involution

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Frontier research (2)

FrontierPaperv0.12026-07-14

Exact Phase-Family Test of FRC Duality v0.1: A Lift Obstruction on the von Mises Manifold

The fixed-mean von Mises family provides an exact test of whether the FRC ledger involutions classified in FRC 830.001 lift from derived coordinates to probability distributions. The family has genuine information-geometric structure: concentration kappa and mean resultant r are natural and expectation coordinates, the log-partition function is strictly convex, and minus the differential entropy is its Legendre dual potential. That structure is not the FRC ledger exchange. For the ledger ell(kappa)=(S(kappa),ln r(kappa)), introduce u=S+ln r and v=S-ln r. The function v is strictly decreasing, while u is strictly unimodal with its unique maximum at kappa r=1. Because D(x,y)=(-y,-x) preserves v, every proposed D lift must fix kappa and can exist only at the two isolated roots of u=0. The broader half-plane involutions R_p(x,y)=(-x-2y+p,y) likewise force kappa to remain fixed and survive at no more than two isolated fixed ledgers. Neither class lifts on any open concentration domain. The point kappa r=1 is a maximum of the system-only ledger total, not a D-fixed point or structural self-duality. FRC therefore retains a P2 coordinate involution and gains a precise P3 lift obstruction in this family; it does not gain a phase-family physical or Majid-style self-duality.

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.