Golden Junction at φ Emerges in One-Mode Gaussian Covariance Metrics

The work compares four local geometries on centered Gaussian covariances while preserving operational distinctions. In a shared normalization the Bures and Fisher-Rao determinants intersect at the golden ratio squared; introducing the action-matching parameter η pulls the Bures-Wasserstein determinant into coincidence solely at that same coordinate pair. At the junction the Bures cometric takes the explicit diagonal form φ⁻²Π₀ + φΠ₂ whose inverse imposes a scalar-to-weight-two penalty ratio of exactly φ³, distinct from both Fisher-Rao and the normalized Bogoliubov-Kubo-Mori metric.

Radial motion corresponds to Gaussian channels or dilations while weight-two motion is generated by system-only unitaries; the mismatch shows that equal determinant density does not imply tensor equality. The Bogoliubov-Kubo-Mori shape eigenvalue 4ℏ²x/log[(x+1)/(x-1)] stiffens tangentially near the pure-state boundary and therefore never joins the determinant crossing, supplying an independent relative-entropy diagnostic.

These geometric distinctions matter for quantum communication and error correction because local penalties dictate the cost of steering Gaussian resources. The explicit sp(4,ℝ) test rules out an unweighted Frobenius gate norm as a proxy for the golden penalty, indicating that circuit-cost models must incorporate the anisotropic cometric rather than assume isotropy.

A natural strengthening would be a finite-dimensional multimode extension that quantifies how the golden anisotropy scales with mode number and whether it survives under realistic loss channels.