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.
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.
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.
The sixty-four hexagrams of the Yijing are modeled here as a combinatorialindex for a rigid projection-invariant tensor — the pseudovolume — associatedto each positive rational scalar. The construction is organized as nine nodes xfive cardinal bodies x 64 hexagram cells, recovering the nine-node He Tu / LuoShu arrangement and identifying the five Wuxing phases with the vertices of aregular 4-simplex in four-dimensional space. The defining structural property is projection invariance: all 2,880 values areconstant across 32 projection planes corresponding to the rotation axes of adodecahedron, partitioned by Hamming weight into four axis classes. Thetransformation group is G = R x A5, where the alternating group A5 acts via anexplicit Mobius construction, connecting the hexagram combinatorics toicosahedral symmetry. The nine-node structure is the three-dimensionalprojection of two paired 4-simplices sharing a center. Each basis b determines a spectral partition M(b) of the 31 non-identityprojection planes into equivalence classes under grid equality; the associatedHilbert space is H_b = C^{|M(b)|}. The first main quantum-structural result isscale-invariance: |M(b)| depends only on the ratio class [b], not on themagnitude S = sum(b_i), so the quantum structure is magnitude-free. This isproved by tracing the linearity of the displacement function through thecardinal body construction and verified across 25,827,165 bases (sums 6--54). The second result is cross-scale indistinguishability: bases from distinct ratioclasses at distinct magnitudes routinely share identical Hilbert spaces andPVMs, with the within-sum collision fraction growing from 31.6% at S=30 to71.8% at S=54. The natural scalar observables are degenerate for effectively allbases; the correct observable structure is the spectral PVM {Pi_i = |m_i><m_i|},always complete without scalar labeling. Approximately 64% of degeneracy casesare permanently unresolvable by any symmetric function of plane indices: theseare A5-conjugate spectral class pairs. The model is fully self-inverse: all instrument parameters are exactlyrecoverable from the pseudovolume's own outputs. Structural results arecomputationally verified; the A5 identification and canonical multiplicationrule admit closed-form proofs. Code is openly available. This is Part I of a two-part work. Part II (Computation and Extension) developsthe exact-arithmetic single-site machinery and H4 multi-site extension.
This paper connects pentagonal geometric frustration, golden-ratio quasi-periodicity, and Fibonacci anyons to practical quantum computing design principles. The fusion rule τ × τ = 1 + τ algebraically forces the quantum dimension d_τ = φ = 1.618... through the equation d² - d - 1 = 0—a mathematical necessity confirmed experimentally at d_τ = 1.598 ± 0.02. The strongest empirical support comes from Fibonacci pulse sequences: Dumitrescu et al. demonstrated that quasi-periodic dynamical decoupling following Fibonacci timing extends qubit coherence from 1.5 to 5.5 seconds (3.7× improvement) on trapped-ion hardware. The mechanism—φ being the "hardest irrational to approximate" (Hurwitz's theorem)—maximally separates all resonances, preventing the constructive interference that degrades periodic sequences. Additional experimental support includes: exciton-polariton condensates on Penrose tiling lattices exhibiting C₁₀ Bragg spectra and 100× extended coherence; X-ray tomography showing decagonal quasicrystals maintaining order through phason-mediated rearrangements; and independent convergence by IBM and other groups toward topologies minimizing frequency collisions through structures the framework interprets as φ-aligned. The paper proposes specific design principles—prime qubit counts to minimize divisor resonances, φ-ratio frequency spacing, Fibonacci temporal protection, prime code distances—and generates testable predictions for current hardware. The constraint geometry framework interprets pentagonal geometry as encoding the φ-sector (diagonal-to-side ratio = φ) and the discrete sector (C₅ × C₂ = C₁₀), explaining why aperiodic structures exhibit extended coherence: they minimize discrete curvature by maximizing distance from rational approximations. The critical exponent ν = 1/ρ* ≈ 0.304 governs how coherence length diverges at organizational phase transitions. While some predictions are confirmed (Fibonacci temporal protection, anyon quantum dimension), others await systematic experimental validation.
Quasicrystals occupy a distinctive position between the translational order of crystals and the disordered amorphous matter. Simulating this physics has remained challenging, since quasiperiodic structures lack the translational symmetry exploited for crystals and generally require costly diagonalization of large finite approximants. Quasicrystalline order admits two equivalent descriptions, a cut-and-project scheme from a higher-dimensional periodic crystal, and a discrete set of substitution rules acting on a finite alphabet. We show that the latter, written in a numeration system adapted to the substitution, defines a deterministic finite automaton with output, the digits of a site index are fed, and the automaton returns the letter occupying that site. We further exploit another equivalence to a different construction, the transition matrices are exactly the tensors of a matrix product state, whose bond dimension is the number of automaton states and is independent of system size. This allows efficient representation of extremely large tight-binding Hamiltonians in the tensor-train language, thereby yielding an exact matrix product operator for the quasicrystal Hamiltonian at any system size. We show how this framework works for two families of one-dimensional quasicrystals, the metallic-mean and k-bonacci families and we explicitly construct the Fibonacci, silver-mean, and Tribonacci quasicrystals. By leveraging efficient tensor-network compression and the kernel polynomial method, we compute spectral densities for chains with more than 109 sites and directly resolve the hierarchical structure of the spectrum.
A quantum error‑correcting code is built from the classical Fibonacci substitution rule, where the code space consists of binary strings without consecutive 1’s, so that the Hilbert space dimension grows as the Fibonacci numbers. Encoding uses a tree tensor network of perfect isometries whose branching follows the inverse Fibonacci substitution. For any contiguous block of boundary qubits, the entanglement entropy equals the number of edges severed by a minimal cut through the tree, multiplied by the natural log of two. The size of that cut behaves as the logarithm base the golden ratio of the block length, giving an entropy scaling with a coefficient involving the golden ratio. The tree carries a discrete causal structure, with a scalar curvature defined from the branching number; the average curvature is twice the reciprocal of the golden ratio, indicating a hyperbolic bulk. A cut‑shift lemma describes how the minimal surface moves under boundary deformations and provides a discrete analogue of the Raychaudhuri equation. Together with the first law of entanglement, this suggests a conjecture that, in the continuum limit, the system is governed by Jackiw‑Teitelboim dilaton gravity, with Newton constant given by one quarter of the natural log of two and a cosmological constant set by the negative of twice the reciprocal of the golden ratio. The proven part of the work immediately applies to fault‑tolerant quantum memories, universal anyonic gates, circuit compression, and concrete protocols for implementation on superconducting and trapped‑ion quantum processors. The code also offers a laboratory for simulating holographic duality and black hole evaporation. The entire framework rests solely on the Fibonacci grammar and standard quantum mechanics.
The non-Abelian topological phase with Fibonacci anyons minimally supports universal quantum computation. In order to investigate the possible phase transitions out of the Fibonacci topological phase, we propose a generic quantum-net wave function with two tuning parameters dual with each other, and the norm of the wave function can be exactly mapped into a partition function of the two-coupled ϕ^{2}-state Potts models, where ϕ=(sqrt[5]+1)/2 is the golden ratio. By developing the tensor network representation of this wave function on a square lattice, we can accurately calculate the full phase diagram with the numerical methods of tensor networks. More importantly, it is found that the non-Abelian Fibonacci topological phase is enclosed by three distinct nontopological phases and their dual phases of a single ϕ^{2}-state Potts model: the gapped dilute net phase, critical dense net phase, and spontaneous translation symmetry breaking gapped phase. We also determine the critical properties of the phase transitions among the Fibonacci topological phase and those nontopological phases.