MATH.9:4.3 - Extend a selected answer over one input orbit

Choose representative data d0 and an answer y0 in B(d0). For input d in the orbit of d0, obtain g with d=g*d0 and define

f(d)=g*y0.

The answer is permitted because the relation in :4.1 carries A(d0) to A(d). It is independent of the transformation used. Indeed, if g1*d0=g2*d0, then k=g2^-1*g1 fixes d0. Since k fixes y0,

g1*y0=g2*(k*y0)=g2*y0.

The rule is also equivariant: for another transformation q,

f(q*(g*d0))=(q*g)*y0=q*f(g*d0).

This constructs the entire rule on that orbit. To evaluate it, retain a way to find g, or an equivalent formula for f(d). An existence argument for g alone may leave the computation unresolved.

If a normalization procedure instead supplies h with h*d=d0, return h^-1*y0. Keeping the direction of this transformation prevents returning a coordinate answer in the normalized frame. The position-selection case in :5.4 carries out both inverse returns.