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 $S_N$ 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 $O(Q^{3.15})$ Pauli terms for atoms and $O(Q^{2.5})$ for composed multi-center molecules (versus $O(Q^{3.9–4.3})$ for Gaussian baselines), with a sub-quadratic Pauli $1$-norm $\lambda = O(Q^{1.69})$ ($R^2=0.997$) 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 $S^3$ 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.
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.
The Geometric Vacuum (GeoVac) framework is a discretization scheme for non-relativistic quantum mechanics on natural geometries: the discrete graph Laplacian on a packed lattice of quantum-number labels replaces the continuous Laplace–Beltrami operator on the corresponding compact manifold. This synthesis paper assembles the framework's foundational arc—the packing construction, the discrete graph and its dynamics, the conformal equivalence with the unit S3, the exchange constant taxonomy, the angular sparsity theorem, the Bargmann–Segal lattice, and the universal/Coulomb partition, together with the reconvergence node where these threads cohere (two-body selection rules from the tensor-product triple, the period classification, the Tannakian substrate and cosmic-Galois injection, and the forced/free seam)—into a single coherent narrative. The arc has a dependency structure worth naming: a single taproot (the packing construction, made physical by the Fock S3 projection) fans out into the framework's machinery, then reconverges at the periods, Tannakian, and forced/free nodes, where the scattered structural findings cohere into one statement about what the framework forces and what it consumes. Four headline structural results frame the program: (a) the discrete Fock graph is conformally equivalent to the unit S3 Laplace–Beltrami operator, verified by 18 independent symbolic proofs; (b) the angular sparsity of two-body matrix elements depends only on lmax and is bit-identical across five distinct central potentials, qualifying as a potential-independent theorem about spherical-fermion bases; (c) the three-dimensional isotropic harmonic oscillator admits a discrete graph encoding on the Hardy space H2(S5) that is bit-exactly π-free in exact rational arithmetic at every finite cutoff; (d) the framework's results split cleanly along an A/D partition—an algebra-only universal sector that transfers free across central potentials, and a Dirac-operator-specific sector containing the spectrum, the calibration π, and the candidate fine-structure-constant combination of Paper 2. A structural reading we make explicit in this synthesis is that the framework's substrate-level scale-freedom—the packing diameter d0 cancelling out of every spectral quantity, and physical scales entering only through the Fock-projection focal length p0—is the geometric expression of the same principle that conformal field theory [ref] embodies at the physics level. GeoVac provides a scale-free mathematical substrate; CFT is a scale-free physics theory living on such a substrate. This is not a claim that GeoVac is a CFT; it is a structural reflection that explains why the framework reproduces continuum CFT-on-sphere observables bit-exactly (documented in the operator-algebra arc, outside the foundations scope of this synthesis). We present GeoVac as a discretization framework that exploits natural geometry, not as a new foundation for quantum mechanics: the discrete graph and the continuous Laplace–Beltrami operator are dual descriptions connected by proven conformal maps, and the interpretive question of which description is more fundamental is left open. The framework's concrete computational advantages—O(V) sparsity, integer eigenvalues on the unit S3, angular momentum selection rules baked into the basis, bit-exactly rational matrix elements on the Bargmann–Segal lattice, and the universal sparsity bound—are independent of that interpretive question. 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.
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.
We present a library of qubit Hamiltonians for molecular quantum simulation that are structurally sparser than conventional Gaussian-basis alternatives by one to three orders of magnitude. The Hamiltonians are constructed from a spectral graph theory framework (GeoVac) that exploits natural coordinate separations in molecular electronic structure, producing block-diagonal electron repulsion integrals and Gaunt-enforced selection rules that persist through Jordan-Wigner encoding. For the composed architecture (PK-partitioned), the composed LiH Hamiltonian is near parity with minimal-basis STO-3G on Pauli count (334 vs. STO-3G's 276 reduced / 907 raw) while using more qubits; the decisive resource advantages are the O(Q2.5) Pauli scaling and a 190× reduction versus cc-pVDZ. The balanced coupled architecture (PK-free) binds LiH with an FCI minimum at a computed Req=3.227 bohr (7.0% above the experimental 3.015 bohr) at nmax=2 (878 Pauli terms, 30 qubits); a single-point evaluation at the experimental geometry reaches 0.20% energy error at nmax=3 (84 qubits) when solved exactly. Chemistry-accuracy scope at nmax=2 is row-conditional: first-row hydrides (LiH, BeH2, H2O, …) bind with documented potential-energy-surface (PES) topology, while second-row hydrides (NaH and below) exhibit monotone overattraction under every tested configuration — frozen-core balanced, multi-zeta valence, screened cross-center, explicit-core Hartree–Fock — with no interior PES minimum at the qubit-Hamiltonian level. We document this scope boundary empirically in \S(ref); the framework's primary value proposition is structural sparsity with quantitative truncation-convergence rates (Paper 38 state-space GH convergence), not energy-accuracy parity with CCSD(T) on second-row systems. The hamiltonian() registry covers 37 systems (35 composed molecules spanning Z=1–56, H through Ba, plus the He and H2 reference atoms), including multi-center systems (CO, N2, F2), mixed frozen-core molecules (NaCl), and the full first transition series as hydrides (ScH through ZnH), with Pauli scaling O(Q2.5) confirmed universal across all three rows, compared to O(Q3.9–4.3) for Gaussian baselines. The composed Pauli count is exactly linear in qubit count (coefficient 11.10, exact, for main-group molecules; 9.23 for d-orbital blocks under the pair-diagonal ERI approximation (Sec. (ref))—the exact global-ML rule makes the d-block coefficient higher, not lower), so adding atoms to a molecule increases quantum cost linearly. Second- and third-row molecules use analytical frozen cores ([Ne], [Ar], [Ar]3d10); the frozen-core energy enters only through the identity operator and does not affect quantum measurement cost. Isostructural molecules with frozen cores produce identical Pauli counts across periodic table rows. The Hamiltonians are available as a Python library with Qiskit, OpenFermion, and PennyLane export. GeoVac Paper 20. 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 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.
The Geometric Vacuum (GeoVac) framework separates into two structurally distinct layers, and that separation is the organizing principle of its precision-observation arc. Layer 1 is a bare combinatorial graph: quantum-number labels (n,ℓ,m,…), integer eigenvalues, rational or algebraic matrix elements, no physical units, and—verified symbolically—no π. Layer 2 is a finite family of named projections from that graph onto continuous physical settings; every physical variable, every dimension, and every transcendental constant that appears in any GeoVac calculation enters through exactly one projection and is traceable to it. This synthesis assembles the four Group 6 papers into a single narrative built on that thesis. The spine is: the projection taxonomy and its three-axis tagging (Paper 34 [ref]); time as the prototype free-side projection—π enters the Klein–Gordon spectrum on S3×R at exactly one step, temporal compactification, while rest mass preserves the algebraic ring (Paper 35 [ref]); entropy as an embedding-class projection artifact, identically zero on the one-body graph and finite only after the metric embedding of the two-body repulsion (Paper 27 [ref]); and the entanglement structure that isolates that repulsion in the angular-momentum eigenbasis (Paper 26 [ref]). A multi-observable focal-length program closes the arc: Roothaan autopsies decompose precision atomic observables into projection-chain-tagged components, exposing literature convention mismatches and framework kernel gaps that single-observable analyses cannot see. Two disciplines run throughout. First, every transcendental named here is tagged to its projection—π via the Hopf and Fock volume measures, π2k via the spectral action, ζ(2k) via even-zeta Dirichlet series, odd ζ and Catalan G via the half-integer-Hurwitz spinor lift, 2π via temporal compactification. Second, the precision matches are sub-percent framework-native agreements that surface convention and kernel issues; they are not first-principles predictions of the constants, and the multi-focal-composition wall—the framework has no native theorem for composing multiple Fock-style projections—is named as an active boundary. The unifying reading is that ``discreteness is compactness'' made operational: the discrete skeleton is the compact/closed regime, and a transcendental is precisely the price of releasing compactness by integrating over a promoted continuous parameter. 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.