28-EFT.WP.Propagation.PathRedshift v1.0 | Chapter 5 — Media & Moving-Media Redshift (Fizeau / Plasma / Turbulence)

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Chapter 5 — Media & Moving-Media Redshift (Fizeau / Plasma / Turbulence)

One-sentence goal: Under a weak-field engineering convention, unify the impacts of moving media (Fizeau), plasma dispersion, and tropospheric/ionospheric time variation on frequency and arrival time, provide a computable decomposition
z_med = z_drag ⊕ z_plasma ⊕ z_trop/iono, specify the companion relation to the two forms of T_arr, and define contracts and manifest publication.

I. Scope & Objects

1. Inputs
   • Media fields: effective refractive index and dispersion n_eff(f,x,t), n_phi(f), n_g(f); medium velocity field v_med(x,t); plasma parameters (electron density N_e(x,t) or integrated TEC).
   • Path & states: gamma(ell) (fiber / free space / deep space segments), TX/RX states state_emit / state_obs (for LOS and relative geometry).
   • Observations: f_obs(t) (PLL / CFO / spectral-line fitting; see Chapter 9), metrology panels, and RefCond (meteorology / ionosphere / geomagnetics / timebase).
2. Outputs
   • Per-term redshift: z_drag (Fizeau drag), z_plasma (plasma dispersion), z_trop/iono (time-varying troposphere/ionosphere); composed z_med.
   • Companion arrival-time correction: ΔT_med with consistency against T_arr^{form1/form2};
   • Manifest: manifest.redshift.med.* with uncertainties u / U.
3. Boundary
   Only medium-induced and medium-motion/time-variation–induced apparent frequency shifts are covered; gravitational terms belong to Chapter 4, pure kinematics to Chapter 3; strong nonlinear refraction/scattering are outside the default scope.

II. Terms & Variables

• Refraction & dispersion: n_phi(f), n_g(f) = n_phi − f * ( d n_phi / d f ), n_eff(f,x,t).
• Medium motion: v_med, with LOS component v_med,los = v_med • n_hat.
• Plasma: plasma frequency
  f_p = √( N_e e^2 / (π m_e) ) (Hz; commonly f_p[kHz] ≈ 8.98 √( N_e[cm^-3] )).
• Atmospheric phase: φ_med(f,t) from troposphere/ionosphere.
• Redshift & arrival time: z, f_emit, f_obs, T_arr^{form1/form2}, ΔT_med, delta_form.
• Dimensions: unit(z) = "1", unit(n_eff) = 1, unit(v_med) = [L]/[T], unit(f_p) = "[1]/[T]", unit(ΔT_med) = [T].

III. Postulates P65-5x

• P65-501 (Component composition):
  1 + z_med = (1+z_drag)(1+z_plasma)(1+z_trop/iono); small-signal z_med ≈ z_drag + z_plasma + z_trop/iono.
• P65-502 (Phase/group consistency): Maintain consistency between phase and group conventions using
  n_g = n_phi − f * ( d n_phi / d f ); publish z alongside T_arr^{form1/form2} and ΔT_med.
• P65-503 (Explicit measures): Any path/time/frequency integral declares ( ∫_{gamma(ell)} • d ell ), ( ∫_{t∈W} • dt ), ( ∫_{f∈B} • df ) and the applicable RefCond.
• P65-504 (Dimensions & sources): All fields pass check_dim( y − f(x) ); record media / meteorology / TEC sources and interpolation windows with hash / Δt.
• P65-505 (No cross-contamination): Do not mix gravitational/kinematic terms; if cross-terms (e.g., moving plasma) are modeled, state the attribution and an upper bound.

IV. Minimal Equations S65-5x

1. Fizeau drag (moving media, small speed)

• S65-501 (Phase / group conventions): For medium velocity v_med along LOS, low-speed:
• z_drag,phi ≈ ( n_phi − 1 ) * ( v_med,los / c_ref ),
• z_drag,grp ≈ ( n_g − 1 ) * ( v_med,los / c_ref ).

For fiber (stationary medium) typically v_med = 0; for FSO with wind/refraction, use statistical/mean v_med.

2. Plasma dispersion (low density)

• S65-502: With
  n_phi(f) ≈ √( 1 − f_p^2 / f^2 ), the apparent frequency is
• f_obs ≈ f_emit √( 1 − f_p^2/f^2 ) ⇒
• z_plasma ≈ ( f_emit / f_obs ) − 1 ≈ (1/2) ( f_p^2 / f^2 ) (f ≫ f_p).

For an inhomogeneous path with f_p = f_p(x):

z_plasma,eff ≈ (1/2) ( ∫_{gamma} f_p^2(x) d ell ) / ( f^2 L_gamma ) (engineering average).

3. Troposphere/ionosphere time variation (frequency drift)

• S65-503: If atmospheric phase varies with time, the instantaneous drift is
• z_trop/iono(t) ≈ − (1/f_emit) ( d φ_med / d t ) (phase convention).

Map to the group convention via n_g / n_phi. Obtain φ_med via GNSS/ionospheric models or fused water-vapor/pressure sensors.

4. Companion arrival-time correction

• S65-504
• ΔT_med = ( 1 / c_ref ) ( ∫_{gamma} Δn_med d ell )

where Δn_med includes temperature/pressure/humidity/ionization contributions. Ensure ΔT_med is consistent with z_trop/iono under the phase/group mapping.

5. Composition & phase/group mapping

• S65-505:
  1 + z_med = (1+z_drag)(1+z_plasma)(1+z_trop/iono); linear sum for small terms.
• S65-506: If the observation/publication uses the group convention, rewrite each component with n_g and group-delay phase φ_g; the phase↔group mapping follows n_g = n_phi − f d n_phi/df.

V. Metrology Pipeline M65-5 (Ready → Modeling → Estimation → Verification → Persistence)

1. Ready: load media sources (meteorology / ionosphere / TEC / EM environment), RefCond, and interpolation strategy Δt / Δx; unify the frequency grid and units.
2. Model / estimate:
   • Compute v_med,los and n_phi / n_g → z_drag;
   • From TEC or N_e fields compute f_p → z_plasma;
   • Fuse GNSS/microwave/water-vapor/pressure sensors to estimate φ_med(f,t) → z_trop/iono;
   • Integrate T_arr^{form1/form2} and ΔT_med in parallel.
3. Verify:
   • check_dim(z) = 1, check_dim(ΔT_med) = [T]; enforce delta_form ≤ tol_Tarr;
   • Baselines: in still air and stationary media, z_drag ≈ 0; on highly stable links, z_plasma matches the model within tolerance;
   • Record uncertainties u/U (e.g., u(TEC), u(v_med), u(φ_med)) and propagate (see Chapter 13).
4. Persist:
   manifest.redshift.med = { sources:{ met.hash, iono.hash, TEC.hash }, z_parts:{ z_drag, z_plasma, z_trop_iono }, z_med, ΔT_med, T_arr_forms, delta_form, u/U, RefCond, contracts.*, signature }.

VI. Contracts & Assertions C65-5x (suggested thresholds)

• C65-501 (Two-form gap): delta_form_p95 ≤ tol_Tarr.
• C65-502 (Source freshness/coverage): age(TEC/met) ≤ Δt_max and spatial coverage ≥ cov_min; record interpolation windows Δt / Δx and clamp them.
• C65-503 (Consistency gate): after phase/group mapping, | z_meas − ( z_kin + z_grav + z_med ) |_p95 ≤ tol_z.
• C65-504 (Attribution / no cross-contamination): if moving plasma produces a cross-term, publish an upper bound and state whether it is attributed to z_drag or z_plasma.
• C65-505 (Dimensional compliance): unit(z) = 1, unit(f_p) = [1]/[T], unit(ΔT_med) = [T].

VII. Implementation Bindings I65-5* (interfaces, I/O, invariants)

• I65-51 fizeau_drag(n_phi|n_g, v_med_los, f) -> { z_drag, meta }
• I65-52 plasma_redshift(TEC|N_e_field, f, path) -> { z_plasma, meta }
• I65-53 trop_iono_drift(phi_med_series, f, window) -> { z_trop_iono, meta }
• I65-54 map_med_to_tarr(n_field_delta, path) -> { ΔT_med, T_arr_forms }
• I65-55 compose_z_med(z_drag, z_plasma, z_trop_iono, mode) -> z_med
• I65-56 assert_med_contracts(ds, rules) -> report
• I65-57 emit_med_manifest(results, policy) -> manifest.redshift.med
  Invariants: two_forms_present = true; check_dim(*) passes; data sources & interpolation policy are traceable; phase/group mapping is fully documented.

VIII. Cross-References

• Redshift baseline & two forms: Chapter 2; kinematic term: Chapter 3; gravitational term: Chapter 4.
• Worldlines / path integrals: Chapter 8; observations / PLL / line fitting: Chapter 9.
• Sensing / ephemerides / environmental fusion: Chapter 10; calibration: Chapter 11; uncertainty & publication: Chapter 13 and Appendices C / E.

IX. Quality & Risk Control

• SLI / SLO: | z_med |_p95, z_resid_p95 = | z_meas − ( z_kin + z_grav + z_med ) |_p95, delta_form_p95, panel_freshness, source_age_p95.
• Fallback strategies: stale/sparse media sources → enlarge guardbands and reduce publication cadence; z_resid breach → shorten windows, strengthen sensor fusion, or revert to a static-media approximation; two-form excess → standardize to form2 and re-estimate media parameters.
• Audit: TEC/met sources and interpolation windows, f_p / φ_med computation evidence, static / windy / ionospheric-storm datasets, and the manifest.redshift.med signature chain with replay scripts.

Summary

• This chapter unifies the frequency and arrival-time impacts of moving media, plasma, and time-varying atmosphere into a decomposition/combination framework for z_med and ΔT_med, paired with the two-form T_arr.
• Via M65-5, C65-5x, and I65-5*, media-induced redshift can be robustly estimated, verified, and manifested, supporting subsequent worldline / observation / fusion chapters and end-to-end use cases.