duality

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A cross-corpus navigation page. Check each item’s declared scope and evidence status.

Frontier research (1)

FrontierPaperv0.12026-07-14

The Reciprocity One-Form v0.1: Affine Classification and Domain Rigidity

This paper classifies every invertible affine transformation of the normalized FRC ledger plane whose derivative preserves the reciprocity distribution ker(dx+dy). In coordinates u=x+y and v=x-y, the complete class is F(u,v)=(lambda u+c, alpha u+beta v+delta), with lambda beta nonzero. The paper derives the group law, inverse, leaf action, all affine involutions, and their fixed sets. Intersecting the classification with the operational half-plane M=R x (-infinity,0] yields a three-parameter family and a one-parameter family of sign-reversing involutions. Intersecting it with the nonnegative-entropy domain M+=[0,infinity) x (-infinity,0] is rigid: every kernel-preserving affine automorphism is either aI or aD, where a>0 and D(x,y)=(-y,-x). Consequently the identity is the unique exact form-preserving automorphism, D is the unique exact form-reversing automorphism, and D is the unique nonidentity involution in this bounded class. The result is an exact coordinate theorem pending independent proof review. It does not lift D to physical states, observables, distributions, dynamics, or operational registers and therefore does not establish a physical, categorical, Born, or Majid-style duality.

Philosophical work (1)

PhilosophicalPaperv0.12026-07-14

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.