Only one speed-square satisfies v^2 = mu / r for a circular orbit; any other value leaves the centripetal and gravitational terms unbalanced, so the check rejects it when the candidate does not match the gravity parameter over the radius. That verdict is exact only for the stated mu, radius, and speed-square in an ideal circular two-body orbit. Real orbits add perturbations, non-circularity, finite body sizes, reference-frame choices, and measurement uncertainty.
A wrong speed-square renders as a rejected circular ledger. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
The wrong candidate still renders as a checked ledger
Try speed-square 3 at the same radius 4.
v2=3
Three over four does not equal the field one
The candidate acceleration is 3 over 4, not the checked field 1.