28-EFT.WP.Propagation.PathRedshift v1.0 | Chapter 1 — Path Redshift: Domain Definition & Scope

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Chapter 1 — Path Redshift: Domain Definition & Scope

One-sentence goal: Delimit the Path Redshift domain—its objects, I/O, and engineering boundaries—and establish a unified publication convention and manifest keys that combine redshift decomposition with the two-form arrival-time framework.

I. Scope & Objects

1. Objects (paths & worldlines)
   Propagation path gamma(ell) (fiber / free space / deep space segments), or spacetime worldline Gamma(tau); loop and rotational geometry (Omega, A).
2. Objects (time–frequency & media)
   Local timebases at TX/RX tau_mono / ts, local clock parameters offset / skew / J; dispersive media fields n_eff(f,x), n_phi(f), n_g(f); gravitational potential phi_grav(x); cosmological scale factor a(t).
3. Inputs
   • TX/RX state: state_emit / state_obs = { r, v, a, att, Omega };
   • Ephemerides & potentials: ephemeris, grav_model;
   • Media & environment: n_field, atmospheric/ionospheric parameters, RefCond;
   • Observations & metrology: carrier / spectral-line streams (PLL / CFO / line-fitting), metrology panels, and calibration evidence.
4. Outputs
   • Redshift decomposition: z_parts = { z_kin, z_grav, z_med, z_cos, z_inst, z_proc } and composite z_path;
   • Two-form arrival time: T_arr^{form1}, T_arr^{form2}, their difference delta_form, and the harmonized T_arr*;
   • Manifest: manifest.redshift.* containing { RefCond, z_parts, z_path, T_arr*, delta_form, u/U, contracts.*, signature }.
5. Boundary (exclusions)
   Higher-order strong-field relativistic and non-perturbative cosmological corrections, and quantum frequency shifts are out of the default engineering scope; include as extended fields with sources and assumptions when required.

II. Terms & Variables

• Path & geometry: gamma(ell), L_gamma, t_hat, A, los / nlos.
• Spacetime & dynamics: t, tau, v, beta = v / c_ref, gamma_L = 1 / sqrt(1 − beta^2), Omega, phi_grav.
• Media & dispersion: n_phi(f), n_g(f) = n_phi − f * ( d n_phi / d f ), n_eff(f,x).
• Redshift & time–frequency: z, f_emit, f_obs, lambda_emit, lambda_obs, a(t).
• Arrival time & timebase: T_arr, c_ref, delta_form, tol_Tarr, tau_mono, ts, offset / skew / J.
• Dimensional examples: unit(z) = "1", unit(T_arr) = "[T]", unit(n_eff) = "1".

III. Postulates P65-1x

• P65-11 (Redshift factorization):
  1 + z_path = (1+z_kin)(1+z_grav)(1+z_med)(1+z_cos)(1+z_inst)(1+z_proc);
  small-signal approximation z_path ≈ ∑ z_i.
• P65-12 (Two forms in parallel): arrival-time publication must present
• T_arr^{form1} = ( 1 / c_ref ) * ( ∫_{gamma} n_eff d ell )

and

T_arr^{form2} = ( ∫_{gamma} ( n_eff / c_ref ) d ell )

with recorded delta_form.

• P65-13 (Explicit measures): every integral/average declares its domain & measure:
  ( ∫_{gamma(ell)} • d ell ), ( ∫_{t∈W} • dt ), ( ∫_{f∈B} • df ).
• P65-14 (Dimensional compliance): every field entering equations passes check_dim( y − f(x) ); log↔linear conversions are recorded as scale.note.
• P65-15 (RefCond traceability): ephemerides / potentials / media / meteorology / timebase recorded in RefCond with hash / validity.

IV. Minimal Equations S65-1x

• S65-11 (Redshift definition):
  1 + z = f_emit / f_obs = lambda_obs / lambda_emit.
• S65-12 (Composite convention):
  z_path = compose( z_kin, z_grav, z_med, z_cos, z_inst, z_proc ) (multiplicative form or linear-sum approximation; specify in publication).
• S65-13 (Two-form arrival-time gap):
  delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) − ( ∫ ( n_eff / c_ref ) d ell ) |.
• S65-14 (Dispersion mapping):
  n_g = n_phi − f * ( d n_phi / d f ), used to keep phase- and group-based arrival-time / frequency shifts consistent.

V. Metrology Pipeline M65-1 (Domain Ready → Modeling → Verification → Persistence)

• Domain ready: unify coordinates and tau_mono / ts; load path & device inventories and RefCond (ephemerides / potentials / media / timebase).
• Model / estimate: compute z_kin / z_grav / z_med / z_cos and the observation z_meas; compute T_arr^{form1/form2} in parallel.
• Verify: run check_dim; enforce the two-form gate delta_form ≤ tol_Tarr; calibrate biases & scale via references (frequency ratios / loopbacks).
• Persist: emit
  manifest.redshift.* = { gamma.hash, RefCond, z_parts, z_path, T_arr_forms, delta_form, u/U, contracts.*, signature }.

VI. Contracts & Assertions C65-1x (suggested thresholds)

• C65-101 (Two-form gap): delta_form_p95 ≤ tol_Tarr (suggest tol_Tarr = 1e−3 * T_arr or link-specific).
• C65-102 (Redshift composition residual): | z_meas − z_path |_p95 ≤ tol_z (SNR/window-dependent).
• C65-103 (RefCond freshness): ephemerides / potentials / media recency ≤ Δt_max; stale → reject or degrade.
• C65-104 (Dimensional compliance): all fields pass check_dim; unit(z) = "1".
• C65-105 (Path monotonicity & geometric consistency): non_decreasing(ell); rotational/loop geometry (Omega / A) consistent (Sagnac term).

VII. Implementation Bindings I65-1* (interfaces, I/O, invariants)

• I65-11 build_refcond(sources) -> RefCond (aggregate ephemerides / potentials / media / timebase)
• I65-12 eval_tarr_forms(n_eff, c_ref, gamma) -> { T_form1, T_form2, delta_form }
• I65-13 compose_z(z_parts, mode) -> z_path (mode ∈ { product, linear })
• I65-14 compare_z(z_path, z_meas) -> { resid, pass }
• I65-15 assert_redshift_contracts(ds, rules) -> report
• I65-16 emit_redshift_manifest(results, policy) -> manifest.redshift
  Invariants: two_forms_present = true; check_dim(*) passes; gamma.hash / RefCond.hash are traceable.

VIII. Cross-References

• Timebase & synchronization: Energies Timebase/Sync volume (tau_mono / ts / offset / skew / J).
• Path integrals & arrival time: EFT.WP.Metrology.PathCorrection v1.0 (two forms & dispersion).
• Instrumentation & calibration: EFT.WP.Metrology.Instrument v1.0.
• Cleaning & contract persistence: EFT.WP.Methods.Cleaning v1.0.

IX. Quality & Risk Control

• SLI / SLO: delta_form_p95, | z_meas − z_path |_p95, panel_freshness, RefCond_coverage.
• Fallback strategies: two-form drift ↑ → standardize to form2 and re-estimate media fields; z_meas mismatch → revert potential/ephemeris versions or shorten windows; stale RefCond → degrade or pause publication.
• Audit: manifest signature chains, source hashes, two-environment recomputations with ε_dual, replay scripts, and panel snapshots.

Summary

• This chapter establishes the domain baseline for Path Redshift: redshift decomposition and two-form arrival time computed and published in parallel, with RefCond traceability and dimensional compliance.
• It provides an executable skeleton P65-1x / S65-1x / M65-1 / C65-1x / I65-1*, with artifacts delivered as manifest.redshift.* to enable reproducible, auditable, and rollback-ready engineering governance.