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Foundations of the Geometric Vacuum Framework: A Synthesis of the Spectral Graph Theory Arc

Josh Loutey

Published Jul 9, 2026• 0 citations
Publisher

Abstract

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 $S^3$, 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 $S^3$ 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 $S^3$ Laplace–Beltrami operator, verified by 18 independent symbolic proofs; (b) the angular sparsity of two-body matrix elements depends only on $l_{\max}$ 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 $H^2(S^5)$ 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 $d_{0}$ cancelling out of every spectral quantity, and physical scales entering only through the Fock-projection focal length $p_{0}$—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 $S^3$, 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.

DOI
10.5281/zenodo.21286861

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Observables as Projections of a π-free Graph: A Synthesis of the GeoVac Precision-Observation Arc

Josh Loutey

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 $(n,\ell,m,\ldots)$, 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 $S^3×ℝ$ 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, $π^{2k}$ 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.

•Jul 9, 2026•0 citations

Structural Sparsity for Quantum Simulation: A Synthesis of the GeoVac Quantum-Computing Arc

Josh Loutey

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.

•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

The Bargmann–Segal Lattice: π-Free Discretization of the Harmonic Oscillator on S^5

Josh Loutey

We construct a discrete graph encoding of the three-dimensional isotropic harmonic oscillator on the holomorphic sector of $S^5$ and verify by exact rational arithmetic that the resulting graph and its projection to physical energies contain no transcendental constants. The Bargmann–Segal transform [ref] maps the harmonic oscillator Hamiltonian to $\hbar\omega(\hat{N} + 3/2)$ on the Fock–Bargmann space, where $\hat{N}$ is the Euler number operator on holomorphic monomials. Restricting to the unit sphere $S^5 \subset \mathbb{C}^3$ identifies the eigenspaces with the symmetric SU(3) irreps $(N, 0)$, of dimension $(N+1)(N+2)/2$, which match the harmonic oscillator shell degeneracies exactly. The graph constructed from these states with dipole transition weights computed by exact fractions.Fraction arithmetic is bit-exactly $π$-free at $N_{\max} = 5$ (56 nodes, 165 edges, zero irrationals encountered). The single $π$ in the entire ecosystem is the Gaussian normalization $π^{-3}$ of the continuous Bargmann measure, which the discrete graph never invokes. The Coulomb problem on $S^3$ and the harmonic oscillator on $S^5$ are both natural sphere quotients, but they live on different spheres with different geometric character (Riemannian vs complex), have natural operators of different order (second vs first), yield spectra of different growth (quadratic vs linear), and demand projections of different type (nonlinear vs linear-affine). This structural asymmetry refines the exchange constant taxonomy [ref]: calibration $π$ is Coulomb-specific. It is the transcendental cost of mapping a quadratic Riemannian eigenvalue to an inverse-square physical energy, and it has no analog for the harmonic oscillator's linear complex-analytic projection. GeoVac Paper 24. 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

Geometric Block Universe

JONATHAN CHARLES DOWNES

We present the Geometric Block Universe (GBU), a framework in which the universe is a single static geometric structure and the probabilistic content of quantum mechanics is derived from that geometry rather than postulated. The central technical task is to obtain the Born rule as a theorem of Riemannian spectral geometry. At the quantum level of description the block is modelled by a compact Riemannian manifold B. The Riemannian volume measure is the natural density on B, and it forces the L2 inner product; the physical states are selected by a Dirichlet variational principle, which we prove to be the unique admissible choice among natural functionals of the appropriate degree under two explicitly stated and minimal hypotheses, in every dimension n ≥ 2. Under these hypotheses the Born rule is not assumed: it follows as a geometric theorem, the squared L2 overlap of states forced by the naturality of the volume measure together with Gleason's theorem [5] applied after holographic truncation. Quantum states are identified with Laplace-Beltrami eigenfunctions, the spectral modes of the metric Laplacian on B. The framework is a no-collapse one, and its ontology is block-universe realist: what is ontic is the geometry, while the state ψ is an indexical description of the transition amplitudes available to an observer from its current block. Nothing collapses and nothing splits; there is a fixed structure and observers who traverse it, and what appears within a traversal as a measurement event is a feature of how a path sits in the block rather than a process the structure undergoes. This places GBU in the no-collapse family, while departing from it ontologically, replacing dynamically generated branches with pre-existing geodesic paths. The holographic bound truncates the spectrum to a finite physical subspace, suggesting a finite-mode regularisation of ultraviolet divergences, with recovery of low-energy quantum field theory deferred to Paper 3. Temporal asymmetry is derived as an observer-selection effect on record-accumulating geodesics rather than a fundamental feature of the geometry. The constants c, ℏ and G are identified as geometric conversion factors between levels of the block structure. We demonstrate the framework on an explicit two-sphere calculation, and outline a five-paper programme. Throughout, the paper is explicit about the boundary between what is proven (the conditional Born-rule theorem), what is a route with a verified proof-of-concept (the dynamics), and what remains conjectural (the origin of the two-level structure, and the constants).

•Jul 2, 2026•0 citations
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 S3S^3S3, 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 S3S^3S3 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 S3S^3S3 Laplace–Beltrami operator, verified by 18 independent symbolic proofs; (b) the angular sparsity of two-body matrix elements depends only on lmax⁡l_{\max}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)H^2(S^5)H2(S5) that is bit-exactly πππ-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 πππ, 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 d0d_{0}d0​ cancelling out of every spectral quantity, and physical scales entering only through the Fock-projection focal length p0p_{0}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)O(V)O(V) sparsity, integer eigenvalues on the unit S3S^3S3, 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.
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,…)(n,\ell,m,\ldots)(n,ℓ,m,…), integer eigenvalues, rational or algebraic matrix elements, no physical units, and—verified symbolically—no πππ. 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—πππ enters the Klein–Gordon spectrum on S3×RS^3×ℝS3×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—πππ via the Hopf and Fock volume measures, π2kπ^{2k}π2k via the spectral action, ζ(2k)ζ(2k)ζ(2k) via even-zeta Dirichlet series, odd ζζζ and Catalan GGG via the half-integer-Hurwitz spinor lift, 2π2π2π 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.
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/111-norm scaling [ref]; the benchmarked resource library across three periodic-table rows [ref]; the SNS_NSN​ 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)O(Q^{3.15})O(Q3.15) Pauli terms for atoms and O(Q2.5)O(Q^{2.5})O(Q2.5) for composed multi-center molecules (versus O(Q3.9–4.3)O(Q^{3.9–4.3})O(Q3.9–4.3) for Gaussian baselines), with a sub-quadratic Pauli 111-norm λ=O(Q1.69)\lambda = O(Q^{1.69})λ=O(Q1.69) (R2=0.997R^2=0.997R2=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 555–2626%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 S3S^3S3 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.
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.
We construct a discrete graph encoding of the three-dimensional isotropic harmonic oscillator on the holomorphic sector of S5S^5S5 and verify by exact rational arithmetic that the resulting graph and its projection to physical energies contain no transcendental constants. The Bargmann–Segal transform [ref] maps the harmonic oscillator Hamiltonian to ℏω(N^+3/2)\hbar\omega(\hat{N} + 3/2)ℏω(N^+3/2) on the Fock–Bargmann space, where N^\hat{N}N^ is the Euler number operator on holomorphic monomials. Restricting to the unit sphere S5⊂C3S^5 \subset \mathbb{C}^3S5⊂C3 identifies the eigenspaces with the symmetric SU(3) irreps (N,0)(N, 0)(N,0), of dimension (N+1)(N+2)/2(N+1)(N+2)/2(N+1)(N+2)/2, which match the harmonic oscillator shell degeneracies exactly. The graph constructed from these states with dipole transition weights computed by exact fractions.Fraction arithmetic is bit-exactly πππ-free at Nmax⁡=5N_{\max} = 5Nmax​=5 (56 nodes, 165 edges, zero irrationals encountered). The single πππ in the entire ecosystem is the Gaussian normalization π−3π^{-3}π−3 of the continuous Bargmann measure, which the discrete graph never invokes. The Coulomb problem on S3S^3S3 and the harmonic oscillator on S5S^5S5 are both natural sphere quotients, but they live on different spheres with different geometric character (Riemannian vs complex), have natural operators of different order (second vs first), yield spectra of different growth (quadratic vs linear), and demand projections of different type (nonlinear vs linear-affine). This structural asymmetry refines the exchange constant taxonomy [ref]: calibration πππ is Coulomb-specific. It is the transcendental cost of mapping a quadratic Riemannian eigenvalue to an inverse-square physical energy, and it has no analog for the harmonic oscillator's linear complex-analytic projection. GeoVac Paper 24. 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 the Geometric Block Universe (GBU), a framework in which the universe is a single static geometric structure and the probabilistic content of quantum mechanics is derived from that geometry rather than postulated. The central technical task is to obtain the Born rule as a theorem of Riemannian spectral geometry. At the quantum level of description the block is modelled by a compact Riemannian manifold B. The Riemannian volume measure is the natural density on B, and it forces the L2 inner product; the physical states are selected by a Dirichlet variational principle, which we prove to be the unique admissible choice among natural functionals of the appropriate degree under two explicitly stated and minimal hypotheses, in every dimension n ≥ 2. Under these hypotheses the Born rule is not assumed: it follows as a geometric theorem, the squared L2 overlap of states forced by the naturality of the volume measure together with Gleason's theorem [5] applied after holographic truncation. Quantum states are identified with Laplace-Beltrami eigenfunctions, the spectral modes of the metric Laplacian on B. The framework is a no-collapse one, and its ontology is block-universe realist: what is ontic is the geometry, while the state ψ is an indexical description of the transition amplitudes available to an observer from its current block. Nothing collapses and nothing splits; there is a fixed structure and observers who traverse it, and what appears within a traversal as a measurement event is a feature of how a path sits in the block rather than a process the structure undergoes. This places GBU in the no-collapse family, while departing from it ontologically, replacing dynamically generated branches with pre-existing geodesic paths. The holographic bound truncates the spectrum to a finite physical subspace, suggesting a finite-mode regularisation of ultraviolet divergences, with recovery of low-energy quantum field theory deferred to Paper 3. Temporal asymmetry is derived as an observer-selection effect on record-accumulating geodesics rather than a fundamental feature of the geometry. The constants c, ℏ and G are identified as geometric conversion factors between levels of the block structure. We demonstrate the framework on an explicit two-sphere calculation, and outline a five-paper programme. Throughout, the paper is explicit about the boundary between what is proven (the conditional Born-rule theorem), what is a route with a verified proof-of-concept (the dynamics), and what remains conjectural (the origin of the two-level structure, and the constants).