Connection and Metric Reconstruction from Gauge-Covariant Relational Kernels Local and Global Reconstruction, Standard Recovery, No-Go Results, and the Reconstruction-Residue Program
We develop a reconstruction framework in which local connection, curvature, holonomy, and positive-semidefinite metric data are extracted from a sufficiently regular complex relational kernel transforming under the endpoint gauge law . For kernels nonvanishing near the diagonal, the first derivative of the phase in the second endpoint defines a local Abelian connection, which obeys. Compatible local kernels patch to a global line-bundle connection, curvature, and holonomy. When is Hermitian and positive semidefinite, diagonal normalization gives , and the mixed logarithmic Hessian of the normalized magnitude defines a gauge-invariant positive-semidefinite relational metric, The full complex relational Hessian decomposes. For state-overlap kernels the construction exactly recovers the Berry connection, Berry curvature, Fubini–Study metric, and quantum-geometric tensor. We also prove local reconstruction stability away from kernel zeros, show that reconstruction is generally noninjective because only finite diagonal jets determine the local geometry, and establish that a smooth second-order scalar-kernel Hessian yields Riemannian rather than genuinely non-Riemannian Finsler structure. The paper separates proven reconstruction, standard recovery, interpretive extension, and open proper-extension targets.
We develop a reconstruction framework in which local connection, curvature, holonomy, and positive-semidefinite metric data are extracted from a sufficiently regular complex relational kernel transforming under the endpoint gauge law . For kernels nonvanishing near the diagonal, the first derivative of the phase in the second endpoint defines a local Abelian connection, which obeys. Compatible local kernels patch to a global line-bundle connection, curvature, and holonomy. When is Hermitian and positive semidefinite, diagonal normalization gives , and the mixed logarithmic Hessian of the normalized magnitude defines a gauge-invariant positive-semidefinite relational metric, The full complex relational Hessian decomposes. For state-overlap kernels the construction exactly recovers the Berry connection, Berry curvature, Fubini–Study metric, and quantum-geometric tensor. We also prove local reconstruction stability away from kernel zeros, show that reconstruction is generally noninjective because only finite diagonal jets determine the local geometry, and establish that a smooth second-order scalar-kernel Hessian yields Riemannian rather than genuinely non-Riemannian Finsler structure. The paper separates proven reconstruction, standard recovery, interpretive extension, and open proper-extension targets.
We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as {\it stretching } and {\it rotation } components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence ∼∣εμ∣−2, while the Berry curvature shows a weaker, subleading divergence ∼∣εμ∣−3/2, with εμ denoting the parameter displacement from the EP along an individual parameter axis μ. Finally, we examine physical realizations of this model, including the non-Hermitian Su--Schrieffer--Heeger (SSH) and Hatano--Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.
We prove a controlled completion of dual-affine entropy geometry connecting a supplied amplitude-carrier complex structure, constructed tangent-bundle Born complex structures, non-Abelian compact transport, and endpoint action through typed relative-entropy reconstruction and defects. On the supplied Hermitian carrier, Umegaki relative entropy reconstructs the entropy potential up to affine gauge and the relative logarithmic cotangent class; diagonal jets yield the Bogoliubov-Kubo-Mori (BKM) metric, Hessian cubic, and dual connections, which construct Born lifts on real doubled statistical carriers. Determinant normalization selects a principal-log traceless Gibbs representative. A quantum channel induces support-reduced output geometry and a Bregman loss whose Hessian and cubic are metric and response losses with sequential cocycles; one CPTP map recovering every state of an embedded faithful model annihilates its restricted loss and intertwines its intrinsic geometry and holonomy. Proximal families yield Rényi and Hopf-Lax identities; BKM relaxation needs mobility and scale. From a separately chosen affine-invariant metric, the compact branch constructs canonical principal SU(n) and associated tangent PSU(n) transport, compared with BKM transport by connection and curvature defects rather than conjugacy. The indexed architecture is dimension-stratified: its general branches are available for each fixed finite n≥2 under their stated inputs, full BKM tangent holonomy is proved only for n≥3, and the qutrit and independently graded AIII packages occur only at n=3 and n=5, respectively. For qutrits, the tracial BKM metric-cubic pair has oriented stabilizer PSU(3); alignment with normalized affine-invariant holonomy lets the central cubic seed a distinct, unique parallel comparison cubic defining a global tangent-frame reduction. A fixed action conversion gives an exact endpoint canonical relation; the parameterized Rényi-Moreau family gives a branch-local scale-Hamiltonian isotopy. Mechanical interpretation needs inertia and a clock. Relative entropy does not construct the amplitude-carrier complex structure or supply physical calibrations.
An isolated group of k bands in an N-level quantum system defines a rank-k spectral projector and hence a map into the complex Grassmannian Gr(k,N). For a smooth gapped Hamiltonian, this projector is globally smooth and periodic even when band topology obstructs a globally smooth periodic, or symmetry-compatible, Bloch frame. We take the globally defined differential $\dd P$ as the central object: it is the tangent field of the Grassmannian map, removes unphysical rotations within the selected subspace, and retains the physical interband transition. The interband block of this tangent data simultaneously determines the quantum metric and Berry curvature; the associated horizontal generator produces finite Grassmannian motion, while wedge products of $\dd P$ enter topological forms. We construct a shortest path between two projectors in the ambient Grassmannian and show that the singular values of its horizontal generator block are the principal angles between the endpoint subspaces. This construction provides a piecewise-geodesic interpretation of discrete geometric phases. In a separate development, we derive a basis-independent expression for the determinant of a multiband Wilson loop from traces of powers of an ordered projector product, without decomposing a degenerate band multiplet into individual bands. Finally, the same tangent-vector calculus organizes Chern characters, chiral winding numbers, and the time-reversal Z2 index. The resulting framework unifies local quantum geometry, finite subspace distance, holonomy, and topology while avoiding global gauge fixing.
We develop a GNS-based construction of geometric tensors on smooth parametric statistical models over C∗-algebras. Since the state space of a C∗-algebra is generally not a smooth manifold, the construction does not rely on pulling back tensors from an ambient state manifold. Instead, the GNS Hilbert spaces and their duals are organized into non-locally-trivial Hilbert fibrations over the state space. For models satisfying a compatibility condition expressing derivatives of expectation values as continuous functionals on the realified GNS fibers, each tangent vector admits a canonical dual GNS representative. Pulling back the dual GNS Hermitian product along the corresponding canonical lift produces a Hermitian tensor K on the complexified tangent bundle of the model, whose real and imaginary parts define, under suitable regularity assumptions, a smooth weak Riemannian metric tensor G and a smooth two-form Ω. In finite-dimensional parameter manifolds the metric is, of course, strong. The construction recovers the Fisher--Rao metric in the commutative dominated case, the Fubini--Study geometry for pure states up to the normalization and sign convention imposed by the dual GNS pairing, and the SLD metric for faithful quantum states. In finite-dimensional faithful models, the two-form Ω is proportional, up to convention, to the expected commutator of the SLD representatives, equivalently to the mean Uhlmann curvature. We show through faithful qubits and displaced thermal states that Ω need not be closed. For bundle-regular models, the associated fiberwise symplectic form on the real dual GNS bundle admits connection-dependent closed extensions to the total space, while closedness of Ω on the parameter manifold is controlled by the covariant exterior derivative of the canonical real dual GNS lift.
This paper develops a constructive local-to-global extension of the Complementary Relational Principle (CRP) and Complementary Relational Dynamics (CRD). It investigates how local two-input relations can be assembled into omnidirectional networks with consistent comparison, observation, reduction, and dynamical evolution. Directional states are indexed by the circle S¹, the sphere S², or the frame-rotation group SO(3). These angular parameter spaces are distinguished from physical spatial dimensions. Positive two-direction kernels provide a common relational representation, while cocycle consistency and connection-based transport specify how locally represented states are compared. The mathematical constructions include positivity-preserving finite directional sampling, gauge-invariant transported relations, reference-routed kernels in the presence of curvature, spherical-harmonic reduction, and quantitative memory corrections under weak anisotropic coupling. Exact block elimination produces a joint direction–time memory kernel, and a contraction argument constructs a causal nonlinear memory functional under stated hypotheses. Two physically distinct applications are developed: visibility-weighted multiview reconstruction with error and covariance bounds, and coherent interference through quadratic detector readouts. An observation-fiber criterion characterizes autonomous reduced dynamics, while a two-state error bound quantifies the predictive consequences of discarded information. A specified kernel–connection feedback model supplies a globally well-posed, positivity- and trace-preserving dynamical core. The paper also distinguishes ordinary 360-degree complex-phase return from spinorial 720-degree frame return. The latter is constructed through an associated SU(2) spinor bundle and becomes observable as a relative sign against a coherent reference; it is not inferred from the imaginary unit alone. These results establish a modular relation–observation–promotion architecture without identifying stereopsis, interference, gauge transport, and fluid-mode reduction as the same physical law. The paper is a companion to “Complementary Relational Dynamics and Three-Dimensional Closure” and preserves its conditional Hodge-degree argument.
Preprint; submitted to Journal of Mathematical Physics. We introduce Self-Fibre, a finite-dimensional mathematical framework in which relational structures between local quantum systems (“fibres”) are not postulated but selected from local fibre data through a variational reconstruction principle. Given a finite collection of local Hilbert-space fibre states, we define a reconstruction functional over candidate relational graphs balancing informational compatibility, pairwise reconstruction consistency, and structural complexity. We prove that this functional attains a minimum on the finite candidate space, and that the minimizer is unique under a reconstruction-gap condition. For finite systems of pure qubit fibres, we explicitly construct induced fibre comparison maps. For every reconstructible pair of fibre states, these maps are the unique minimal Hilbert–Schmidt-distance unitaries consistent with the pair. We further prove that the associated closedcycle holonomy satisfies an exact discrete identity expressing the global reconstruction obstruction as a sum of local composition defects transported around the cycle. This obstruction quantifies the failure of pairwise-consistent local reconstructions to form a globally consistent comparison structure. All results are established within the finite pure-qubit setting. We do not assume or derive a spacetime manifold, metric structure, or continuum limit. The reconstruction obstruction is proposed only as a candidate discrete precursor to geometric curvature, with any continuum correspondence left as an open problem.