Parallax asks where a thing is. Vortex asks what path it takes. This asks the question prior to both — could you know it is there at all, and by what means? — and answers it with a controlled scaling model. Size enters every detection budget squared; observer range enters it to whatever exponent the geometry forces. Within that boundary each modality is a straight line, and where two lines cross is a rung of the same χ ladder as everything else here.
Model boundary. The plot changes observer range while holding heliocentric distance, phase, bandpass, cadence and signal-to-noise assumptions fixed. It is a scaling comparison, not a survey-completeness curve: moving a target from 1 AU to the Kuiper belt also changes its illumination and temperature, so the optical and thermal lines must be recomputed rather than extrapolated.
FIG 12 the slope column
The slopes follow what illuminates the target and how often the signal pays the inverse square law, under the fixed-state boundary above. A shallower observer-range slope loses ground more slowly in that slice; actual survey reach also moves with illumination, temperature, phase, bandpass, cadence and sensitivity.
| Modality | What lights the target | Observer-range law | Slope | Model size at χ 6 | at χ 9 | Model boundary |
|---|
FIG 13 crossings
Two lines of different slope meet exactly once inside one fixed-state model, and one division finds it. Because the answer is a χ, it is a rung like any other — it gets a link, and it can be sent to someone. Parallel slices never trade places; their fixed offset reflects both target assumptions and configured instrument limits.
FIG 14 the other χ
Radar's own literature writes Woodward's ambiguity function as χ(τ, ν) — the same letter this suite spent forty-seven decades building a ladder out of. The coincidence is not only notational: the delay axis τ is that ladder. An echo returns after 2r/c seconds, so a delay is a distance is a rung, and the surface below is drawn over the same coordinate as everything above it.
Everything so far has been about detection. This is the question immediately after. Range resolution is c/2B, and on the ladder that is an uncertainty in χ itself — so the suite can finally say how sharply its own coordinate can be measured.
Two results carry the picture, and both are asserted in the unit suite rather than described here. A chirp's surface is the plain pulse's, sheared — ν → ν + kτ, unit Jacobian — and that tilt is the coupling between range and velocity, which is why one look at a moving target reads it at the wrong distance. And the volume under |χ|² is invariant: ambiguity can be pushed around the plane but never removed. A waveform is a decision about where to put it, not about how much to have.