What Would Make FRC Reciprocity a Duality? v0.1
FRC uses dS + k* d ln C = 0 as an operational bookkeeping relation and proposes a wider open-system interpretation as a conjecture. This paper asks a prior mathematical question: what additional structure would make that reciprocity a duality? It separates five claim classes—coordinate transform, bookkeeping identity, balance law, duality, and self-duality—and supplies exact counterexamples showing that none implies the next. On the normalized ledger plane, the map D(x,y)=(-y,-x) is a sign-reversing involution satisfying D^2=id and D*omega=-omega for omega=dx+dy. On the operational domain y<=0 it is a self-map only when x>=0, and no present construction lifts it from derived ledger coordinates to independently defined FRC states, observables, dynamics, or operational registers. Present FRC therefore reaches an exact coordinate-level sign-reversing involution on a restricted domain, but not a physical, categorical, or Majid-style self-duality. The result is a terminology lock, a promotion test, and a research gate for the 830 series.