If Earth, together with our Sun, were in orbit of another star, there will be a hub and 90* perpendicular to the plane of its orbit, in the star studded canopy overhead, a pole -the still point around which, indirect motion of those stars MUST ensue.
an exercise: suspend our Sun from that spot on a tether, swing it (with velocity in compliance of the precession rate :o) suffice to say greater radius, would require greater orbital velocity. Its circuit is surrounded by the 12 constellations of the Zodiac, in turn within the greater Galaxy. That movement of the Sun induces perception of motion in the actually fixed stars, creating a pattern, discernable over long periods by tracking their shifting co-ordinates.
This is the missing reference frame and it is lacking in your binary considerations.
'The math', essentially communicating relationships, may work to all intensive purposes but relating what? a motion that does not take place -the perceived projected onto a motionless celestial body resulting in misinterpretation and the outlandish claim that our Sun orbits another star, of undisclosed location and consequently, absence of orbital and parameter data.
Ok let’s continue, changing star coordinates should attest the motion of our Sun & Earth around the (fictitious) solar mass... the Sun's current location, you'd agree, has to be part of a point in progress along the bogus orbit -already in swing, on the tether. For the surrounding Zodiac to proceed by, the way it does, the plane of orbit would have to correspond with the actual occurrence. Earth's orbit of the Sun is slightly inclined by comparison but an irrelevant detail.
The Sun, in orbit of the other star, will have to be in tandem with established star motion -discernible around the acknowledged EP, which correlates with the centre of Earth’s ecliptic orbit of the Sun. So, let’s attempt to apply this motion data to justify a binary... in order for the supposed orbital motion of our Sun to comply with the pattern of changing star co-ordinates, the location of the object in question has to be aligned on the plane of the Zodiac, between the North and South poles of the Ecliptic.
In this case, the binary mass should be easily detectable, just haul out and dust off the old spying tube... a bit of a paradox, since our Sun is already in that neighborhood but who knows perhaps its own Holy Spirit. Orbital dynamics dictate the Lost Star, as the governing force of our Sun, being an inner object and the star motion, likewise, necessitates it. The familiar planetary order from the Sun: Mercury - Venus - Earth - Mars - Jupiter - Saturn - Uranus - Neptune - Pluto thus, places the MBM drum roll... (wait for it)... Mystery Binary Mass!! in the space between our Sun and Mercury, which makes for a somewhat crowded locale -difficult to miss there...
It brings to mind ‘an aberration’, as phrased by a contemporary era professor in astronomy referring to Keplers’ structure of the SolarSystem.
Little wonder you receive flack, now a strike on ‘the empire’… truth could undermine an enterprise... the exulted clan should be weary of a gift horse ;-]
Your binary scenario suggests Earth’s axis is of relative fixed orientation and does not perform a 360* retrograde motion. The star motion data instead, complies with precession of the Earth and its axis orientation change can be witnessed in local SS motion: Sun-Earth relationship and accumulative time retardation / advance Solstice -eventually shifting the Northern hemisphere summers to December -the product of Earth rotating retrograde and presenting different topography to face the Sun.
The time is what it is
“every dog has its day”