stepping, explicit Jacobians w.r.t. segment node positions/normals can be built for uncertainty propagation (Chapter 13).

V. Metrology Pipeline M65-8 (Ready → Solve → Integrate → Verify → Persist)

1. Ready
   Unify coordinates with DEM/maps; load n_phi / n_g or beta(ω) and RefCond; set boundary conditions
   bc = { r_emit, r_obs, Σ, tol_geo } and prior gamma0.
2. Solve the ray
   • Choose convention: eikonal / Fermat / Hamilton;
   • Iterate / ray-trace: Runge–Kutta / ray stepping + interface Snell corrections;
   • Convergence & tolerances: terminal error || gamma(L_gamma) − r_obs || ≤ tol_geo, tangential continuity at interfaces ≤ tol_snell.
3. Path integrals
   • Compute T_phi, T_g, and T_arr^{form1/form2}, recording delta_form;
   • For band handling (narrow/wide) and phase/group mapping, log ΔT_map per Chapter 7 if needed.
4. Verification
   • check_dim(T_*) = "[T]"; delta_form ≤ tol_Tarr;
   • Straight-line sanity: || gamma − line ||_max ≤ tol_line (fiber straight segment / FSO line-of-sight);
   • Uncertainty: construct J_T, estimate u(T) and coverage U = k•u_c (Chapter 13).
5. Persist

manifest.redshift.ray = { gamma.hash,

segments:[{ Σ_j, n_jump, snell_resid }],

solver:{ mode, step, tol },

T_phi, T_g,

T_arr_forms:{ form1, form2, delta_form },

ΔT_map?, u/U, RefCond, contracts.*, signature }

VI. Contracts & Assertions C65-8x (suggested thresholds)

• C65-801 (Monotonicity & endpoints): non_decreasing(ell) and || gamma(L_gamma) − r_obs || ≤ tol_geo.
• C65-802 (Interface consistency): tangential continuity ≤ tol_snell; illegal reflection events must be tagged or rejected.
• C65-803 (Two-form gap): delta_form_p95 ≤ tol_Tarr; for wideband mapping, ΔT_map_p95 ≤ tol_map.
• C65-804 (Linearity / plausibility): for FSO line-of-sight, curvature_max ≤ tol_curv; for fiber straight segments, bend_radius ≥ R_min (from manifests/drawings).
• C65-805 (Dimensional compliance): unit(T_*) = [T], unit(n_*) = 1; conversions and interpolation strategies are persisted.

VII. Implementation Bindings I65-8* (interfaces, I/O, invariants)

• I65-81 estimate_geometry(anchors, dem, constraints) -> gamma0
• I65-82 solve_ray(n_field, gamma0, bc) -> gamma (Fermat/Hamilton; convergence/step/interface list)
• I65-83 integrate_path(n_eff, gamma, c_ref) -> { T_form1, T_form2, delta_form }
• I65-84 check_interfaces(gamma, Σ) -> { snell_resid[], tags }
• I65-85 jacobian_T(n_field, gamma) -> { J_T, u_T } (GUM/MC interface)
• I65-86 assert_ray_contracts(ds, rules) -> report
• I65-87 emit_ray_manifest(results, policy) -> manifest.redshift.ray
  Invariants: two_forms_present = true; check_dim(*) passes; gamma.hash / RefCond.hash traceable; convergence and interface tolerances satisfy contracts.

VIII. Cross-References

• Redshift baseline & two forms: Chapter 2; kinematic / gravitational / media terms: Chs. 3–5; dispersion mapping: Chapter 7; observations & PLL/spectral fitting: Chapter 9; fusion & calibration: Chs. 10/11; uncertainty & publication: Chapter 13 and Appendices C/E.
• Fiber/FSO propagation compensation: see EFT.WP.Packets.Light v1.0 Chapter 6 (CD/PMD/nonlinearity) and Chapter 7 (FSO).

IX. Quality & Risk Control

• SLI / SLO: rate of tol_geo satisfaction, snell_resid_p95, delta_form_p95, curvature_max, bend_radius_min, panel_freshness.
• Fallback: non-convergence → relax initial guess/step or switch solver; interface violations → rebuild segments / correct material data / tag reflection branches; over-limit two-form gap → standardize to form2 and back-write to n_* sources.
• Audit: gamma version & interface list, straightness & curvature checks, convergence logs, and the manifest.redshift.ray signature chain with replay scripts.

Summary

• This chapter unifies worldline/ray solving and path integration into an executable engineering framework, standardizing T_* computation in phase/group conventions with two-form cross-checks.
• With M65-8 / C65-8x / I65-8* and manifest.redshift.ray.*, geometry and integrals become traceable, auditable, and rollback-ready, providing a stable geometric baseline for the observation convention (Chapter 9) and fusion/calibration (Chs. 10/11) that follow.