Spectral theory for the projected-probability framework

Develop a complete spectral theory for the projected-probability framework associated with the bicomplex scalar extension, including clarification of the status of ideal-valued eigenvalues of the form kE.

Background

The paper restricts the Born-rule interpretation to the embedded complex sector and treats the ideal sector only algebraically. Consequently, the formal ideal-valued eigenvalue relation obtained in the ideal sector is not integrated into a complete spectral framework.

The authors identify the absence of such a theory as an unresolved mathematical issue and specifically state that the status of eigenvalues kE requires clarification beyond the formal relation derived in the paper.

References

A complete spectral theory for the specific projected-probability framework considered here remains to be developed. In particular, the status of ideal-valued eigenvalues of the form kE requires clarification beyond the formal algebraic relation considered here.

— A Minimal Bicomplex Extension of the Complex Scalar Algebra of Quantum Mechanics with an Ideal-Valued Sector  (2610.09747 - Otte, 7 Oct 2026) in Section 11.5, Open Mathematical Questions