Pseudovolumes III: Scale-Invariant Spectral Hilbert Spaces and Cross-Scale Indistinguishability

We develop a complete quantum information formalization of the spectral
structure of a rigid projection tensor (pseudovolume), establishing two
structural results that go strictly beyond Khrennikov's quantum integer
framework (2010).

The spectral Hilbert space H_b = C^|M(b)| assigns to each basis shape b (a
vector of positive rational components) a Hilbert space whose dimension equals
the number of spectral classes — equivalence classes of 32 projection planes
under a grid-equality relation. The first main result is scale-invariance:
|M(b)| depends only on the ratio class [b], not on the scalar magnitude
S = sum(b_i). In Khrennikov's framework the partition count p(n) is strictly
increasing; here the quantum structure is magnitude-free.

The second result is cross-scale indistinguishability: basis shapes from
distinct ratio classes at distinct magnitudes routinely share identical Hilbert
spaces and projection-valued measures, with hundreds of thousands of non-trivial
collision pairs across small magnitudes and onset of asymptotic universality at
higher sums. In Khrennikov's framework no two distinct integers are ever
quantum-mechanically indistinguishable; here indistinguishability is the norm.

The natural scalar observables are degenerate for 99.8% of basis shapes. The
correct observable structure is the spectral PVM {Pi_i = |m_i><m_i|}, always
complete without scalar labeling. Approximately 50% of degeneracy cases are
permanently unresolvable by any symmetric function of plane indices: these are
A5-conjugate spectral class pairs, a symmetry-protected structural feature
absent from the integer partition model.