Invert parallax without inventing symmetric distance errors

1. Define the stellar inference

Annual stellar parallax is a geometric angular displacement caused by the observer’s changing position around the Sun. The parallax angle convention uses a one-AU baseline, not the full peak-to-peak displacement across opposite orbital positions. With parallax p in arcseconds, distance in parsecs is 1/p; with milliarcseconds it is 1000/p.

2. Connect mechanism and assumptions

The original records stipulate a positive central parallax and a hard symmetric bound. Because inverse distance decreases monotonically with parallax, the distance limits are 1000/(p+b) and 1000/(p−b). The endpoints reverse order and the two distance deviations are generally unequal. This is interval propagation, not a Gaussian confidence statement.

Proper motion is a longer-term angular drift and can coexist with the annual signature. Orbital motion in an unresolved binary, calibration zero points and instrument systematics can affect a real astrometric fit. Repeated measurements are useful only when the time model and correlated errors are retained.

When measurements approach zero or include negative parallaxes, a simple reciprocal and symmetric interval no longer provide this well-behaved construction. Negative measured estimates can arise from noise without representing negative physical distances. A real statistical distance inference needs its likelihood, calibration and prior assumptions; the positive teaching grid does not solve that broader problem.

3. Work the supplied case

At p=20 mas with hard bound±2 mas, central inverse distance is 50 pc. Bounds are 1000/22≈45.4545 and 1000/18≈55.5556 pc. The lower deviation≈4.54545 differs from the upper≈5.55556.

4. Predict and investigate

Does a symmetric angular bound remain symmetric after taking a reciprocal?

Invert both endpoints before subtracting the central value. Compare larger and smaller fractional parallax bounds, and distinguish hard limits from a probabilistic uncertainty claim.

Source and boundaries

Primary source. Checked 31 October 2026. All numerical records and diagrams here are original supplied models, not measured stellar populations or a full stellar-evolution solver. No direct observing or physical experiment is requested.