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.
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.
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.
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.
We show that the angular momentum quantum numbers (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−1 for shell k, which is isomorphic to the spherical harmonic basis Yℓm with ℓ=k−1 and m=−ℓ,…,+ℓ. The principal quantum number n corresponds to a cumulative grouping of the first n shells, producing the n2 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 (S2), corresponding to spin multiplicity. When the packing is interpreted as a graph, its Laplacian decomposes into (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) 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 S3 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 construct a discrete graph encoding of the three-dimensional isotropic harmonic oscillator on the holomorphic sector of S5 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) on the Fock–Bargmann space, where N^ is the Euler number operator on holomorphic monomials. Restricting to the unit sphere S5⊂C3 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 Nmax=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 S3 and the harmonic oscillator on S5 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.
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 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 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∼10−14 for He at nmax=2,3 in the angular momentum eigenbasis. All ground-state entanglement is generated by the two-body interaction Vee, with the non-degeneracy qualifier essential: open-shell and dissociated-bond states escape the floor. Second, the holographic area-law scaling Sn=klogAn with An∝n4, asserted on one-particle grounds in earlier framework papers, is rigorously recovered as a pair-counting statement: An=gn2, where 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) pair-state of the 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)γ, where w~B is the dimensionless Vee mass off-diagonal in the one-body eigenbasis and δB the dimensionless first H1 gap; the local exponent extrapolates to γ∞≈1.96 in the joint Z,nmax→∞ limit (Richardson on nmax=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/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.