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 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.
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.
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.
Infinity is central to deriving macroscopic irreversibility from reversible microscopic laws across mathematics, theoretical computer science and physics. In analysis, infinite processes—such as Dedekind cuts and Cauchy sequences—construct real numbers as equivalence classes of rational approximations, bridging discrete rationals to the continuous real line. In quantum mechanics, infinite tensor products model nested measurements, where sectorization partitions the Hilbert space into equivalence classes, reconciling unitary evolution with wavefunction collapse. In statistical mechanics, macrostates emerge as equivalence classes of microstates sharing identical macroscopic properties, providing the statistical basis for thermodynamic irreversibility despite reversible dynamics. Equivalence relations formalize For-All-Practical-Purposes (FAPP) indistinguishability, reflecting operational limits on precision and observation. Together, these examples reveal a unified framework where infinity and equivalence underpin emergent macroscopic behavior from microscopic reversibility.
This work develops a quantitative theory of how much information about a quantum parameter can and cannot be erased by unitary operations, and applies it to the θ = π quantum critical point of the massive lattice Schwinger model. For two nearby parameter-dependent quantum states, the distinguishability stored in eigenvalue populations cannot be changed by unitary rotations, whereas information stored in eigenbasis orientation can be moved by such operations. This motivates the erasability floor: the minimum fidelity-based distinguishability remaining after optimal unitary erasure. For smooth full-rank state families with simple spectra, the paper proves E_spec / E → F_cl / F_Q, where F_Q is the quantum Fisher information and F_cl is its eigenvalue-population contribution. Thus the locally uneraseable fraction of parameter information is exactly the population share of the quantum Fisher information, while the erasable fraction is its coherence contribution. The result is further extended to eigenvalue crossings and degenerate subspaces. A transversal crossing produces an explicit correction to the floor that is parabolic in the crossing position within the sampling window. Degenerate-point slopes are determined by the eigenvalues of the projected derivative within the degenerate subspace rather than by basis-dependent diagonal entries. Symmetric crossings can drive the erasability floor exactly to zero. The theoretical results are verified numerically on analytic and engineered state families, including crossing, degeneracy, additivity, and random-family tests. The framework is then applied to the massive lattice Schwinger model at θ = π with the Wilson-line zero mode retained as a dynamical rotor. Matrix-free GPU Krylov calculations extend the finite-size tower to N = 30 sites, corresponding to a Hilbert-space dimension of approximately 1.4 × 10^9. The global fidelity-susceptibility peak grows approximately as N^s with s = 1.87–1.89 over N = 22–30, corresponding to ν = 1.06(1), consistent with the two-dimensional Ising universality class. The pseudo-critical peak locations extrapolate to a lattice critical mass near 0.278 at a = 1/2, with a fit-form range extending to 0.296. The central value lies within 0.007 of the independently predicted value obtained by transporting the established continuum critical mass through the known additive lattice mass shift. The erasability floor reveals the information structure of the transition. Near criticality, only about 6–8% of the mass-parameter information carried by a gauge-invariant six-site region is population-borne, implying that more than 92% is coherence-borne and therefore movable by local unitary rotations. Away from the transition, the population contribution rises substantially. A regional susceptibility constructed from the same reduced states locates the critical point to within approximately 0.009 of the global susceptibility across all five large system sizes, demonstrating that the critical point remains locally detectable even as the total Hilbert space grows by more than two orders of magnitude. The study also reports unresolved finite-grid questions rather than forcing conclusions: monotonic scaling of the floor minimum and of the dip–peak separation cannot be established on the present mass grid and require finer sampling. Together, the results connect quantum Fisher information, unitary erasability, eigenvalue-crossing geometry, and quantum criticality within a single reproducible framework. The accompanying package includes theorem-verification scripts, the matrix-free lattice solver, validation gates, analysis code, figure-generation code, and raw large-N data.