We compare four local geometries of centered one-mode bosonic Gaussiancovariances while keeping their operational meanings distinct. In a commontrace-orthonormal normalization, the Bures and covariance Fisher--Raocometric determinants cross intrinsically at \(x^2=\varphi\). Afterintroducing a constant action-matching parameter \(\eta\), theBures--Wasserstein determinant joins them only at\((x,\eta)=(\sqrt\varphi,\sqrt\varphi)\). Equal determinant density doesnot mean tensor equality: relative to Fisher--Rao, the Bures cometric at thejunction is\[ Q_{\rm gold}=\varphi^{-2}\Pi_0+\varphi\Pi_2, \qquad \det Q_{\rm gold}=1,\]so its inverse has the exact scalar-to-weight-two quadratic penalty ratio\(\varphi^3\). The normalized Bogoliubov--Kubo--Mori metric supplies anindependent relative-entropy diagnostic. It agrees radially with Bures buthas shape cometric eigenvalue\(4\hbar^2x/\log[(x+1)/(x-1)]\); it therefore does not form a fourthdeterminant junction and becomes tangentially stiff at the pure-stateboundary. Operationally, weight-two motion is generated by a system-onlyGaussian unitary, whereas radial motion requires a Gaussian channel orancillary dilation. The known Uhlmann--Ruan purification quotient realizesthe local Bures state metric, but an explicit\(\mathfrak{sp}(4,\mathbb R)\) test shows that an unweighted right-invariantFrobenius gate norm does not reproduce the golden penalty. Conditionalbranch selection and finite-rate Bures--Wasserstein cost separation arepresented as secondary statements, not as dynamical phase transitions.
We compare four local geometries of centered one-mode bosonic Gaussiancovariances while keeping their operational meanings distinct. In a commontrace-orthonormal normalization, the Bures and covariance Fisher--Raocometric determinants cross intrinsically at (x^2=\varphi). Afterintroducing a constant action-matching parameter (\eta), theBures--Wasserstein determinant joins them only at((x,\eta)=(\sqrt\varphi,\sqrt\varphi)). Equal determinant density doesnot mean tensor equality: relative to Fisher--Rao, the Bures cometric at thejunction is[ Q_{\rm gold}=\varphi^{-2}\Pi_0+\varphi\Pi_2, \qquad \det Q_{\rm gold}=1,]so its inverse has the exact scalar-to-weight-two quadratic penalty ratio(\varphi^3). The normalized Bogoliubov--Kubo--Mori metric supplies anindependent relative-entropy diagnostic. It agrees radially with Bures buthas shape cometric eigenvalue(4\hbar^2x/\log[(x+1)/(x-1)]); it therefore does not form a fourthdeterminant junction and becomes tangentially stiff at the pure-stateboundary. Operationally, weight-two motion is generated by a system-onlyGaussian unitary, whereas radial motion requires a Gaussian channel orancillary dilation. The known Uhlmann--Ruan purification quotient realizesthe local Bures state metric, but an explicit(\mathfrak{sp}(4,\mathbb R)) test shows that an unweighted right-invariantFrobenius gate norm does not reproduce the golden penalty. Conditionalbranch selection and finite-rate Bures--Wasserstein cost separation arepresented as secondary statements, not as dynamical phase transitions.
We construct the covariance-sector Bures geometry of centered bosonic Gaussian states from the Gaussian symmetric-logarithmic-derivative equation and determine its quantum-admissible domain. At the Williamson uncertainty floor, the boundary geometry is anisotropic: spectrum-changing radial coefficients diverge, whereas squeezing and rotation directions along the pure Gaussian orbit remain finite. Dually, the covariance cometric acquires an exact m2-dimensional kernel when m modes become pure, yet the boundary remains at finite radial Bures distance. We derive the potential Φ=−21∑klog(νk2−41), which generates fixed-frame covariance dilation and equals βF on fixed-Hamiltonian thermal families. At large symplectic eigenvalue, the relative quantum correction is O(ℏ2/ν2), recovering covariance Fisher--Rao geometry. This intrinsic classical regime differs from the pure-state boundary, where classical statistics require a specified measurement channel. As a secondary comparison, the normalized Bures, Fisher--Rao, and action-matched Bures--Wasserstein determinant densities possess an exact one-mode junction at (x,η)=(φ,φ),x=2ν/ℏ. This junction is kinematic, and its branch interpretation is conditional. At a prescribed finite nonzero compression rate, the Bures action diverges while the Bures--Wasserstein cost remains finite; a bounded Bures budget instead enforces radial deceleration.
Information distance and circuit complexity are both obtained by minimizing lengths, but they minimize over different objects. We make this distinction explicit for faithful one-mode Gaussian states. First, invariant-form uniqueness implies that no positive-definite quadratic gate cost can be invariant under the full adjoint action of the noncompact symplectic group; a positive Cartan majorant necessarily introduces additional reference data. The Uhlmann purification quotient realizes the Bures metric, and the radial covariance direction requires a system-ancilla coupling because system-only Gaussian unitaries preserve the Williamson eigenvalue. We then minimize fixed right-invariant quadratic norms on the minimal two-mode Gaussian gate algebra sp(4,R). For the unweighted Frobenius norm, the quotient coefficients for radial and traceless covariance tangents are G0=[ℏ2(u−1)]−1 and G2=[ℏ2(3u−1)]−1, where u=(2ν/ℏ)2. Their ratio does not equal the Bures ratio. The radial coefficient, however, reproduces the Bures value exactly at every u; the mismatch is confined to the traceless sector. More generally, a constant block-diagonal two-weight schedule gives G0/G2=1+2(β/α)u/(u−1); matching Bures throughout the isotropic family would require the state-dependent relation β/α=1/u. At the Bures-Fisher determinant crossing u=φ, pointwise matching is possible only by inserting β/α=φ−1. Thus the Bures purification quotient is an exact state-geometric cost, but it is neither an unweighted symplectic gate cost nor a member of this fixed two-weight Nielsen family. The existence of a more general fixed positive gate norm realizing the quotient remains open.
We construct the covariance-sector Bures geometry of centered bosonic Gaussianstates directly from the Gaussian symmetric-logarithmic-derivative equationand determine its global quantum domain. The Robertson--Schr\"odingercondition bounds this domain below by the Williamson uncertainty floor. Itsboundary geometry is anisotropic: spectrum-changing radial coefficientsdiverge, whereas squeezing and rotation coefficients remain finite along thepure Gaussian orbit. Dually, the covariance cometric develops an exactm2-dimensional kernel when m Williamson modes are pure, although thefloor remains at finite radial Bures distance. We derive the radial potentialΦ=−21∑klog(νk2−41), which generates covariancedilation on fixed-Williamson-frame radial submanifolds and satisfiesΦ=βF on fixed-Hamiltonian thermal families. At large symplecticeigenvalue the relative symplectic correction is suppressed asO(ℏ2/ν2), yielding an intrinsic high-noise recovery of covarianceFisher--Rao geometry. This limit is distinct from the Williamson floor, whichis a quantum pure-state boundary rather than a classical limit. A classicalmodel at the floor instead requires an explicitly supplied measurement channelor commutative restriction; the boundary geometry constrains the structure onwhich that operational map acts but does not select the map itself. As asecondary comparison, we place the Gaussian Bures geometry alongside theFisher--Rao and Bures--Wasserstein covariance geometries. After introducing aconstant action-matching parameter η, their normalized determinantdensities possess an exact one-mode junction at(x,η)=(φ,φ). The equality is kinematic: it isneither an additional boundary nor a dynamical transition, and branch selectionrequires an explicitly stated minimum-log-determinant hypothesis. For the sameprescribed finite-rate radial compression toward the floor, the Bures kineticdensity and action diverge while the Bures--Wasserstein cost remains finite;under a bounded Bures kinetic budget, the intrinsic consequence is radialdeceleration rather than a forced change of geometry.
For centered bosonic Gaussian states, the covariance pullback of the Bures cometric differs from the lift-normalized classical covariance Fisher--Rao cometric by a state-independent bilinear form. Under the canonical identification of symmetric covariance covectors with sp(2N,R), this form is the trace form and hence a fixed multiple of the Killing form. We prove that, within the normalized symmetric Petz family, Bures is uniquely selected by requiring such an additive state-independent symplectic correction. We determine the full Williamson-frame spectrum and show that a boundary stratum with m pure modes has an m2-dimensional cometric kernel isomorphic to u(m), while the pure Gaussian orbit remains nondegenerate. For radial one-mode estimation, ideal heterodyne detection accesses the exact fraction (ν−ℏ/2)/(ν+ℏ/2) of the SLD quantum Fisher information. Its vanishing boundary limit reflects finite heterodyne information relative to a divergent radial quantum Fisher information, not zero measurement information. Finally, a minimal Schur realization fixes the auxiliary inertia and identifies the second Schur complement as the true covariant vertical block.
We formulate a geometric hypothesis for state-dependent writing and readoutas the operational step of a constructive chain. Prior geometric work identifiesthe covariance-independent quantum correction absent from classicalFisher--Rao covariance geometry as the trace/Killing form of the symplecticalgebra. A related thermodynamic construction shows that a symmetry-constrainedGaussian GENERIC neural--flow cell can realize the resulting Bures geometry,admit inverse Bogoliubov--Kubo--Mori (BKM) mobility, and maintain anon-Planckian dynamical Williamson floor. Taking that realized cell asinput, we derive an active transverse-processing window. Around an isotropicone-mode covariance, the aforementioned quantum correction generates Buresscalar/weight-two selectivity while inverse BKM supplies complementarymobility. Their product has a unique interior optimum atx0≃1.124307. The imported exact golden junction retains 92.34% ofthis raw maximum while matching normalized fidelity, inferential, andtransport costs. Under the Optimal Cognitive Bottleneck Postulate and anindependently calibrated transport scale η=φ, the junctionovertakes the raw optimum for mismatch sensitivityλ>λc≃0.08942 and becomes the unique fully transmittedstate in the hard-matching limit. At the imported junction(x,η)=(φ,φ), the relative Bures cometric isφ−2Π0+φΠ2, giving the inversescalar-to-weight-two cost ratio φ3. The claim concernsquantum-geometric organization of an active macroscopic register, notmicroscopic neural coherence, quantum-computational speedup, or persistentmemory.
We develop an operator-algebraic framework for modular self-duality, symmetrized relative entropy, and Bogoliubov--Kubo--Mori (BKM) susceptibility of local states in quantum field theory. In finite dimensions, modular self-duality singles out fixed points at which a state coincides with its modularly reflected partner. At such points, the natural comparison functional is the symmetrized Umegaki relative entropy. It vanishes at coincidence, and its Hessian is governed by the BKM quantum Fisher information along the reflected tangent direction. We then extend this fixed-point construction to the local type~III von Neumann algebras that arise in quantum field theory. Here, a local state is compared with the modular pullback of its commutant restriction, and the intrinsic comparison functional is the symmetrized Araki relative entropy. For sufficiently regular state deformations, the fixed-localization Hessian at the self-dual point defines a type~III BKM susceptibility. This coefficient is obtained by evaluating the BKM bilinear form on the tangent selected by the modular pairing. Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral U(1) current on half-line algebras. In both examples, the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations. We also apply the construction to a ball-shaped boundary region in the vacuum state of a conformal field theory with a semiclassical anti-de Sitter (AdS) gravitational dual. For linear bulk coherent excitations in an exact complementary-recovery model of this sector, bulk--boundary modular intertwining identifies the modular partner with the AdS--Rindler-reflected coherent state, and the symmetrized relative entropy is exactly quadratic.