exact-mathematics

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Frontier research (2)

FrontierPaperv0.12026-07-15

Dimension Threshold for FRC Reciprocity v0.1: One-Parameter and Qubit Lift Obstructions

The FRC ledger involution D(s,y)=(-y,-s), with normalized entropy s=S_mu/k*_mu and y=ln C, cannot act nontrivially on any one-parameter family whose coherence is strictly monotone and whose normalized entropy decreases with log coherence. The preserved coordinate v=s-y is then injective, so every lift is the identity restricted to u=s+y=0. Exact normalized reciprocity ds+dy=0 sharpens the result: a connected curve lies on u=c; if c is nonzero there is no same-family lift, while c=0 permits only the trivial identity on a ledger-injective family. The wrapped-Cauchy family supplies an exact Poisson-kernel example with stationary coherence 1/sqrt(3) and algebraic fixed ledgers. The full qubit state space crosses the dimension threshold: its two-dimensional ledger image contains a nonempty D-invariant region K with a one-dimensional fixed leaf and admits explicit inequivalent set-theoretic fiber lifts. Those lifts are not physical promotions. No incoherent operation, no unital qubit CPTP channel, and no unitary or antiunitary Wigner symmetry realizes D on all of K. A general nonunital coherence-generating CPTP or resource-assisted lift remains open.

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.