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

We develop an exact two-access formulation in which a common reciprocal hyperbolic source is mapped into the same complex projective state space CP¹ through two distinct physical access definitions. Reference–idler access yields the bounded readout law δ_A(r) = 2 gd(r), while signal–idler access yields δ_B(r) = gd(2r). Both normalized families lie on the same Bloch-sphere great-circle meridian, but trace different arcs as functions of the common source parameter r. The resulting Fubini–Study pullback metrics are g_A(r) = sech²(r) and g_B(r) = sech²(2r). We derive exact inverse-reconstruction sensitivities and demonstrate that the two readout laws agree to linear order near r = 0 but first separate at cubic order. Fixed, calibrated, lossless, r-independent SU(2) receiver transformations preserve the projective geometry of each encoded family, whereas changing the physical access alters the source-to-projective map. A conditional quantum Fisher information interpretation follows if the normalized doublets are physically realized as pure quantum-state families. We also identify failure conditions involving parameter-dependent receiver transformations, differential gain, detuning, loss, and decoherence. The contribution is a precise mathematical characterization of access-dependent encoding, information-geometric pullbacks, and reconstruction sensitivity, accompanied by falsification and reproducibility criteria. It does not establish a new projective geometry, a universal Fibonacci preference, or an experimentally verified quantum transducer.