28-EFT.WP.Propagation.PathRedshift v1.0 | Chapter 2 — Postulates & Minimal Equations (Redshift Baseline)

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Chapter 2 — Postulates & Minimal Equations (Redshift Baseline)

One-sentence goal: Establish an integrated, computable, auditable, and persistable baseline that links the decomposition and composition of path redshift z_path with the two-form arrival time T_arr, and provide equations and processes that anchor all subsequent implementations and releases.

I. Scope & Objects

1. Inputs
   • Path / worldline: gamma(ell) (fiber / free space / deep-space segments), or TX/RX 4-velocity worldlines u^μ_emit / u^μ_obs;
   • Media / potential / cosmology: n_eff(f,x), phi_grav(x), a(t);
   • Kinematics & rotation: v_emit/obs, Omega, loop oriented area A;
   • Observations: carrier / spectral-line streams, PLL / CFO / line-fit derived f_obs(t);
   • Reference conditions: RefCond (ephemerides / potentials / media / meteorology / time zone / timebase).
2. Outputs
   • Redshift decomposition & composition: z_parts = { z_kin, z_grav, z_med, z_cos, z_inst, z_proc }, z_path = compose(z_parts);
   • Two-form arrival time: T_arr^{form1}, T_arr^{form2}, gap delta_form, and harmonized T_arr*;
   • Manifest keys: manifest.redshift.* recording { z_parts, z_path, T_arr_forms, delta_form, u/U, contracts.*, signature }.
3. Boundary
   Default to first-order engineering for weak fields / small angular rates / small media drifts; strong-field or higher-order corrections are persisted as extended fields with sources and assumptions.

II. Terms & Variables

• Redshift & time–frequency: z, f_emit, f_obs, lambda_emit, lambda_obs, 1+z = f_emit / f_obs = lambda_obs / lambda_emit.
• Path & geometry: gamma(ell), L_gamma, t_hat, A, Omega.
• Media & dispersion: n_phi(f), n_g(f) = n_phi − f * ( d n_phi / d f ), n_eff(f,x).
• Arrival time & timebase: T_arr, c_ref, delta_form, tau_mono, ts, offset / skew / J.
• Dimensions: unit(z) = "1", unit(T_arr) = "[T]", unit(n_eff) = "1".

III. Postulates P65-2x

• P65-201 (Decompose & compose). Decompose the path redshift into physical and engineering components and compose multiplicatively
• 1 + z_path = (1+z_kin)(1+z_grav)(1+z_med)(1+z_cos)(1+z_inst)(1+z_proc)

with small-signal approximation z_path ≈ ∑ z_i.

• P65-202 (Two-form arrival time). Compute in parallel and constrain by contract:
• T_arr^{form1} = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell )
• T_arr^{form2} = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell )

Record delta_form and assert ≤ tol_Tarr.

• P65-203 (Explicit measures). Every integral/average declares its domain and measure, e.g.
  ( ∫_{gamma(ell)} • d ell ), ( ∫_{t∈W} • dt ), ( ∫_{f∈B} • df ).
• P65-204 (Dimensions & timebase). All fields pass check_dim( y − f(x) ); computation runs on tau_mono, publication at ts; log↔linear conversions are recorded as scale.note.
• P65-205 (RefCond traceability). Ephemerides / potentials / media / meteorology / timebase are persisted in RefCond with hash / validity.

IV. Minimal Equations S65-2x

1. General basis & 4-vector form

• S65-201 (Definition). 1 + z = f_emit / f_obs = lambda_obs / lambda_emit.
• S65-202 (4-vector, abstract). 1 + z = ( u_μ k^μ )_{emit} / ( u_μ k^μ )_{obs}, where u^μ are TX/RX 4-velocities and k^μ the wave 4-vector; engineering implementations use the component approximations below.

2. Kinematic redshift (Doppler / Sagnac / non-inertial)

• S65-203 (Relativistic Doppler, single LOS). Let β = v_los / c_ref, γ_L = 1 / √(1 − β^2),
• 1 + z_kin ≈ γ_L ( 1 − β )

(source approaching observer; bidirectional motion composes multiplicatively). Small-speed limit: z_kin ≈ − v_los / c_ref.

• S65-204 (Sagnac time/frequency). Loop geometry produces a time bias
• Δt_Sag = ( 2 * Omega • A ) / c_ref^2

with relative frequency bias z_Sag ≈ − Δt_Sag / T_meas (averaged over measurement time T_meas).

3. Gravitational redshift (weak field) & path term

• S65-205 (Weak-field potential difference). z_grav ≈ − Δphi_grav / c_ref^2, with Δphi_grav = phi_grav(obs) − phi_grav(emit).
• S65-206 (Path/Shapiro, first order). Path-traversal delay Δt_Sh ≈ ( 2GM / c_ref^3 ) ln( … ), with instantaneous shift z_Sh ≈ − d(Δt_Sh)/dt (often negligible or folded into z_grav per mission domain).

4. Media redshift (moving media / plasma / troposphere & ionosphere)

• S65-207 (Fizeau drag, small speed). z_med,drag ≈ ( n_eff − 1 ) * v_med,los / c_ref.
• S65-208 (Plasma). f_obs = f_emit √( 1 − f_p^2 / f^2 ), low-density approximation z_plasma ≈ (1/2) ( f_p^2 / f^2 ).
• S65-209 (Neutral / ionized atmosphere drift). z_med,t ≈ − (1/f) dφ_med/dt, where φ_med is the media phase; estimate via corr_env(x; RefCond).

5. Cosmological redshift

• S65-210 (FRW). 1 + z_cos = a(t_obs) / a(t_emit); low-z approximation z_cos ≈ H_0 Δt (H_0 Hubble constant).

6. Two-form arrival time & dispersion mapping

• S65-211 (Two-form gap).
  delta_form = | ( 1/c_ref ) ( ∫ n_eff d ell ) − ( ∫ ( n_eff / c_ref ) d ell ) |.
• S65-212 (Phase/group mapping). n_g = n_phi − f * ( d n_phi / d f ), yielding group arrival time T_g = ( ∫ ( n_g / c_ref ) d ell ); use this to keep group/phase redshift and arrival time consistent.

7. Composition & publication convention

• S65-213 (Composition).
  z_path = compose( z_kin, z_grav, z_med, z_cos, z_inst, z_proc ) (multiplicative; or publish the linear-sum approximation with a note).
• S65-214 (Harmonized arrival time).
  T_arr* = T_arr^{form2} + ΔT_geom + ΔT_med + ΔT_inst + ΔT_proc + ΔT_sync (see Chapter 8).

V. Metrology Pipeline M65-2 (Ready → Modeling → Verification → Persistence)

1. Ready: unify coordinates and tau_mono / ts; load path & device inventories, RefCond, and ephemerides / potentials / media profiles; declare unit/dimension mappings.
2. Model / estimate: compute z_kin / z_grav / z_med / z_cos using S65-203…210; compute T_arr^{form1/form2} and delta_form in parallel.
3. Verify:
   • check_dim(z) = "1", check_dim(T_arr) = "[T]";
   • calibrate bias/scale via references (frequency ratios / loopbacks / reference lines);
   • record uncertainties u / U and provenance (model / measurement / environment / approximation).
4. Persist:
   manifest.redshift.* = { gamma.hash, RefCond, z_parts, z_path, T_arr_forms:{ form1, form2, delta_form }, u/U, contracts.*, signature }.

VI. Contracts & Assertions C65-2x (suggested thresholds)

• C65-201 (Two-form gap): delta_form_p95 ≤ tol_Tarr (suggest tol_Tarr = 1e−3 • T_arr, or link-specific).
• C65-202 (Decomposition consistency): | z_meas − z_path |_p95 ≤ tol_z (SNR/window-dependent).
• C65-203 (RefCond freshness/coverage): ephemerides/potentials/media recency ≤ Δt_max, coverage ≥ cov_min.
• C65-204 (Path monotonic / geometry): non_decreasing(ell); when loops exist, (Omega / A) consistent and persisted.
• C65-205 (Dimensional compliance): all published fields pass check_dim; log↔linear conversions recorded in the manifest as scale.note.

VII. Implementation Bindings I65-2* (interfaces, I/O, invariants)

• I65-21 compose_z(parts, mode) -> z_path (mode ∈ { product, linear })
• I65-22 doppler_kinematic(state_emit, state_obs, los, frame) -> z_kin
• I65-23 sagnac_time(Omega, A) -> { Δt_Sag, z_Sag }
• I65-24 grav_weakfield(potential_model, worldlines) -> z_grav
• I65-25 medium_fizeau_plasma(n_field, v_med, f, gamma) -> z_med
• I65-26 cosmo_frw(a_t, t_emit, t_obs) -> z_cos
• I65-27 tarr_two_forms(n_eff, c_ref, gamma) -> { T_form1, T_form2, delta_form }
• I65-28 z_to_tarr_map(n_phi, n_g, f) -> mapper (phase/group mapping helper)
• I65-29 assert_baseline_contracts(records, rules) -> report / emit_redshift_manifest(results, policy) -> manifest.redshift
  Invariants: two_forms_present = true; check_dim(*) passes; gamma.hash / RefCond.hash traceable.

VIII. Cross-References

• Two-form arrival time & dispersion: EFT.WP.Metrology.PathCorrection v1.0.
• Timebase / sync / instrument delays & loopback: Energies volumes on TimeBase/Sync/Instrumentation.
• Observation conventions & LLR/PLL: Chapter 9 of this volume.
• Fusion & calibration: Chapters 10/11.
• Uncertainty & publication: Chapter 13 and Appendices E/C.

IX. Quality & Risk Control

• SLI / SLO: delta_form_p95, | z_meas − z_path |_p95, panel_freshness, RefCond_coverage.
• Fallback: rising two-form gap → standardize to form2; z_meas mismatch → revert potentials/ephemerides or shorten windows; stale RefCond → degrade or pause publication.
• Audit: manifest signature chains, source hashes, two-environment recomputation with ε_dual, replay scripts, and panel snapshots.

Summary

• This chapter delivers the engineering baseline for z_path decomposition—composition and two-form T_arr, codifies minimal equations for Doppler / Sagnac / gravitational / media / cosmological effects, and makes dimensions / dual forms / RefCond hard contractual conditions for publication.
• With M65-2 and I65-2*, teams can compute, verify, and manifest z_path and T_arr* across scenarios, providing a unified anchor for the chapters and use cases that follow.