zero

In the Primitive Recognition Calculus, integers are not postulated; they are assembled from orbit displays. A signed orbit is a pair (pos, neg) of distinction naturals, with intended meaning the difference pos − neg (K4.6). Distinction naturals themselves arise from the logic-level Peano structure (zero and successor as generator iterates), so the two zeros in the pair are the same identity element that is never a successor.

This module sits after orbit arithmetic and builds the integer and rational layers: signed orbits quotient to PRCInt, and ratio orbits (signed numerator over nonzero distinction-natural denominator) quotient to PRCRat. The zero signed orbit is the canonical neutral display before any quotient identification.

Pure definition: apply the SignedOrbit constructor to DistinctionNat.zero in both the positive and negative slots. No lemmas, no tactics, no rewriting.

Zero is the base point of the signed-orbit lattice that underlies K4.8 PRC integers. Downstream constructions (integer embedding toInt, the balanced-length setoid, ratio-orbit numerators, and the eventual display into classical $\mathbb{Z}$) all need a named neutral element so that additive structure and the cross-multiplication equivalence for rationals can be stated without ad-hoc pairs. In the broader Recognition stack this is scaffolding for arithmetic forced from logic (Peano theorems such as zero-not-successor), not a physical constant; it feeds the foundation layer that later supports the forcing chain and mass ladder only indirectly.