The QFI Machine: Quotient-Flow Invariants and Universal Transport

We present a unified framework for transporting invariants across represen- tational idioms. The Quotient-Flow Invariant (QFI) schema—not to be confused with quantum Fisher information, which is one instantiation—axiomatizes three properties: isomorphism invariance (S1), refinement monotonicity (S2), and gap-damped stability (S3). Under this schema, QFI acts as a Lyapunov functional for admissible dynamics, decaying toward fixed points with rates controlled by spectral gaps. We develop spec- tral instruments (tilted operators, cumulant generating functions, large deviation rates) that convert path costs into computable horizons—the Murphy horizon Lc specifying the minimum observations needed to achieve a given confidence. The framework enables idiom translation: different syntaxes (sets, categories, quantum channels, economic sys- tems) realize the same relational invariant through different denominators. We formalize this as a Langlands-style reciprocity, with idioms corresponding to automorphic presen- tations and quotient flows to universal L-objects. The QFI Machine integrates these components into a thermodynamic cycle: Intake (idiom projection), Core (invariant ex- traction), Transmission (drift handling), Exhaust (diagnostic validation). The machine instantiates the Elliott Motive S= R/D, where Ris the relational invariant (shape) and D is the denominator (tempo). Specifying (R,D) determines all observable structure. Slogan. The engine exists to weigh the motive