Bures--Wasserstein covariance dynamics admits a conservative Hamiltonian parent whose Eisenhart--Duval lift defines a Bargmann geometry. After passing to the conformal frame associated with the parabolic $\mathfrak z=2$ scaling of the free covariance flow, we study a Schr\"odinger-type temporal deformation of this lifted geometry. The null charge of the Bargmann lift acts as an informational mass. For nonzero temporal deformation and nonzero informational mass, accessible massive null geodesics cannot reach the boundary of absolute certainty, where the covariance scale vanishes, but instead possess a finite inner turning point. Reinstating a logarithmic spectral-floor potential shows that this kinetic mass gap and the substrate capacity floor combine into a single effective radial potential, with the operative cutoff set by the larger of the two radii. Transverse squeezing momentum supplies an outer confinement and yields a loading bound: a confined orbit exists only while the conserved energy dominates the transverse momentum, and when that bound fails the radial well disappears through the merger of its turning points. We interpret this result, conditionally, as a geometric model of finite stable processing intervals in information-processing substrates, while restricting the formal claims to centred covariance dynamics in a local neighbourhood of the isotropic ray of the positive-definite cone.
Bures--Wasserstein covariance dynamics admits a conservative Hamiltonian parent whose Eisenhart--Duval lift defines a Bargmann geometry. After passing to the conformal frame associated with the parabolic z=2 scaling of the free covariance flow, we study a Schr"odinger-type temporal deformation of this lifted geometry. The null charge of the Bargmann lift acts as an informational mass. For nonzero temporal deformation and nonzero informational mass, accessible massive null geodesics cannot reach the boundary of absolute certainty, where the covariance scale vanishes, but instead possess a finite inner turning point. Reinstating a logarithmic spectral-floor potential shows that this kinetic mass gap and the substrate capacity floor combine into a single effective radial potential, with the operative cutoff set by the larger of the two radii. Transverse squeezing momentum supplies an outer confinement and yields a loading bound: a confined orbit exists only while the conserved energy dominates the transverse momentum, and when that bound fails the radial well disappears through the merger of its turning points. We interpret this result, conditionally, as a geometric model of finite stable processing intervals in information-processing substrates, while restricting the formal claims to centred covariance dynamics in a local neighbourhood of the isotropic ray of the positive-definite cone.
Covariance dynamics on the positive-definite cone are commonly described by gradient flows, which encode dissipative relaxation but obscure the underlying phase-space structure. We construct a finite-dimensional Hamiltonian lift of covariance dynamics on Symn+ equipped with the Bures--Wasserstein metric. The natural mechanical Lagrangian yields canonical momentum Π=21LΣ[Σ˙], where LΣ is the Lyapunov operator, and explicit Hamiltonian H(Σ,Π)=2tr(ΠΣΠ)+V(Σ). Adding Rayleigh dissipation recovers the Bures--Wasserstein gradient flow in the overdamped limit. For a spectral-floor and trace-control potential, the quadratic fluctuation Hamiltonian around the isotropic equilibrium separates trace and traceless modes; the baseline stiffness diverges as (s−ν)−2 as the equilibrium covariance approaches the floor. The construction identifies a conservative parent system for constrained Bures--Wasserstein covariance relaxation and fixes the local stiffness scale induced by the spectral floor.
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.
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.
The Bures--Wasserstein (BW) covariance programme developed in a series of recent preprints presents its dynamics as capacity-constrained gradient flows on the cone of positive-definite matrices. We show that the same dynamics admit a variational formulation with explicit symplectic structure. We construct a conservative action on the BW manifold, extended by capacity parameters $(\numin, \Qmax, \Dfixed)$ entering as couplings, and show that its Euler--Lagrange equations reproduce the BW dynamics after a standard Legendre transform. The symplectic form on the associated phase space restricts cleanly to the double-quantum (DQ) and zero-quantum (ZQ) subspaces previously identified in the algebraic sector classification of the programme. We prove that the quadratic part of the Hamiltonian contains a†a†+aa pairing terms in the DQ sector and bilinear mixing terms in the ZQ sector, and that these algebraic structures force SU(1,1) and SU(2) respectively as the kinematic symmetry groups. The sector classification is therefore not ad hoc but a consequence of the variational structure. The resulting action remains symplectically well posed across admissible capacity couplings and provides the conservative foundation for later dissipative and boundary-theoretic extensions.
Building on the recently established decomposition of the Gaussian Bures cometric into a calibrated Fisher term and a state-independent Killing-form correction, we investigate the material realization and thermodynamic stabilization of constitutive Bures geometry. The associated Schur realization permits an independently specified action scale, whose material calibration remains a separate physical requirement. First, we show that passive static elimination of a positive symmetric auxiliary block cannot produce the required mixed-sign correction to the fixed Fisher baseline.A successful realization must therefore depart from at least one assumption of this construction. We then formulate identification of the Bures cometric as an explicit control-matching problem demanding an exact tensor match. For an active neural--flow candidate combining a cardiac pressure command, flow feedback and vascular stiffness modulation, a specified quadratic-probe diagnostic reveals limited local reachability and persistent response asymmetry in the tested parameter range. This motivates a search for independently effective controls, consistent with Ashby's principle of requisite variety.Separately, on the prescribed constitutive covariance domain, we construct an explicit GENERIC stabilization framework. Combining Hamiltonian Bures-Wasserstein kinetics, reservoir friction and a separate positive regulatory mobility, we prove local exponential stability of a prescribed interior covariance under explicit positivity and coupling conditions.While a matching physiological realization remains to be established, the framework provides a thermodynamically admissible architecture for regulating covariance dynamics on a prescribed constitutive Bures domain, opening a concrete route to investigate biological information processing without assuming a microscopic quantum substrate. Compact rotation is retained separately from covariance motion, distinguishing reversible circulation from driven dissipative circulation. Finally, a golden operating window is motivated by determinant equality and conditional gate-cost matching; a computational advantage remains a task-dependent hypothesis.
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.