For any $N$-fermion system whose single-particle orbitals are products of spherical harmonics and arbitrary radial functions, the fraction of nonzero two-body matrix elements is determined solely by angular momentum selection rules and is independent of the radial potential. Exact enumeration of the Gaunt coefficients yields the angular ERI density $D(l_{\max})$ under the Coulomb selection rule (conservation of total magnetic quantum number, $m_a+m_b=m_c+m_d$): $14.84%$ at $l_{\max}=1$, $8.52%$ at $l_{\max}=2$, $6.06%$ at $l_{\max}=3$, $4.83%$ at $l_{\max}=4$, and $3.99%$ at $l_{\max}=5$. At $l_{\max}=2$, $91.48%$ of all possible two-body matrix elements are guaranteed to vanish by the Wigner–Eckart theorem before any property of the potential is invoked. A stricter pair-diagonal restriction ($m_a=m_c$, $m_b=m_d$), applicable to axially-symmetric or $m$-decoupled bases, gives a further reduction ($2.76%$ at $l_{\max}=2$, $1.44%$ at $l_{\max}=3$). The bound is independent of the number of radial quantum numbers retained at fixed $l_{\max}$, and therefore independent of qubit count. Numerical verification across five potentials (Coulomb, harmonic oscillator, Woods–Saxon, square well, Yukawa) confirms the theorem: angular sparsity patterns are bit-identical across all potentials at matched orbital counts. The radial-angular factorization and the Gaunt selection rules are standard results from Condon & Shortley and Slater; the contribution of this paper is the explicit quantitative bound at each $l_{\max}$, framed as a universal resource guarantee for qubit Hamiltonian sparsity that applies to electronic structure, nuclear shell models, quantum dots, and any other spherical-potential fermion system. GeoVac Paper 22. Part of the GeoVac research corpus (mathematical foundations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
For any N-fermion system whose single-particle orbitals are products of spherical harmonics and arbitrary radial functions, the fraction of nonzero two-body matrix elements is determined solely by angular momentum selection rules and is independent of the radial potential. Exact enumeration of the Gaunt coefficients yields the angular ERI density D(lmax) under the Coulomb selection rule (conservation of total magnetic quantum number, ma+mb=mc+md): 14.84 at lmax=1, 8.52 at lmax=2, 6.06 at lmax=3, 4.83 at lmax=4, and 3.99 at lmax=5. At lmax=2, 91.48 of all possible two-body matrix elements are guaranteed to vanish by the Wigner–Eckart theorem before any property of the potential is invoked. A stricter pair-diagonal restriction (ma=mc, mb=md), applicable to axially-symmetric or m-decoupled bases, gives a further reduction (2.76 at lmax=2, 1.44 at lmax=3). The bound is independent of the number of radial quantum numbers retained at fixed lmax, and therefore independent of qubit count. Numerical verification across five potentials (Coulomb, harmonic oscillator, Woods–Saxon, square well, Yukawa) confirms the theorem: angular sparsity patterns are bit-identical across all potentials at matched orbital counts. The radial-angular factorization and the Gaunt selection rules are standard results from Condon & Shortley and Slater; the contribution of this paper is the explicit quantitative bound at each lmax, framed as a universal resource guarantee for qubit Hamiltonian sparsity that applies to electronic structure, nuclear shell models, quantum dots, and any other spherical-potential fermion system. GeoVac Paper 22. Part of the GeoVac research corpus (mathematical foundations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
We characterize the entanglement structure of multi-electron atoms in the angular momentum eigenbasis—the basis whose orbitals are products of hydrogenic radial functions and spherical harmonics, labeled by quantum numbers (n,l,m). Four computational results are presented. First, energy-entanglement decoupling: the off-diagonal one-body Hamiltonian (graph Laplacian topology, coupling constant κ=−1/16) contributes 10–39 of the correlation energy but less than 0.2 of the entanglement entropy; the off-diagonal electron repulsion contributes 100 of the entanglement. Entanglement entropy scales as S∼Z−2.56 across He-like ions. Second, basis-intrinsic sparsity: among orbital bases for systems with spherical symmetry, the angular momentum eigenbasis is the unique basis producing sparse electron repulsion integrals (ERIs). Any unitary rotation of orbital indices, no matter how small, fills the ERI tensor from 42 to 99 density in a step-function transition at the identity. The nonzero ERI count is Z-independent, confirming that sparsity is purely geometric. Third, entanglement network topology: orbital mutual information maps reveal a hub migration pattern—1s→2s→2p—that tracks the partially filled shell across the first row of the periodic table. Fourth, core-valence decoupling: the core-valence mutual information falls to ≈2×10−3 by Z=4 and below 10−3 for Z≥5, quantitatively justifying the composed fiber bundle factorization used in the GeoVac molecular framework. In the composed molecular architecture, per-block entanglement is R-independent, with core/bond entropy ratios of 50× at matched qubit count. These results establish that the angular momentum eigenbasis decouples one-body energy transport from two-body entanglement, and that the O(Q2.5) Pauli scaling of the GeoVac qubit Hamiltonians is a direct consequence of this basis choice. This is Paper 26 in the GeoVac series. GeoVac Paper 26. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
We present structurally sparse qubit Hamiltonians for the nuclear shell model, constructed by applying the GeoVac angular-sparsity framework to the harmonic oscillator (HO) single-particle basis with a Mayer–Jensen spin–orbit term. The HO shell closures 2,8,20,40,70,112 emerge directly from graph state counting, and the physical magic numbers 2,8,20,28,50,82,126 are recovered with a spin–orbit strength vls/ℏω≈0.17 and a Nilsson ℓ(ℓ+1) correction dℓℓ/ℏω≈0.02. We build the first two-species qubit encoding for the deuteron (1p+1n, 16 qubits, 592 non-identity Pauli terms, 1-norm ≈342 MeV) using the Minnesota nucleon–nucleon potential and Moshinsky–Talmi brackets, and extend the architecture to helium-4 (2p+2n, 16 qubits, 712 Pauli terms) with intra-species antisymmetrization and proton–proton Coulomb repulsion. Despite a Hilbert-space dimension 12.25× larger than the deuteron, He-4 requires only 1.20× more Pauli terms — a direct consequence of the angular selection rules that carry through to multi-particle systems. A composed nuclear–electronic deuterium encoding (26 qubits) validates the multi-block architecture and quantifies the ∼1013 coefficient ratio between nuclear, electronic, and hyperfine scales. We state and apply the Fock projection rigidity theorem: the S3 conformal projection that makes electronic Hamiltonians equivalent to a graph Laplacian is unique to the Coulomb potential and does not extend to HO or Woods–Saxon systems. For nuclear Hamiltonians, the GeoVac contribution is therefore the angular machinery and qubit encoding, not the conformal projection. All nuclear physics inputs (Minnesota parameters, spin–orbit strength, ℏω) follow the standard shell-model literature; the quantitative binding energies at Nshells=2 are far from experiment and should be read as encoding-validation benchmarks, not as a nuclear-structure calculation. GeoVac Paper 23. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
We show that qubit Hamiltonians derived from spectral graph theory lattices exhibit structurally favorable Pauli decompositions for quantum simulation. After Jordan–Wigner transformation, the GeoVac lattice Hamiltonian yields O(Q3.15) Pauli string terms in the number of qubits Q, compared to O(Q4+) for conventional Gaussian-basis Hamiltonians (our own log-log fit of the published Pauli counts of Trenev et al. [ref] gives Q4.25 for LiH and Q3.92 for H2O, spanning Q3.9–Q4.3). The advantage originates in angular momentum selection rules—enforced by Gaunt integrals at the two-electron integral level—which cause the electron repulsion integral (ERI) tensor density to decay as ∼1/M (empirically ≈M−0.85–M−1) with the number of spatial orbitals M, while Gaussian ERI tensors remain near-dense at O(1). Beyond term counts, we quantify two operationally relevant cost metrics: the number of qubitwise-commuting (QWC) measurement groups scales as O(Q3.36), and the Pauli 1-norm λ=∑i∣ci∣—which controls both Trotter step counts and qubitization query complexity—scales as O(Q1.69) with R2=0.997. The sub-quadratic 1-norm scaling is the key result for fault-tolerant simulation: it implies that Trotter circuit depth grows far more slowly with system size than term counts alone would suggest. All Gaussian baselines are validated against published integral values (Szabo and Ostlund for H2; analytical STO for He). For composed multi-center systems, the pseudopotential approximation of the composed natural geometry framework [ref] yields block-diagonal electron repulsion integrals with a universal Pauli scaling of O(Q2.5), independent of molecular topology (exponent spread 0.02 across three molecules). At equal qubit counts, this represents two or more orders of magnitude fewer Pauli terms than Gaussian baselines (published counts of Trenev et al. [ref], Q3.9–Q4.3 span fitted by us)—an advantage that grows with system size due to the ∼1.5 exponent gap. These comparisons are at matched qubit counts, not matched accuracy; the composed basis carries classical equilibrium-geometry errors of 5–26% (LiH/BeH2/H2O; Sec. (ref)), making the framework optimal for fixed-geometry quantum simulation rather than geometry optimisation. Across the composed molecular library (35 molecules spanning Z=1–56, H through Ba, including the first transition series ScH–ZnH), the composed Pauli count is exactly linear in the qubit count: NPauli=11.10×Q (exact, non-identity terms; d-orbital blocks give 9.23). These counts are realized under a pair-diagonal angular approximation (ma=mc, mb=md) adopted deliberately for sparsity; the exact global-ML Coulomb rule (used in the framework's precision-physics paths, and considered but left on the table here) gives ≈2.5× more terms with the same scaling, under which the d-block coefficient is in fact higher, not lower (Sec. (ref)). GeoVac Paper 14. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
The Geometric Vacuum (GeoVac) framework discretizes electronic structure on natural geometries, producing sparse graph Hamiltonians whose angular content is fixed by Gaunt selection rules. This synthesis assembles the framework's quantum-computing arc—the four Group 4 papers—into a single resource-oriented narrative. The spine is: the qubit encoding and its Pauli/1-norm scaling [ref]; the benchmarked resource library across three periodic-table rows [ref]; the SN representation-theoretic foundation that classifies the atomic building blocks [ref]; and the nuclear shell-model extension that delineates exactly which part of the framework transfers [ref]. The unifying value proposition is structural, not accuracy-based: after the Jordan–Wigner transformation the GeoVac lattice Hamiltonian yields O(Q3.15) Pauli terms for atoms and O(Q2.5) for composed multi-center molecules (versus O(Q3.9–4.3) for Gaussian baselines), with a sub-quadratic Pauli 1-norm λ=O(Q1.69) (R2=0.997) that controls Trotter depth and qubitization query cost. Two disciplines run through the arc. First, we separate the robust scaling exponents—unchanged under the choice of electron-repulsion selection rule—from the absolute multipliers, which depend on a stated pair-diagonal angular approximation and are disclosed as such. Second, every comparison is made at matched qubit count, not matched accuracy: the composed basis carries classical equilibrium-geometry errors of 5–26, so the framework is positioned for fixed-geometry NISQ/VQE simulation (Pauli count, QWC measurement groups), not for fault-tolerant geometry optimisation. The nuclear extension is honest in the same way: its binding energies are encoding-validation benchmarks far from experiment, and the Fock projection rigidity theorem states precisely that the conformal S3 structure does not transfer to the harmonic-oscillator basis—only the angular machinery and the qubit encoding do. GeoVac synthesis document. Part of the GeoVac research corpus (synthesis / reading guide group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
In this paper, we continue a line of research initiated by Basu, Brakensiek, and Putterman [2026] studying the sparsifiability of Hamiltonians. We focus particularly on the sparsifiability of the widely-studied Quantum Cut (QC) Hamiltonians. Our main result is that in an n-qubit system, any n-qubit QC Hamiltonian can be sparsified to O(n/ε2) many terms while preserving the energy of every state up to a factor of 1±ε. Our result can be interpreted as giving an importance sampling scheme for the edges of an arbitrary graph G such that the \emph{Kikuchi} graph at level ℓ of the sampled graph is a spectral approximation to the Kikuchi graph of G. Importantly, the \emph{same} sampling scheme works simultaneously for all ℓ.
The natural approach of leverage score sampling, analyzed via matrix concentration inequalities, yields a polynomially worse bound in our setting because the underlying matrices have dimension ∼2n. Instead, our approach relies on decomposing the action of these matrices into invariant subspaces. Then, by using an operator-valued inequality of Alon and Kozma [Ann. Henri Poincaré, 2020], itself building on an \emph{octopus inequality} of Caputo, Liggett, and Richthammer [J. AMS, 2010], we extend our sparsification technique to all expander graphs. We then invoke expander decomposition to extend our sparsifier to all graphs.
We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch-Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II.