28-EFT.WP.Propagation.PathRedshift v1.0 | Chapter 10 — Fusion of Sensors / Ephemerides / Environment (RefCond & Synchronization)

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Chapter 10 — Fusion of Sensors / Ephemerides / Environment (RefCond & Synchronization)

One-sentence goal: Fuse ephemerides, gravity, media, meteorology, and local clock data into a unified reference condition RefCond, solve for synchronization parameters offset / skew / J, and thereby provide a single computational time axis and coherent data plane for computing z_parts / z_path and T_arr^{form1/form2}—with fully auditable publication.

I. Scope & Objects

1. Inputs
   • Ephemerides / attitude & dynamics: ephemeris(t), { r(t), v(t), a(t), att(t), Ω(t) } for station / payload / target.
   • Potentials & parameters: phi_grav(x,t), GM, model degree/order and versioning.
   • Media & environment: n_eff / n_phi / n_g(f,x,t), TEC / N_e(x,t), met(t) = { T, P, RH }, iono, trop.
   • Local clock & time-transfer: oscillator priors (offset / skew / J), sync observations τ_sync (PPS / Two-Way / reference chains).
   • Observation-side references: z_meas(t) (Ch. 9), prior path gamma0 (Ch. 8).
2. Outputs
   • Fused reference condition RefCond (with hash / validity / coverage);
   • Synchronization solution offset / skew / J with uncertainty u/U; unified publication time ts and monotone computation axis tau_mono;
   • Consistent mappings for z_pred / z_parts and T_arr^{form1/form2}; manifest manifest.redshift.refcond.*.
3. Boundary
   This chapter addresses data fusion and synchronization; component models are in Chs. 3–6; path & integrals in Ch. 8; observations in Ch. 9.

II. Terms & Variables

• State & noise: x = [ r, v, a, att, Ω, offset, skew, J_params ]^T, process noise w ~ N(0,Q).
• Measurements: y = [ ephem_meas, att_meas, z_meas, τ_sync, met, TEC, … ]^T, measurement noise v ~ N(0,R).
• Time-transfer model: t_rx = (1+skew) t_tx + offset + J(t); J(t) may include white/pink noise (AR/MA).
• Reference condition:
  RefCond = { ephemeris.hash, gravity.hash, iono.hash, trop.hash, nfield.hash, osc.hash, tz, Δt/Δx, coverage, valid_from/to }.
• Dimensions: unit(offset) = [T], unit(skew) = 1, unit(J) = [T], unit(coverage) = 1.

III. Postulates P65-10x

• P65-1001 (Single time axis): All z / T_arr computations map to the unified computational axis tau_mono and are published at ts; the solved offset / skew / J must be persisted.
• P65-1002 (Traceable RefCond): Provenance for ephemerides / gravity / media / meteorology / clocks—sources, versions, interpolation, coverage—must be recorded in RefCond with hash / validity / Δt / Δx / coverage.
• P65-1003 (Two-form pairing): Fusion & synchronization publication must include T_arr^{form1/form2} with delta_form ≤ tol_Tarr, explicitly stating interpolation/mapping effects on both forms.
• P65-1004 (Explicit measures): Any fusion / smoothing / resampling declares the window and measure ( ∫_{t∈W} • dt ), grid Δt / Δx, and frames.
• P65-1005 (Dimensional compliance): All fields pass check_dim( y − f(x) ); record any log↔linear transforms via scale.note.

IV. Minimal Equations S65-10x

1. State-space / factor-graph fusion (engineering form)

• S65-1001 (Discrete state):
• x_{k+1} = F_k x_k + G_k w_k, y_k = h_k(x_k) + v_k

Typical sub-mappings:

• Observation → redshift: z_meas = z_pred(x, RefCond) + ε_z (composed from Chs. 3–6)
• Media → fields: TEC_meas = TEC(x,t) + ε_TEC (and analogously for met)
• Time-transfer → clock: τ_sync ≈ (1+skew) t_tx + offset + J(t) − t_rx + ε_τ
• Ephemerides → position/velocity: r_meas = r + ε_r, v_meas = v + ε_v
• S65-1002 (Filtering / smoothing): EKF/UKF/RTS or factor-graph least squares
• x̂ = argmin_x ∥x − x_0∥_{P_0}^{2} + ∑ ∥ f_i(x) − y_i ∥_{R_i}^{2}

2. Clock & jitter model

• S65-1003 (Linear clock)
• t_rx = (1+skew) t_tx + offset + J(t)

with J(t) = Σ J_i(t) (e.g., white PM, white FM, random-walk FM). For engineering, approximate with AR(1)/MA(1).

3. Interpolation / resampling & windows

• S65-1004 (Time resampling)
• x(t_k) = Interp( x(t_i); Δt )

(spline / piecewise linear); interpolation error ε_interp enters Q/R.

• S65-1005 (Spatial mapping)
  Line/surface/volume integrals along gamma(ell) for n_eff(f,x) and TEC(x): ( ∫_{gamma} • d ell ); persist discretization step Δell.

4. Unified RefCond and z_pred

• S65-1006
• RefCond = pack( hashes, versions, Δt/Δx, coverage, frames, tz, osc )
• z_pred = compose( z_kin(x̂), z_grav(RefCond, x̂), z_med(RefCond, γ), z_cos(RefCond) )

(see Chs. 2–6).

5. Arrival-time harmonization

• S65-1007
• T_arr^{form1/form2} ← parallel integrals with synchronized n_eff and gamma(ell)
• ΔT_sync = offset + skew • T_window + E[ J(t) ]
• T_arr* = T_arr^{form2} + ΔT_geom + ΔT_med + ΔT_inst + ΔT_proc + ΔT_sync

(per Chs. 2 / 8).

V. Metrology Pipeline M65-10 (Ready → Fuse → Sync → Verify → Persist)

1. Ready: align coordinates & timescales; load ephemerides / gravity / media / meteorology / clock sources; set window W, grids Δt/Δx, and EKF/UKF/factor-graph configs; fix thresholds and SLOs.
2. Fuse: run EKF/UKF/factor-graph to obtain { x̂_k, P_k }; generate RefCond (with coverage/freshness, hashes, interpolation policy).
3. Sync: solve offset / skew / J; map observation & models to tau_mono; compute ΔT_sync.
4. Verify:
   • Dimensions & two forms: check_dim(*), delta_form ≤ tol_Tarr;
   • Consistency: resid_z = z_meas − z_pred; clock-noise PSD vs model fit (KS/χ²);
   • Freshness/coverage: age(source) ≤ Δt_max, spatial coverage ≥ cov_min.
5. Persist:

manifest.redshift.refcond = { RefCond,

sync:{ offset, skew, J_model, u/U },

fusion:{ method, Q/R, P },

z_pred,

T_arr_forms,

delta_form,

coverage,

sources.hashes,

contracts.*, signature }

VI. Contracts & Assertions C65-10x (suggested thresholds)

• C65-1001 (Two-form gap): delta_form_p95 ≤ tol_Tarr; and | T_arr* − t̂_cont |_p95 ≤ tol_align.
• C65-1002 (Freshness / coverage): age(ephemeris/grav/iono/trop/n_field) ≤ Δt_max, coverage ≥ cov_min; interpolation steps Δt/Δx within allowed bands.
• C65-1003 (Clock consistency): |J|_p95 ≤ J_max, |skew| ≤ skew_max (ppm), and offset drift rate ≤ rate_max.
• C65-1004 (Residual gate): |resid_z|_p95 ≤ tol_z (consistent with Ch. 9 SNR/CRLB).
• C65-1005 (Dimensional compliance): all published fields pass check_dim; record any log↔linear conversions in scale.note.

VII. Implementation Bindings I65-10* (interfaces, I/O, invariants)

• I65-101 fuse_refcond(sources, cfg) -> { RefCond, coverage, hashes }
• I65-102 solve_sync(rx_ts, tx_ts, priors) -> { offset, skew, J_model, u/U }
• I65-103 interp_sources(sources, Δt, Δx, policy) -> { streams_interp, ε_interp }
• I65-104 compose_z_pred(x̂, RefCond, gamma) -> { z_pred, meta }
• I65-105 compose_Tarr_forms(n_eff, gamma, c_ref) -> { T_form1, T_form2, delta_form }
• I65-106 assert_fusion_contracts(ds, rules) -> { report, pass }
• I65-107 emit_refcond_manifest(results, policy) -> { uri, status }
  Invariants: two_forms_present = true; check_dim(*) passes; timeframes/coordinates/source hashes traceable; freshness/coverage meet thresholds.

VIII. Cross-References

• Kinematic / gravitational / media / cosmological components: Chs. 3–6 (for z_pred).
• Dispersion & convention mapping: Ch. 7; path integrals: Ch. 8; observation conventions: Ch. 9.
• Calibration & invariants: Ch. 11; uncertainty/guardband: Ch. 13 and Appendices C/E; runtime & panels: Ch. 14.

IX. Quality & Risk Control

• SLI / SLO: delta_form_p95, |resid_z|_p95, age(source)_p95, coverage, |skew|_p95, |J|_p95, panel_freshness.
• Fallback strategies: stale/sparse sources → enlarge guardbands & lower publication cadence; unstable clock → switch to a better reference / widen sync bandwidth; large residuals → shorten windows / update models; two-form excess → standardize to form2.
• Audit: source & version hashes, fusion/sync parameters & covariances, residuals & PSD evidence, and the manifest.redshift.refcond signature chain with replay scripts.

Summary

• This chapter fuses multi-source sensors / ephemerides / environment and solves synchronization into computable RefCond + offset/skew/J, firmly paired with the two-form arrival time and z_pred.
• With M65-10 / C65-10x / I65-10* and manifest.redshift.refcond.*, it establishes a single time axis and unified data plane—traceable, auditable, and rollback-ready—as the foundation for publishing path redshift.