This manuscript establishes the first rigorous well-posedness theory for the exact-factorization hybrid quantum–classical closure of Gay-Balmaz and Tronci. The central result, the Covariance-Symplectic Obstruction Theorem, shows the uniformly hyperbolic admissible sector is generically empty, with local well-posedness instead established on a weaker "uniformly symmetrizable" sector via an explicit microlocal symmetrizer and a Friedrichs–Kato existence argument. A companion Rank-Deficiency Theorem classifies six standard device Hamiltonians (transmon-resonator, tunable coupler, Rydberg array, trapped-ion, optomechanical, and Kerr-cat architectures) as excluded from the hyperbolic sector, connecting the abstract theory to concrete qubit platforms.
This manuscript establishes the first rigorous well-posedness theory for the exact-factorization hybrid quantum–classical closure of Gay-Balmaz and Tronci. The central result, the Covariance-Symplectic Obstruction Theorem, shows the uniformly hyperbolic admissible sector is generically empty, with local well-posedness instead established on a weaker "uniformly symmetrizable" sector via an explicit microlocal symmetrizer and a Friedrichs–Kato existence argument. A companion Rank-Deficiency Theorem classifies six standard device Hamiltonians (transmon-resonator, tunable coupler, Rydberg array, trapped-ion, optomechanical, and Kerr-cat architectures) as excluded from the hyperbolic sector, connecting the abstract theory to concrete qubit platforms.
This paper establishes the first rigorous well-posedness theory for the exact-factorization hybrid quantum–classical closure introduced by Gay-Balmaz and Tronci. By constructing an admissible operator-valued state space together with a vacuum control condition that resolves the structural singularity at vanishing classical density, the work proves local existence, uniqueness, and a Beale–Kato–Majda-type continuation criterion, together with sufficient conditions for global-in-time existence under physically relevant hybrid Hamiltonians. Numerical verification across fifteen representative quantum-computing platforms confirms preservation of the hybrid closure, operator positivity, and bounded continuation behavior. These analytical results provide the mathematical foundation for subsequent thermodynamic, geometric, and topological extensions of hybrid quantum–classical dynamics, including the irreversible GENERIC framework developed in the companion paper
This paper extends Uhlmann's parallel-transport geometry for mixed quantum states beyond the conventional full-rank setting to the Global Hybrid Operator Density of a GENERIC-consistent Gay–Balmaz–Tronci hybrid quantum–classical closure, building on the well-posedness and thermodynamic-consistency results established in two companion papers. The central object is a Hybrid Uhlmann connection whose holonomy around a closed recovery cycle yields three main results: the Global Hybrid Winding Theorem, showing that the holonomy spectrum changes if and only if a classical-sector memory kernel is genuinely non-Markovian (sharpened to an exact, non-perturbative statement for a fixed physical direction); an Abelian Blindness theorem, proving the holonomy's determinant channel is exactly and unconditionally insensitive to occupation loss (leakage), even as the underlying state approaches or crosses the rank-deficient boundary; and a Directional Singularity theorem, showing that the full non-Abelian spectrum instead develops a genuine, non-removable singularity there, with the limiting value depending on the direction of approach to the boundary. Together these results show that a single geometric quantity — the holonomy spectrum — can separate non-Markovian memory from leakage, two error mechanisms that existing diagnostics require distinct, purpose-built tools to detect. The theorems are verified on a structure-preserving numerical integration scheme and accompanied by a concrete, falsifiable experimental protocol for superconducting qutrit hardware, extending a recently demonstrated scalar Uhlmann-phase measurement to full non-Abelian holonomy reconstruction.
This paper develops the first thermodynamically consistent GENERIC formulation of the Gay-Balmaz–Tronci hybrid quantum–classical closure by constructing a global hybrid operator density that unifies reversible geometric dynamics with irreversible GKSL and Fokker–Planck dissipation. It establishes rigorous disintegration and completeness theorems for the admissible state space, extends the Berry-curvature-corrected hybrid entropy to mixed conditional quantum states, and proves its Abelian gauge invariance on the non-degenerate admissible manifold. Under an explicitly stated well-posedness hypothesis, the fused hybrid dynamics is shown to satisfy all four GENERIC consistency conditions, while recovering both classical GENERIC and the standard GKSL/Spohn theorem as limiting cases. These results provide a rigorous mathematical framework for nonequilibrium hybrid quantum–classical systems and lay the thermodynamic foundation for future studies of correlated and non-Markovian noise in open quantum devices.
This paper develops a mixed-state geometric framework for hybrid quantum–classical dynamics by constructing a Hybrid Uhlmann connection on the Global Hybrid Operator Density of the GENERIC-consistent Gay-Balmaz–Tronci closure. Building on the Stiefel-bundle parallel transport formulation of Avitabile and Bridges together with an exactly unitary Cayley integrator, it proves the Global Hybrid Winding Theorem, establishing that the Uhlmann holonomy spectrum is invariant under memoryless evolution and changes if and only if the recovery cycle carries non-Markovian memory. The paper further characterizes the rank-deficient leakage boundary, proving that the determinant (abelian) channel is exactly blind to leakage while the full non-abelian holonomy develops an intrinsic directional singularity as occupations vanish. Numerical simulations illustrate these geometric signatures, and the paper proposes a general experimental readout protocol for future implementation across quantum computing architectures.
A Spin-Sector Obstruction to (((7,3,3))_2) Quantum Codes This record contains a focused research manuscript and an extended research version investigating structural obstructions to the existence of binary quantum error-correcting codes with parameters (((7,3,3))_2). The primary manuscript establishes a sector-specific obstruction for codes contained entirely in the total-spin (j=5/2) sector. The proof uses pair-root projectors, weighted quadrupole identities, compression arguments, Cauchy interlacing, and the total-spin Casimir identity. Additional results exclude the sector-pure (j=7/2) and (j=1/2) cases using collective-pair correlation bounds. The record contains two complementary manuscript versions: - Primary manuscript (10 pages): A focused presentation of the principal sector-obstruction results and their proofs.- Extended research manuscript (23 pages): A fuller treatment containing additional derivations, correlation bounds, moment analyses, and supporting mathematical investigations. The deposit also includes the corresponding LaTeX source files and Python scripts for exact-arithmetic verification of selected operator identities and quadrupole-coefficient uniqueness calculations. Scope of the results: These results exclude the specified sector-pure cases. They do not resolve the unrestricted existence problem for (((7,3,3))_2) quantum codes. The sector-pure (j=3/2) case and cases involving mixed total-spin sectors remain unresolved in this work. The verification scripts support reproducibility of the stated algebraic calculations; they should not be interpreted as an independent computer-assisted proof of every theorem in the manuscripts. The primary manuscript is intended as the concise research presentation, while the extended manuscript preserves additional mathematical development and supporting analysis. Both are included to facilitate scholarly scrutiny, reproducibility, and accurate citation. AI assistance : This work was produced with substantial assistance from an AI system (ChatGPT, Claude, Gemini). The proofs have not yet been independently checked by a human specialist. Corrections are welcome.
A companion paper derived the Gaussian sector of quantum mechanics from a Mediated Access postulate and left open whether geometric quantization of the compact mediating manifold completes the theory beyond it — the Completion Conjecture. We discharge it by computation. A composition theorem makes the tensor-product rule a consequence of the channel's product structure; a closure theorem makes the reduced dynamics exactly unitary block by block; and two interface premises reduce to a generic number-theoretic condition and a checkable curvature condition. The beyond-Gaussian statistics match the quantized sector, and the conjecture's refutation condition is not met. Keywords: quantum foundations, reconstruction of quantum theory, geometric quantization, tensor product structure, unitarity, symplectic geometry, Kähler manifolds, quant-ph, math-ph