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 at\(x_0\simeq1.124307\). 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 \(\eta=\sqrt\varphi\), the junctionovertakes the raw optimum for mismatch sensitivity\(\lambda>\lambda_c\simeq0.08942\) and becomes the unique fully transmittedstate in the hard-matching limit. At the imported junction\((x,\eta)=(\sqrt\varphi,\sqrt\varphi)\), the relative Bures cometric is\(\varphi^{-2}\Pi_0+\varphi\Pi_2\), giving the inversescalar-to-weight-two cost ratio \(\varphi^3\). The claim concernsquantum-geometric organization of an active macroscopic register, notmicroscopic neural coherence, quantum-computational speedup, or persistentmemory.
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 at(x_0\simeq1.124307). 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 (\eta=\sqrt\varphi), the junctionovertakes the raw optimum for mismatch sensitivity(\lambda>\lambda_c\simeq0.08942) and becomes the unique fully transmittedstate in the hard-matching limit. At the imported junction((x,\eta)=(\sqrt\varphi,\sqrt\varphi)), the relative Bures cometric is(\varphi^{-2}\Pi_0+\varphi\Pi_2), giving the inversescalar-to-weight-two cost ratio (\varphi^3). The claim concernsquantum-geometric organization of an active macroscopic register, notmicroscopic neural coherence, quantum-computational speedup, or persistentmemory.
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.
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 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 x2=φ. Afterintroducing a constant action-matching parameter η, theBures--Wasserstein determinant joins them only at(x,η)=(φ,φ). Equal determinant density doesnot mean tensor equality: relative to Fisher--Rao, the Bures cometric at thejunction isQgold=φ−2Π0+φΠ2,detQgold=1,so its inverse has the exact scalar-to-weight-two quadratic penalty ratioφ3. The normalized Bogoliubov--Kubo--Mori metric supplies anindependent relative-entropy diagnostic. It agrees radially with Bures buthas shape cometric eigenvalue4ℏ2x/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 explicitsp(4,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.
We formulate a conditional geometric framework for a phase bypass of a zero sector. The calibrated order parameter is a smooth section of a Hermitian complex line bundle, not a complete normalized quantum state. At a transverse zero, its normalized section has no canonical continuous extension to a full neighbourhood in the original space, whereas its pullback extends uniquely and smoothly to the exceptional boundary of the real oriented blow-up. The boundary value is determined by the vertical derivative and records the incoming normal direction; it does not select an outgoing path. On a controlled scattering component, a Hamiltonian flow supplies the missing dynamical datum under explicit transversality, two-sided non-trapping, completeness, and uniqueness assumptions. The resulting first-exit map is a symplectomorphism and is C1 persistent under small C2 perturbations of shifted Hamiltonians with uniform C3 control and quantitative regularity, flight-time, boundary-separation, and intersection-isolation margins. A Koopman–cocycle operator transports an internal fibre over this map but remains a classical construction unless a separate quantum realization is supplied. For a symmetry-constrained cubic exceptional-point slice, analytic continuation gives exact C3 monodromy, and a companion generator admits a formal GKSL-compatible no-jump embedding after a uniform loss shift. An engineered retain/erase benchmark provides programmed positive control and independently calibrated erase and lifted-EP3 nulls. The register-free bridge forbids deliberate injection of the orientation label, defines a natural carrier as minimal process memory across a certified causal break, and separates Hilbert-space spectral subspaces from Liouville-space operator modes. Exact memory identification is stated as a reversible full-algebra sufficiency criterion. A separate approximate theorem converts finite recovery and predictive-factorization errors into quantitative information and output-distribution bounds. A unified analytic benchmark derives the section, first-exit dynamics, resolved-to-control map, spectral family, and Kato transport from one calibrated joint model. These results define a falsifiable certification programme; they do not claim a completed hardware realization or a universal law of phase bypass.
We extend a finite-dimensional Araki–Bogoliubov–Kubo–Mori (BKM) responsecost construction to an arbitrary real Hilbert response space, normal quantumcoarse-graining channels between von Neumann algebras, composable channel chains, and continuous coarse-graining flows. The response map is a boundedoperator Rτ : hτ → K on the BKM tangent Hilbert space at resolution τ , withcovariance operator Γτ = RτR∗τ. For the accessible tangent vτ and responsebτ = Rτ vτ , the exact minimum resolved cost is the closed quadratic form qΓ†τ(bτ ) = ∥Γ−1/2τbτ∥2, rather than the operator expression ⟨bτ , Γ†τ bτ ⟩ unlessthe stronger operator-domain condition holds. A normal channel induces,under explicit second-order regularity assumptions, a contraction of BKMtangent Hilbert spaces. Its defect is positive and composes by an exact chain rule. For a right-continuous coarse-graining evolution the retained BKMinformation is monotone and determines a positive Stieltjes loss measure. These ingredients give the master relationg0(v, v) = qΓ†τ(bτ ) + ∥Pker Rτ vτ∥2hτ + µlossv((0, τ ]).When the loss measure is absolutely continuous, the last term is R τ0 Jinfo,v(s) ds with Jinfo,v ≥ 0. BKM-continuous response functionals allow the same theoremto treat suitable unbounded or smeared QFT observables without requiringthe observables themselves to be bounded operators. We give explicit specializations and first quantitative applications in four sectors. In Yang–Mills