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TOPOLOGICAL VALENCY Extracting Low-Dimensional Bonding Structure from the Electronic Hilbert Space via the Bethe-Projection Principle

Michael Aksman

Published Sep 23, 2026• 0 citations
Publisher

Abstract

The chemical bond should be regarded not primarily as a local feature of the full electron density, but as a low-dimensional correlation structure embedded in the many-electron state.Standard quantum chemistry approaches the many-body problem by expanding the state vector in an exponentially large continuous configuration space (R3MNc ), wherein the discrete nature of valency is obscured by the coordinate representation required for high-accuracy numerical resolution. By applying the Bethe-Projection Principle, we demonstrate that physical interpretability requires extracting a low-dimensional invariant manifold from this ambient space.The bond structure is isolated strictly via the structural projection of the full many-body state onto a chemically selected subspace of entangled atomic modes (MMEAO). Within this framework, valency is rigorously redefined via the number and structure of these selected entangled bonding channels, utilizing von Neumann entropy as a diagnostic of bond entanglement and quantifying structural rigidity through the separation of the occupation spectrum.

DOI
10.5281/zenodo.22928108

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The chemical bond should be regarded not primarily as a local feature of the full electron density, but as a low-dimensional correlation structure embedded in the many-electron state.Standard quantum chemistry approaches the many-body problem by expanding the state vector in an exponentially large continuous configuration space (R3MNc ), wherein the discrete nature of valency is obscured by the coordinate representation required for high-accuracy numerical resolution. By applying the Bethe-Projection Principle, we demonstrate that physical interpretability requires extracting a low-dimensional invariant manifold from this ambient space.The bond structure is isolated strictly via the structural projection of the full many-body state onto a chemically selected subspace of entangled atomic modes (MMEAO). Within this framework, valency is rigorously redefined via the number and structure of these selected entangled bonding channels, utilizing von Neumann entropy as a diagnostic of bond entanglement and quantifying structural rigidity through the separation of the occupation spectrum.
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The fundamental barrier to understanding quantum many-body systems is the exponential explosion of the exact Hilbert space. We demonstrate that structural reality cannot be directly observed in the full wavefunction, but must be extracted by identifying a lowdimensional invariant manifold. We unite four historically distinct mathematical solutions to this dimensionality crisis—the Bethe Ansatz for 1D integrable spin chains, MEAO extraction of the chemical bond, compressed sensing, and the DSM-861 Heegner Ansatz for 3D turbulence—and add a fifth: protein folding on the G861 spectral manifold. All five are instances of the Bethe-Projection Principle. The DSM-861 instance (June 2026) is now fully explicit: the Duality Operator T maps the R2583 vorton phase space onto the 2D Heegner orbifold Γ0(163)\H; the Biot-Savart kernel is the S-matrix; the modular orbit closure h(−163) = 1 plays the role of the Bethe equations; the -1/12 Casimir residue is the conserved charge enforcing the invariant manifold; and ∆j = 7.499 × 10−13 is the quantitativeeigenvalue gap measuring the robustness of the sparse representation.
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This paper proposes a projection-geometric reinterpretation of electronic structure in quantum chemistry. We model the electronic state not as a purely three-dimensional wavefunction, but as a high-dimensional internal state Ψ∈CN\Psi \in \mathbb{C}^NΨ∈CN, whose observable properties arise only after projection into three-dimensional space. Within this framework, the familiar constraints of quantum chemistry---such as the two-electron occupancy limit of an orbital, the discrete shapes of atomic and hybrid orbitals, and the fixed angles of molecular bonds---are understood as consequences of information loss induced by dimensional projection. To formalize this idea, we introduce an analogy with operating-system (OS) state models, in which rich internal process states are compressed into limited external interfaces, producing apparent resource constraints. We show that the structure of electronic orbitals, Pauli exclusion, and molecular bonding can be interpreted as manifestations of a general internal--projection--observation architecture. This structure is closely related to that previously developed in the Twin-Complex Cosmology framework, where cosmological observables arise from the projection of internal dynamical degrees of freedom. The result is a unified conceptual model in which diverse physical and computational phenomena share a common pattern: high-dimensional internal dynamics, a projection map that discards information, and observable behavior that appears constrained or quantized as a result. This suggests that projection geometry may serve as a general explanatory mechanism across multiple domains of physics and information theory.
Quantum chemistry is one of the major potential applications in quantum computation. Currently there is a considerable focus on relatively small active spaces as a consequence of hardware noise and exponential bottlenecks in simulations. In the long run, there will be an increasing demand in reliable approximations for larger systems -- both, as initial states for projective algorithms like the quantum phase estimation or for the evaluation of dynamical properties. While numerous approaches to select active spaces and extrapolate basis set accuracy exist, there is currently no consistent approach that results in a single quantum circuit for the total system. In this work, we combine hybrid Fermionic-Bosonic encodings with the structured approach of Quantum Valence Bond Theory to directly construct quantum circuits for comparably large molecular systems. With this approach we are able to push simulability barrier of variational quantum eigensolvers towards chemically relevant systems and demonstrate circuit designs that outperform active space counterparts and achieve good approximations with respect to the exact solutions.