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