Natural spectra and arithmetic of lens spaces

The lens spaces L(49; 1, 6, 15) and L(49; 1, 6, 20) have identical natural spectra precisely for five irreducible SO(5)-types. This proves the finiteness conjecture of Lauret, Miatello and Rossetti. The types admitting bounded or compact self-adjoint corrections are classified, and the least operator norm and essential norm are determined for every one-row type on the bounded-correction locus. The distinction is arithmetic: exact equality requires equality of each Casimir eigenvalue multiplicity, whereas bounded correction requires equality of the total multiplicities in each separated quadratic cluster. Periodic character identities give both conditions and the asymptotic number of unmatched frequencies. For L(p²; 1, 1 ± p, 1 ± 3p), the bounded-correction classification holds for every prime p ≥ 5. For flat bundles on L(q; 1ᵐ, a), with q ≥ 3 odd, m ≥ 1, and a a unit modulo q, equality of connection spectra is characterized in terms of the holonomies. For effective prime-square lens orbifolds, the scalar spectrum determines the gcd classes of the rotation weights and implies isometry of the corresponding proper covers at orders 9 and 25.