Indexing the archive…
Your Universe of Digital Possibilities
A receiver is hidden somewhere on Earth; only its signals arrive. Each is a satellite’s clock reading, compared with the receiver’s own. GPS is a clock problem: four satellites for four unknowns — where you are, and how wrong your clock is — and the geometry of the sky multiplies every nanosecond into metres. Fix the receiver from its signals alone, then turn around.
The delay between a satellite’s time stamp and the receiver’s clock, times c: the true range, plus the receiver clock’s error b — the same for every satellite — plus noise. Four of them fix x, y, z and b; a nanosecond of b is thirty centimetres of every range.
H holds, for each satellite, minus the unit vector toward it and a 1 for the clock. Range noise σ becomes position-and-time error σ·GDOP: satellites spread over the sky keep it near 2; bunched together, the rows of H nearly agree, HᵀH nearly loses its inverse, and the error ellipse swells.
Twenty thousand kilometres up, the gravitational potential is weaker, and a clock there runs fast by GM/c²·(1/R − 1/a) ≈ 5.3 × 10⁻¹⁰ against one on the ground. Net of the speed term: +38.5 µs a day, removed before launch by setting the clocks slow.
At 3.87 km/s a satellite clock runs slow by v²/2c² ≈ 8.3 × 10⁻¹¹ — the first term of 1/γ. Small, and every day.
The Orfeo Edition’s eighth shadow hides a point on a sphere behind four clocks. The rebuild sees only ranges that are all wrong by the same amount; the residuals are all it can count before turning around; and where the ranges cannot choose — three satellites, four unknowns — it draws the family and names the prior it picks with. Light’s fixed speed is The Cone’s; the clock that ticks slower deeper in a well is The Well’s; and the bending of the signal itself is The Lens’s.