Two Fibonacci Butterflies: Covariant Gudermannian Readout, Projective Geometry and Conditional Quantum Fisher Information

Abstract

We develop a precise and falsifiable two-route formulation in which the same real hyperbolic source, r ∈ R, is accessed through two physically distinct encodings and mapped into the same projective geometry. The two access laws are δ_A(r) = −2 gd(r) and δ_B(r) = −gd(2r), where gd(r) is the Gudermannian function. Both routes therefore lie on the same great-circle meridian of the Bloch sphere, but they traverse different arcs of that meridian with different r-parameterizations. A fixed calibrated receiver-frame change acts as a fixed SU(2) rotation that preserves the projective point and the Fubini–Study geometry, whereas a change of physical access can modify the encoding law itself and thereby change the readout law, the pullback metric and the inverse reconstruction. For both routes we compute the pullback Fubini–Study metric, derive exact inverse-reconstruction formulas, and analyze the associated conditioning. The two accesses are indistinguishable at linear order around r = 0 but separate at cubic order, providing a direct nonlinear discriminator between the two encodings. When the normalized doublets are physically realized as pure quantum-state families, the corresponding quantum Fisher information is conditionally given by F_Q(r) = 4g_FS(r). The manuscript also examines fixed and parameter-dependent receiver transformations, differential gain, phase detuning, asymptotic conditioning, mixed-state limitations and related failure modes. The result is an exact mathematical characterization within the declared two-port model rather than an exhaustive classification of all possible physical accesses.