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self-duality

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Relatedborn-reciprocityFRC 800 seriesmajidCoherencedualityEntropyframework-notehopf-algebra

Frontier research (1)

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.

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.