lift-obstruction

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Frontier research (3)

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.

FrontierPaperv0.22026-07-15

FRC and Quantum Born Reciprocity v0.2: A Present-Structure Obstruction to Majid-Style Self-Duality

FRC has an exact normalized entropy-coherence ledger involution D(s,y)=(-y,-s), a typed category of operational registers, and now both one-parameter and full-qubit lift obstructions. This paper asks whether those results constitute Born reciprocity or Majid-style representation-theoretic self-duality. They do not. The canonical Born exchange B_a(q,p)=(ap,-q/a) is order four and symplectic, whereas D is order two and anti-symplectic on the ambient ledger plane; indeed their different orders prevent conjugacy even by a bijection. The ledger datum also does not reconstruct a state-space action, and FRC 830.004 shows that this non-reconstruction occurs in natural qubit fibers. A primitive Hopf algebraization makes D only a Hopf automorphism of one chosen ledger algebra, without an independently defined dual object, pairing, semidualisation, representation exchange, or physical state-observable map. The general one-parameter identity-lift obstruction excludes every nonidentity lift under its monotonicity hypotheses, while the full qubit ledger excludes incoherent operations, unital qubit CPTP channels, and unitary or antiunitary Wigner symmetries as global lifts. A general nonunital, coherence-generating CPTP lift remains open. Present FRC therefore has exact coordinate, typed-morphism, dimension-threshold, and scoped no-go mathematics, but not a Born- or Majid-style self-duality.

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.