We develop a complete quantum information formalization of the spectralstructure of a rigid projection tensor (pseudovolume), establishing twostructural results that go strictly beyond Khrennikov's quantum integerframework (2010). The spectral Hilbert space H_b = C^|M(b)| assigns to each basis shape b (avector of positive rational components) a Hilbert space whose dimension equalsthe number of spectral classes — equivalence classes of 32 projection planesunder a grid-equality relation. The first main result is scale-invariance:|M(b)| depends only on the ratio class [b], not on the scalar magnitudeS = sum(b_i). In Khrennikov's framework the partition count p(n) is strictlyincreasing; here the quantum structure is magnitude-free. The second result is cross-scale indistinguishability: basis shapes fromdistinct ratio classes at distinct magnitudes routinely share identical Hilbertspaces and projection-valued measures, with hundreds of thousands of non-trivialcollision pairs across small magnitudes and onset of asymptotic universality athigher sums. In Khrennikov's framework no two distinct integers are everquantum-mechanically indistinguishable; here indistinguishability is the norm. The natural scalar observables are degenerate for 99.8% of basis shapes. Thecorrect observable structure is the spectral PVM {Pi_i = |m_i><m_i|}, alwayscomplete without scalar labeling. Approximately 50% of degeneracy cases arepermanently unresolvable by any symmetric function of plane indices: these areA5-conjugate spectral class pairs, a symmetry-protected structural featureabsent from the integer partition model.
We develop the computational structure of the pseudovolume tensor of Part Iin two sections. Single-site machinery establishes exact-arithmetic foundations. Thepseudovolume's five simplex vertices, lying in the integer lattice H_Z := Z^4,are projected onto the A5-invariant parallel subspace V_par ≅ R^3 to produce a131-point icosahedral quasicrystal; its A5 symmetry is verified algebraically(Procrustes error ~10^{-16}) and via golden-ratio pair-correlation spacing.Exact arithmetic is carried by the quadratic field Q(phi) = {a + b*phi : a,b inQ}, which closes under the G-metric inner products of H. The QPhiPEPS is aprojected entangled pair state on the 131 quasicrystal sites with Q(phi)-valuedvirtual bond indices and bond dimension chi = 2. The bond contraction theoremestablishes that every G-metric contraction of adjacent bond vectors is a purerational, so the MERA coarse-graining hierarchy terminates in Q. Additionalresults: the ring F_2[x]/(x^6-1) contains a [6,2,4] constant-weight code(W(z) = 1 + 3z^4); all four bulk parameters are exactly recoverable fromboundary observables with a 45:1 compression ratio. H4 multi-site extension lifts the single-site machinery to a multi-sitearchitecture. The subgroup index [2I:2T] = 5 — the unique algebraic origin ofevery occurrence of five in the model — forces a 5-site network with H4 asglobal symmetry group. The A5-equivariant inter-site bond maps force a GHZground state whose entanglement entropy equals log|M(b)| uniformly across allbipartitions. This identifies the invariant Phi(b) = log|M(b)| exactly: it issimultaneously the unnamed spectral invariant of Part I, the entanglemententropy of the 5-site network, and the contextuality measure of the local PVM.The Zimba-Penrose parallel is thereby promoted from analogy to structure. A120-cell tiling extends the network to 600 sites, introducing a qualitativelydistinct inter-cell bond algebra (intra-cell: 16 of 60 A5 elements; inter-cell:12 of 60, equal to |A4|) at each coarsening level. A robustness theorem givesthe exact ground-state fidelity threshold p*(m) = 1 - m^{-5} = 1 -exp(-5*Phi(b)) for preferred bases under independent depolarizing noise. This is Part II of a two-part work. Part I (Foundation, Construction, andProperties) is deposited at https://doi.org/10.5281/zenodo.21038657.
We extend the pseudovolume framework through three new constructions and fiveindependent structural validations spanning quasicrystal physics, geometriccoding theory, holographic coding, topological phases, and structuralbioinformatics. The construction produces a 131-point icosahedral quasicrystal via projectionof the integer lattice Z^4 onto the parallel subspace via an explicitA5-equivariant lifting matrix. Exact arithmetic is carried by the quadraticfield Q(phi) = {a + b*phi : a,b in Q} (phi = (1+sqrt(5))/2), which closesunder G-metric inner products. Building on this, we construct the QPhiPEPS: aprojected entangled pair state whose physical indices are the 131 quasicrystalsites and whose virtual bond indices lie in Q(phi)^4. The bond contractiontheorem establishes that every G-metric contraction of adjacent bond vectors isa pure rational, so the MERA coarse-graining hierarchy terminates in Q. Five structural validations confirm connections to independent scientificdomains: (1) Quasicrystal/icosahedral materials: A5 symmetry of the 131-pointsample verified algebraically and via golden-ratio pair-correlation spacing.(2) Geometric coding theory: the ring F2[x]/(x^6-1) contains a [6,2,4]constant-weight code; three MDS codes are identified. (3) Holographic coding:all four bulk parameters are exactly recoverable from boundary observables; theMERA circuit is the explicit bulk encoding map. (4) Tensor networks/topologicalphases: QPhiPEPS realizes the Fibonacci string-net model on the 131-sitequasicrystal with bond dimension chi=2, quantum dimension d_tau=phi; thetopological phase is non-abelian and universal for topological quantumcomputation. (5) Structural bioinformatics: A5 acting on the twenty standardamino acids produces exactly two orbits of size 10.
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.