28-EFT.WP.Propagation.PathRedshift v1.0 | Chapter 7 — Dispersion and Phase/Group Conventions (n_phi / n_g)

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Chapter 7 — Dispersion and Phase/Group Conventions (n_phi / n_g)

One-sentence goal: Distinguish the meaning and proper use of phase vs group conventions in dispersive media; provide the engineering relationships between n_phi(f) and n_g(f), define consistent mappings between redshift z and arrival time T_arr across the two conventions, and enforce auditable publication via contracts and manifests.

I. Scope & Objects

1. Inputs
   • Medium dispersion: n_phi(f,x,t) (phase refractive index), n_g(f,x,t) (group refractive index), or equivalently beta(omega) (propagation constant).
   • Path / worldline: gamma(ell) (fiber / free space / deep-space segments), observation window W = [ t_0, t_1 ].
   • Observation conventions: phase/carrier tracking (PLL/CFO) yielding φ(t), f_obs(t); group-arrival/envelope tracking yielding t̂_cont (see Chs. 3/9).
   • References: RefCond (temperature / pressure / humidity / TEC / ephemerides / timebase).
2. Outputs
   • Cross-convention mapping: conversions from phase-convention frequency shift z_φ to group-convention frequency shift / arrival time z_g, T_g, and the inverse mapping.
   • Segment-wise arrival times:
     T_phi = ( ∫ n_phi/c_ref d ell ), T_g = ( ∫ n_g/c_ref d ell ), and their relation to T_arr^{form1/form2}.
   • Manifest: manifest.redshift.disp.* with uncertainty u/U.
3. Boundary
   Default weak dispersion and narrowband engineering conditions; strong nonlinearity / strong dispersion are handled in Chapter 6 (compensation) and appendices as extensions.

II. Terms & Variables

• Frequency & wavenumber: f, omega = 2π f, k = beta(omega) = n_phi(omega) * omega / c_ref.
• Phase/group indices:
  n_phi(f), n_g(f) = n_phi - f * ( d n_phi / d f ) = n_phi + omega * ( d n_phi / d omega ).
• Phase/group velocities & delays:
  v_p = c_ref / n_phi, v_g = c_ref / n_g;
  T_phi = ( ∫_{gamma} n_phi/c_ref d ell ), T_g = ( ∫_{gamma} n_g/c_ref d ell ).
• Redshift & conventions: phase-convention z_φ = f_emit/f_obs − 1; group-convention z_g derived from arrival-time changes or group-frequency tracking.
• Dimensions: unit(n_phi) = unit(n_g) = 1, unit(T_*) = [T], unit(z) = 1.

III. Postulates P65-7x

• P65-701 (Convention separation): the phase convention governs carrier/spectral-line frequency shift and phase accumulation; the group convention governs envelope/energy arrival time. The two must not be mixed. Publication must include a consistent mapping note.
• P65-702 (Two-form pairing): publication of T_arr continues to follow Chapter 2’s two-form scheme T_arr^{form1/form2}, explicitly stating which n_{•} (n_phi or n_g) is used; enforce delta_form ≤ tol_Tarr.
• P65-703 (Explicit measures): any ( ∫_{gamma(ell)} • d ell ), ( ∫_{t∈W} • dt ), ( ∫_{f∈B} • df ) must annotate its domain and measure.
• P65-704 (Dimensions & provenance): pass check_dim( y − f(x) ); record the sources and fit ranges of n_phi / n_g and beta(omega) (Sellmeier / plasma / empirical tables) with hashes.
• P65-705 (Narrowband consistency): under narrowband, weak-dispersion conditions, mapping error between conventions shall be covered by the guardband and its uncertainty u(delta_map) must be recorded.

IV. Minimal Equations S65-7x

1. Phase/group index relation and arrival times

• S65-701:
  n_g(f) = n_phi(f) - f ( d n_phi / d f ) = n_phi + omega ( d n_phi / d omega ).
• S65-702: Phase/group arrival times
• T_phi = ( ∫ n_phi/c_ref d ell ), T_g = ( ∫ n_g/c_ref d ell )

When publishing T_arr^{form1/form2} per Chapter 2, explicitly declare whether n_eff uses n_phi or n_g, and report delta_form.

2. Mapping frequency shift between conventions (narrowband, weak dispersion)

• S65-703 (phase → group): with narrowband center f_0, instantaneous phase-convention shift
  z_φ = − (1/f_emit) ( d φ / d t ), the group-convention shift is approximately
• z_g ≈ z_φ − (1/n_g) ( d n_g / d ln f ) • z_φ

(first-order Taylor; used to size mapping differences and guardbands).

• S65-704 (group → phase):
• z_φ ≈ z_g + (1/n_phi) ( d n_phi / d ln f ) • z_g

Note: when | d ln n / d ln f | ≪ 1, z_g ≈ z_φ; the residual goes into uncertainty.

3. Plasma special case (cold, collisionless)

• S65-705:
  n_phi = √( 1 − f_p^2/f^2 ), v_g = c_ref √( 1 − f_p^2/f^2 ) ⇒ n_g = c_ref/v_g = 1 / n_phi.
  Hence
  T_g = ( ∫ ( 1 / n_phi ) / c_ref d ell ); the phase↔group mapping can be taken as n_g = 1 / n_phi (state applicability and sources in the manifest).

4. Fiber special case (Sellmeier and beta(omega))

• S65-706: given beta(omega):
• n_phi = c_ref β / ω, n_g = c_ref ( d β / d ω ),
• β_2 = d^2 β / d ω^2 (dispersion parameter), group delay τ_g = d β / d ω

consistent with Chapter 6 compensation parameters.

5. Arrival-time consistency mapping

• S65-707: when f_obs(t) is measured in the phase convention but T_g is to be published, use (S65-701) to map T_phi ↔ T_g and report
• ΔT_map = | T_g − T_phi | and u(ΔT_map)

Include ΔT_map in delta_form or the guardband.

V. Metrology Pipeline M65-7 (Ready → Modeling → Mapping → Verification → Persistence)

1. Ready: load/fit n_phi(f) or beta(omega), derive n_g(f) via (S65-701/706); unify frequency grid and units; fix RefCond.
2. Model / estimate:
   • Segment-wise integration to obtain T_phi, T_g, and T_arr^{form1/form2};
   • From PLL/CFO/line-fitting (Ch. 9) obtain z_φ(t) and derive z_g(t) (or inverse) via (S65-703/704).
3. Verify:
   • check_dim(T_*) = "[T]", check_dim(z) = 1; enforce delta_form ≤ tol_Tarr;
   • Evaluate ΔT_map and u(ΔT_map)—under weak dispersion they should be well below the published guardband;
   • Record source/model hashes and the applicable frequency band.
4. Persist:

manifest.redshift.disp = { n_phi.hash | beta.hash, band, T_phi, T_g, ΔT_map, map.method,

z_{φ,g} (windowed), T_arr_forms, delta_form, u/U, RefCond, contracts.*, signature }

VI. Contracts & Assertions C65-7x (suggested thresholds)

• C65-701 (Two-form gap): delta_form_p95 ≤ tol_Tarr (Chapter 2).
• C65-702 (Mapping consistency): ΔT_map_p95 ≤ tol_map (suggest tol_map = 0.1 • tol_Tarr); u(ΔT_map) is persisted.
• C65-703 (Band & provenance): band ⊂ band_spec, and the sources of n_phi / n_g / beta and their applicable bands are persisted; out-of-band → reject.
• C65-704 (Dimensional compliance): unit(n_*) = 1, unit(T_*) = [T], unit(z) = 1.
• C65-705 (Weak-dispersion premise): if | d ln n / d ln f |_max > τ_disp, adopt dual publication (phase & group) or enlarge and mark the guardband.

VII. Implementation Bindings I65-7* (interfaces, I/O, invariants)

• I65-71 fit_nphi_or_beta(samples) -> { n_phi(f) | beta(ω), band, meta }
• I65-72 derive_ng(n_phi|beta) -> { n_g(f), meta }
• I65-73 integrate_Tphi_Tg(n_phi, n_g, gamma) -> { T_phi, T_g, T_arr_forms }
• I65-74 map_z_phi_to_z_g(z_phi, n_phi, n_g, band) -> { z_g, ΔT_map, u(ΔT_map) }
• I65-75 assert_dispersion_contracts(ds, rules) -> report
• I65-76 emit_disp_manifest(results, policy) -> manifest.redshift.disp
  Invariants: two_forms_present = true; check_dim(*) passes; band/RefCond/hash are traceable; mapping method and applicability are documented.

VIII. Cross-References

• Redshift baseline & two forms: Chapter 2; kinematics: Chapter 3; gravity: Chapter 4; media: Chapter 5.
• Worldlines & path integrals: Chapter 8; observation conventions: Chapter 9; fusion & calibration: Chs. 10 / 11; uncertainty & publication: Ch. 13 and Appendices C / E.
• Propagation compensation & dispersion parameters: see EFT.WP.Packets.Light v1.0 Chapter 6 (CD/PMD/nonlinearity).

IX. Quality & Risk Control

• SLI / SLO: delta_form_p95, ΔT_map_p95, | z_resid |_p95 = | z_meas − ( z_kin + z_grav + z_med + z_cos ) |_p95, panel_freshness, band_coverage.
• Fallback strategies: mapping error ↑ → switch to dual-convention publication and/or widen guardband; out-of-band → shrink bandwidth or select a new model; two-form excess ↑ → standardize to form2 and update media models.
• Audit: sources & fit ranges for n_phi / n_g / beta, evidence for ΔT_map and u(ΔT_map), and the manifest.redshift.disp signature chain with replay scripts.

Summary

• This chapter establishes the engineering relationship between n_phi and n_g, codifies the use of phase vs group conventions for redshift and arrival time, and provides a consistent mapping with manifest-grade publication.
• Through M65-7 / C65-7x / I65-7* and manifest.redshift.disp, path-redshift and arrival-time publication in dispersive media remain comparable, traceable, and rollback-ready.