ratioOrbitSetoid

In the Primitive Recognition Calculus integer/rational layer, a ratio orbit is a display with a signed-orbit numerator and a nonzero distinction-nat denominator (K4.7). Two such displays are identified when their cross-multiplied numerators and denominators agree at the orbit level: that is the content of ratio-orbit equivalence (K4.8).

Mathlib constructs quotients from a Setoid (a type plus a proved equivalence relation). This definition is the setoid instance on ratio orbits. Upstream, the relation is defined as cross-equality, and ratioOrbitEquiv_equivalence already shows it is reflexive, symmetric, and transitive by reducing to the cross-equality equivalence theorem on ratio orbits.

The local module builds PRC integers and rationals as quotient-native objects rather than importing classical $\mathbb{Z}$ and $\mathbb{Q}$ as primitives, so that arithmetic stays inside orbit and distinction structure.

One-line setoid package. The relation field is set to ratio-orbit equivalence (cross-multiplication). The iseqv field is discharged by the existing theorem that this relation is an equivalence, which itself is a thin wrapper around cross-equality equivalence on ratio orbits. No new algebraic work occurs here.

This is the setoid substrate for K4.8 PRC rationals. Downstream, PRCRat is defined as the quotient by this setoid, and the ratio-orbit constructor feeds that quotient. Without it, rationals cannot be formed as nonzero-denominator ratio-orbit classes identified by cross-multiplication.

In the Recognition foundation stack this keeps rational arithmetic native to orbit displays rather than smuggling in classical $\mathbb{Q}$. That matters for later cost and ladder constructions that must stay inside PRC-native types. It is scaffolding for the rational layer, not a physics forcing step (T5–T8), but it is required infrastructure before mass-ladder or constant derivations can speak in PRC rationals.