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Entanglement Structure of the Angular Momentum Eigenbasis: Energy-Entanglement Decoupling and Basis-Intrinsic Sparsity

Josh Loutey

Published Jul 9, 2026• 0 citations
Publisher

Abstract

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 $\kappa = -1/16$) 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 $S \sim Z^{-2.56}$ 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—$1s \to 2s \to 2p$—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 $\approx 2×10^{-3}$ by $Z = 4$ and below $10^{-3}$ for $Z \geq 5$, 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 $O(Q^{2.5})$ 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.

DOI
10.5281/zenodo.21286827

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Entropy as a Projection Artifact: One-Body Operators are Entanglement-Inert on Sparse Lattices

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We characterize the operator-theoretic origin of entanglement entropy in the GeoVac Coulomb lattice, a finite spectral graph realization of the hydrogenic angular momentum eigenbasis on $S^{3}$. Five results are presented. First, the von Neumann entropy of any single-particle reduced density matrix derived from a non-degenerate ground state of a purely one-body fermionic Hamiltonian on a finite lattice is identically zero; numerically, we confirm $S_{\mathrm{kin}}/S_{\mathrm{full}} \sim 10^{-14}$ for He at $n_{\max}=2,3$ in the angular momentum eigenbasis. All ground-state entanglement is generated by the two-body interaction $V_{ee}$, with the non-degeneracy qualifier essential: open-shell and dissociated-bond states escape the floor. Second, the holographic area-law scaling $S_n = k\log A_n$ with $A_n\propto n^{4}$, asserted on one-particle grounds in earlier framework papers, is rigorously recovered as a pair-counting statement: $A_n=g_n^{2}$, where $g_n=2n^{2}$ is the shell degeneracy. The factor of four is the two-body signature. Third, in the Coulomb lattice the cusp concentrates as a single ``hot node'' on the $(1s,1s)$ pair-state of the $V_{ee}$ graph; entanglement is co-located with cusp mass. Fourth, on the Bargmann–Segal harmonic oscillator lattice the ground-state entropy of any two-fermion closed-shell state interacting via any central two-body potential is identically zero, a consequence of Moshinsky–Talmi total-quanta conservation; this is a rigidity statement structurally parallel to the Fock and Bargmann–Segal theorems of Paper 24. Fifth, Coulomb-lattice block entropy obeys a universal scaling $S_B = A (\tilde{w}_B/\delta_B)^{\gamma}$, where $\tilde{w}_B$ is the dimensionless $V_{ee}$ mass off-diagonal in the one-body eigenbasis and $\delta_B$ the dimensionless first $H_1$ gap; the local exponent extrapolates to $\gamma_{\infty} \approx 1.96$ in the joint $Z, n_{\max}\to\infty$ limit (Richardson on $n_{\max}=2,3,4,5$), slightly below the non-degenerate second-order Rayleigh–Schrödinger value of $2$, with the residual gap attributed to multi-shell aggregation in the basis-extension limit. Entropy is slotted into the exchange-constant taxonomy as a projection-class observable: zero on the combinatorial graph, finite only after the metric embedding of $1/r_{12}$. GeoVac Paper 27. 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.

•Jul 9, 2026•0 citations

Angular Momentum as Information Geometry: Isotropic Packing and the Universal Angular Cross-Section

Josh Loutey

We show that the angular momentum quantum numbers $(\ell, m)$ emerge from a purely geometric construction: uniform-density packing of distinguishable states on concentric isotropic shells in two dimensions. Starting from two axioms—discrete binary distinguishability and geometric indifference (maximum entropy)—the construction yields per-shell angular capacities $2k-1$ for shell $k$, which is isomorphic to the spherical harmonic basis $Y_{\ell m}$ with $\ell = k - 1$ and $m = -\ell, \ldots, +\ell$. The principal quantum number $n$ corresponds to a cumulative grouping of the first $n$ shells, producing the $n^2$ orbital count, but this grouping is not forced by the axioms alone. Similarly, a factor of 2 arises from the orientability of the compactified packing manifold ($S^2$), corresponding to spin multiplicity. When the packing is interpreted as a graph, its Laplacian decomposes into $(\ell, m)$ sectors whose ground-state eigenvalue spectrally converges to the hydrogen ground-state energy, as shown in subsequent papers; the full per-shell spectrum $E_n = -Z^2/(2n^2)$ requires an additional node-weight term (Paper 18 \S iii). This angular cross-section recurs as a universal structural factor inside every natural geometry developed in the Geometric Vacuum series—from $S^3$ for atoms through composed fiber bundles for multi-electron molecules—making it a foundational motif of the framework. This is Paper 0 in the series. GeoVac synthesis document. 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.

•Jul 9, 2026•0 citations

Structurally Sparse Qubit Hamiltonians from Spectral Graph Theory

Josh Loutey

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 $O(Q^{3.15})$ Pauli string terms in the number of qubits $Q$, compared to $O(Q^{4+})$ for conventional Gaussian-basis Hamiltonians (our own log-log fit of the published Pauli counts of Trenev et al. [ref] gives $Q^{4.25}$ for LiH and $Q^{3.92}$ for H$_2$O, spanning $Q^{3.9}$–$Q^{4.3}$). 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 $\sim 1/M$ (empirically $\approx M^{-0.85}$–$M^{-1}$) 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 $O(Q^{3.36})$, and the Pauli 1-norm $\lambda = \sum_i |c_i|$—which controls both Trotter step counts and qubitization query complexity—scales as $O(Q^{1.69})$ with $R^2 = 0.997$. 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 H$_2$; 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 $O(Q^{2.5})$, 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], $Q^{3.9}$–$Q^{4.3}$ span fitted by us)—an advantage that grows with system size due to the $\sim 1.5$ exponent gap. These comparisons are at matched qubit counts, not matched accuracy; the composed basis carries classical equilibrium-geometry errors of 5–26% (LiH/BeH$_2$/H$_2$O; 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: $N_{\mathrm{Pauli}} = 11.10 × Q$ (exact, non-identity terms; $d$-orbital blocks give 9.23). These counts are realized under a pair-diagonal angular approximation ($m_a=m_c$, $m_b=m_d$) adopted deliberately for sparsity; the exact global-$M_L$ Coulomb rule (used in the framework's precision-physics paths, and considered but left on the table here) gives $\approx 2.5×$ 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.

•Jul 9, 2026•0 citations

Potential-Independent Angular Sparsity for Qubit Hamiltonians in Spherical Fermion Systems

Josh Loutey

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.

•Jul 9, 2026•0 citations

Nuclear Shell Model Hamiltonians on the Hyperspherical Lattice

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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 $v_{ls}/\hbar\omega \approx 0.17$ and a Nilsson $\ell(\ell+1)$ correction $d_{\ell\ell}/\hbar\omega \approx 0.02$. We build the first two-species qubit encoding for the deuteron ($1p+1n$, $16$ qubits, $592$ non-identity Pauli terms, $1$-norm $\approx 342$ 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 $\sim 10^{13}$ coefficient ratio between nuclear, electronic, and hyperfine scales. We state and apply the Fock projection rigidity theorem: the $S^3$ 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, $\hbar\omega$) follow the standard shell-model literature; the quantitative binding energies at $N_{\mathrm{shells}}=2$ 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.

•Jul 9, 2026•0 citations

Structured High-Angular-Momentum Coulomb Tensors from Real and Complex Solid-Harmonic Integral Engines: A Perspective

Bo Peng

Electron-repulsion integrals describe the Coulomb interaction between charge distributions built from orbital basis functions. Most integral algorithms generate these quantities through Cartesian Gaussian functions, whose angular shapes are written as powers of $x$, $y$, and $z$, and then transform the result to spherical functions. This route is effective, but from $d$ shells onward the Cartesian representation contains more functions than the spherical space required by the calculation. Direct real or complex solid-harmonic engines work in that target space from the beginning. They therefore produce a smaller final Coulomb tensor while preserving the ordering, phase, and magnetic-quantum-number labels that describe its angular structure. Following this structure beyond integral evaluation reveals direct connections to the algorithms that use the tensor. Simple analytical counts quantify tensor size, angular blocks, radial Slater--Condon parameters, and pair-space work. These quantities guide low-rank factorization, local Hamiltonian construction, quantum simulation, and transformations to spinor or effective-model bases. In this way, solid-harmonic integral engines provide a direct bridge between efficient integral generation and structured many-electron computation.

physics.chem-phCross-listedarXiv:2607.25245•Jul 29, 2026
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)(n, l, m)(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\kappa = -1/16κ=−1/16) contributes 101010–3939%39 of the correlation energy but less than 0.20.2%0.2 of the entanglement entropy; the off-diagonal electron repulsion contributes 100100%100 of the entanglement. Entanglement entropy scales as S∼Z−2.56S \sim Z^{-2.56}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 4242%42 to 9999%99 density in a step-function transition at the identity. The nonzero ERI count is ZZZ-independent, confirming that sparsity is purely geometric. Third, entanglement network topology: orbital mutual information maps reveal a hub migration pattern—1s→2s→2p1s \to 2s \to 2p1s→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\approx 2×10^{-3}≈2×10−3 by Z=4Z = 4Z=4 and below 10−310^{-3}10−3 for Z≥5Z \geq 5Z≥5, quantitatively justifying the composed fiber bundle factorization used in the GeoVac molecular framework. In the composed molecular architecture, per-block entanglement is RRR-independent, with core/bond entropy ratios of 50×50×50× 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)O(Q^{2.5})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 characterize the operator-theoretic origin of entanglement entropy in the GeoVac Coulomb lattice, a finite spectral graph realization of the hydrogenic angular momentum eigenbasis on S3S^{3}S3. Five results are presented. First, the von Neumann entropy of any single-particle reduced density matrix derived from a non-degenerate ground state of a purely one-body fermionic Hamiltonian on a finite lattice is identically zero; numerically, we confirm Skin/Sfull∼10−14S_{\mathrm{kin}}/S_{\mathrm{full}} \sim 10^{-14}Skin​/Sfull​∼10−14 for He at nmax⁡=2,3n_{\max}=2,3nmax​=2,3 in the angular momentum eigenbasis. All ground-state entanglement is generated by the two-body interaction VeeV_{ee}Vee​, with the non-degeneracy qualifier essential: open-shell and dissociated-bond states escape the floor. Second, the holographic area-law scaling Sn=klog⁡AnS_n = k\log A_nSn​=klogAn​ with An∝n4A_n\propto n^{4}An​∝n4, asserted on one-particle grounds in earlier framework papers, is rigorously recovered as a pair-counting statement: An=gn2A_n=g_n^{2}An​=gn2​, where gn=2n2g_n=2n^{2}gn​=2n2 is the shell degeneracy. The factor of four is the two-body signature. Third, in the Coulomb lattice the cusp concentrates as a single ``hot node'' on the (1s,1s)(1s,1s)(1s,1s) pair-state of the VeeV_{ee}Vee​ graph; entanglement is co-located with cusp mass. Fourth, on the Bargmann–Segal harmonic oscillator lattice the ground-state entropy of any two-fermion closed-shell state interacting via any central two-body potential is identically zero, a consequence of Moshinsky–Talmi total-quanta conservation; this is a rigidity statement structurally parallel to the Fock and Bargmann–Segal theorems of Paper 24. Fifth, Coulomb-lattice block entropy obeys a universal scaling SB=A(w~B/δB)γS_B = A (\tilde{w}_B/\delta_B)^{\gamma}SB​=A(w~B​/δB​)γ, where w~B\tilde{w}_Bw~B​ is the dimensionless VeeV_{ee}Vee​ mass off-diagonal in the one-body eigenbasis and δB\delta_BδB​ the dimensionless first H1H_1H1​ gap; the local exponent extrapolates to γ∞≈1.96\gamma_{\infty} \approx 1.96γ∞​≈1.96 in the joint Z,nmax⁡→∞Z, n_{\max}\to\inftyZ,nmax​→∞ limit (Richardson on nmax⁡=2,3,4,5n_{\max}=2,3,4,5nmax​=2,3,4,5), slightly below the non-degenerate second-order Rayleigh–Schrödinger value of 222, with the residual gap attributed to multi-shell aggregation in the basis-extension limit. Entropy is slotted into the exchange-constant taxonomy as a projection-class observable: zero on the combinatorial graph, finite only after the metric embedding of 1/r121/r_{12}1/r12​. GeoVac Paper 27. 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 show that the angular momentum quantum numbers (ℓ,m)(\ell, m)(ℓ,m) emerge from a purely geometric construction: uniform-density packing of distinguishable states on concentric isotropic shells in two dimensions. Starting from two axioms—discrete binary distinguishability and geometric indifference (maximum entropy)—the construction yields per-shell angular capacities 2k−12k-12k−1 for shell kkk, which is isomorphic to the spherical harmonic basis YℓmY_{\ell m}Yℓm​ with ℓ=k−1\ell = k - 1ℓ=k−1 and m=−ℓ,…,+ℓm = -\ell, \ldots, +\ellm=−ℓ,…,+ℓ. The principal quantum number nnn corresponds to a cumulative grouping of the first nnn shells, producing the n2n^2n2 orbital count, but this grouping is not forced by the axioms alone. Similarly, a factor of 2 arises from the orientability of the compactified packing manifold (S2S^2S2), corresponding to spin multiplicity. When the packing is interpreted as a graph, its Laplacian decomposes into (ℓ,m)(\ell, m)(ℓ,m) sectors whose ground-state eigenvalue spectrally converges to the hydrogen ground-state energy, as shown in subsequent papers; the full per-shell spectrum En=−Z2/(2n2)E_n = -Z^2/(2n^2)En​=−Z2/(2n2) requires an additional node-weight term (Paper 18 \S iii). This angular cross-section recurs as a universal structural factor inside every natural geometry developed in the Geometric Vacuum series—from S3S^3S3 for atoms through composed fiber bundles for multi-electron molecules—making it a foundational motif of the framework. This is Paper 0 in the series. GeoVac synthesis document. 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 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)O(Q^{3.15})O(Q3.15) Pauli string terms in the number of qubits QQQ, compared to O(Q4+)O(Q^{4+})O(Q4+) for conventional Gaussian-basis Hamiltonians (our own log-log fit of the published Pauli counts of Trenev et al. [ref] gives Q4.25Q^{4.25}Q4.25 for LiH and Q3.92Q^{3.92}Q3.92 for H2_22​O, spanning Q3.9Q^{3.9}Q3.9–Q4.3Q^{4.3}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\sim 1/M∼1/M (empirically ≈M−0.85\approx M^{-0.85}≈M−0.85–M−1M^{-1}M−1) with the number of spatial orbitals MMM, while Gaussian ERI tensors remain near-dense at O(1)O(1)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)O(Q^{3.36})O(Q3.36), and the Pauli 1-norm λ=∑i∣ci∣\lambda = \sum_i |c_i|λ=∑i​∣ci​∣—which controls both Trotter step counts and qubitization query complexity—scales as O(Q1.69)O(Q^{1.69})O(Q1.69) with R2=0.997R^2 = 0.997R2=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_22​; 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)O(Q^{2.5})O(Q2.5), independent of molecular topology (exponent spread 0.020.020.02 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], Q3.9Q^{3.9}Q3.9–Q4.3Q^{4.3}Q4.3 span fitted by us)—an advantage that grows with system size due to the ∼1.5\sim 1.5∼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_22​/H2_22​O; Sec. (ref)), making the framework optimal for fixed-geometry quantum simulation rather than geometry optimisation. Across the composed molecular library (35 molecules spanning Z=1Z = 1Z=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×QN_{\mathrm{Pauli}} = 11.10 × QNPauli​=11.10×Q (exact, non-identity terms; ddd-orbital blocks give 9.23). These counts are realized under a pair-diagonal angular approximation (ma=mcm_a=m_cma​=mc​, mb=mdm_b=m_dmb​=md​) adopted deliberately for sparsity; the exact global-MLM_LML​ Coulomb rule (used in the framework's precision-physics paths, and considered but left on the table here) gives ≈2.5×\approx 2.5×≈2.5× more terms with the same scaling, under which the ddd-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.
For any NNN-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⁡)D(l_{\max})D(lmax​) under the Coulomb selection rule (conservation of total magnetic quantum number, ma+mb=mc+mdm_a+m_b=m_c+m_dma​+mb​=mc​+md​): 14.8414.84%14.84 at lmax⁡=1l_{\max}=1lmax​=1, 8.528.52%8.52 at lmax⁡=2l_{\max}=2lmax​=2, 6.066.06%6.06 at lmax⁡=3l_{\max}=3lmax​=3, 4.834.83%4.83 at lmax⁡=4l_{\max}=4lmax​=4, and 3.993.99%3.99 at lmax⁡=5l_{\max}=5lmax​=5. At lmax⁡=2l_{\max}=2lmax​=2, 91.4891.48%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=mcm_a=m_cma​=mc​, mb=mdm_b=m_dmb​=md​), applicable to axially-symmetric or mmm-decoupled bases, gives a further reduction (2.762.76%2.76 at lmax⁡=2l_{\max}=2lmax​=2, 1.441.44%1.44 at lmax⁡=3l_{\max}=3lmax​=3). The bound is independent of the number of radial quantum numbers retained at fixed lmax⁡l_{\max}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⁡l_{\max}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 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 2,8,20,40,70,1122, 8, 20, 40, 70, 1122,8,20,40,70,112 emerge directly from graph state counting, and the physical magic numbers 2,8,20,28,50,82,1262, 8, 20, 28, 50, 82, 1262,8,20,28,50,82,126 are recovered with a spin–orbit strength vls/ℏω≈0.17v_{ls}/\hbar\omega \approx 0.17vls​/ℏω≈0.17 and a Nilsson ℓ(ℓ+1)\ell(\ell+1)ℓ(ℓ+1) correction dℓℓ/ℏω≈0.02d_{\ell\ell}/\hbar\omega \approx 0.02dℓℓ​/ℏω≈0.02. We build the first two-species qubit encoding for the deuteron (1p+1n1p+1n1p+1n, 161616 qubits, 592592592 non-identity Pauli terms, 111-norm ≈342\approx 342≈342 MeV) using the Minnesota nucleon–nucleon potential and Moshinsky–Talmi brackets, and extend the architecture to helium-4 (2p+2n2p+2n2p+2n, 161616 qubits, 712712712 Pauli terms) with intra-species antisymmetrization and proton–proton Coulomb repulsion. Despite a Hilbert-space dimension 12.25×12.25×12.25× larger than the deuteron, He-4 requires only 1.20×1.20×1.20× more Pauli terms — a direct consequence of the angular selection rules that carry through to multi-particle systems. A composed nuclear–electronic deuterium encoding (262626 qubits) validates the multi-block architecture and quantifies the ∼1013\sim 10^{13}∼1013 coefficient ratio between nuclear, electronic, and hyperfine scales. We state and apply the Fock projection rigidity theorem: the S3S^3S3 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\hbar\omegaℏω) follow the standard shell-model literature; the quantitative binding energies at Nshells=2N_{\mathrm{shells}}=2Nshells​=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.
Electron-repulsion integrals describe the Coulomb interaction between charge distributions built from orbital basis functions. Most integral algorithms generate these quantities through Cartesian Gaussian functions, whose angular shapes are written as powers of xxx, yyy, and zzz, and then transform the result to spherical functions. This route is effective, but from ddd shells onward the Cartesian representation contains more functions than the spherical space required by the calculation. Direct real or complex solid-harmonic engines work in that target space from the beginning. They therefore produce a smaller final Coulomb tensor while preserving the ordering, phase, and magnetic-quantum-number labels that describe its angular structure. Following this structure beyond integral evaluation reveals direct connections to the algorithms that use the tensor. Simple analytical counts quantify tensor size, angular blocks, radial Slater--Condon parameters, and pair-space work. These quantities guide low-rank factorization, local Hamiltonian construction, quantum simulation, and transformations to spinor or effective-model bases. In this way, solid-harmonic integral engines provide a direct bridge between efficient integral generation and structured many-electron computation.