Six-Field: A Constrained Eigenvalue Problem Underlying the Particle Mass Spectrum

This document formulates a precisely defined mathematical problem arising from the Six-Field framework: the formalization of a generalized, pre-geometric eigenvalue structure whose discrete spectrum empirically reproduces the full hierarchy of known fundamental particle masses. Rather than proposing a new theory ad hoc, it specifies a constrained operator-admissibility challenge—identify operator structures that necessarily yield a quadratic eigenvalue spectrum λn∼n2\lambda_n \sim n^2λn∼n2 under strict bottom-up, single-substrate, and pre-geometric conditions. The work consolidates the mathematical structures already established, including a generalized eigenvalue formulation at substrate level and a finite relational propagation bound normalized to unity (c=1c = 1c=1), and clearly separates them from elements that remain to be formally derived. A concrete set of open formalization tasks is enumerated—explicit operator forms, proof of quadratic spectral necessity, mode-selection rules, quantum-number constraints, and the status of the scale parameter—each framed as a tractable problem for mathematicians and mathematical physicists. Importantly, the framework is particularly suited to early validation through mathematical admissibility studies and re-analysis of existing datasets, without requiring new experiments or new experimental infrastructure. The problem specification presented here forms the mathematical core of a broader operator-first research trajectory, whose continuation depends explicitly on the outcome of the formalization challenges identified in this document.