ONAR9 — Orientated Numbers Arithmetic Relations of 9

Abstract

ONAR9 is formulated as a finite relational frame with a separate compositional payload. A declared balance between complete source/branch relations and complementary position roles selects four branches and ten complete relations. Fixing one oriented reference edge in K5 leaves nine nonreference edges in an intrinsic nondegenerate 3+3+3 class decomposition; choosing orientations gives a 19-state signed coordinatization. In the decimal reversal-subtraction realization, the nonzero local magnitudes {1,…,9} generate capacity-m payloads as multisets, in bijection with monotone mirrored-gap states. The conventional reversal map is written as a borrow expansion (CAST) followed by sorting and mirrored-gap extraction (PAIR). The fixed-width identity 2r+s=n is retained, with s interpreted as an inactive/central-width parameter rather than a monotone closure count. A type-B signed-permutation orbit describes one formal order/orientation lift of a gap state, while a separate body-reconstruction theorem makes explicit that this orbit is not the complete history of a raw digit word. Classical Kaprekar difference coordinates, multiset counting, Coxeter groups, and reversible-computation embeddings are treated as background. The proposed contribution is the combined relational frame/payload decomposition and its explicit information-loss bookkeeping; priority remains subject to external review.