The axiom is that the Earth orbits the Sun along with the other planets while turning to the Sun each day as a function of daily rotation.

The original geocentric astronomers founded their hypotheses on the observation that the Sun passed directly through the constellations while the planets displayed direct/retrograde motions with the Earth as stationary and central.

In a moving Earth/Sun centred system, the Sun's motion around the Earth daily is due to the rotation of the Earth hence the axiom is proved. The orbital motion of the Earth is proved by an alternative hypothesis based on the observation that stars, in a line-of-sight transition, move from left to right of the Sun.


Now the false hypothesis and the basis for a base/step mathematical induction -

"That the fixed stars being at rest, the periodic times of the five primary planets, and (whether of the sun about the earth, or) of the earth about the sun, are in the sesquiplicate proportion of their mean distances from the sun.... for the periodic times are the same, and the dimensions of the orbits are the same, whether the sun revolves about the earth, or the earth about the sun." Newton

There is no geocentric/heliocentric equivalency as the "sun about the earth" refers to daily rotation and can only be answered in rotational terms so it cannot be equated with the Earth's orbital motion around the central Sun..