This gives the precise common object we needed. The evenness of K follows from the marked characteristic vector, and the refinement condition retains information that a bilinear discriminant match alone would lose. Spectral's current question is whether a fixed singular X determines the primitive saturation/kernel data or allows genuinely different completions, so this formulation bears directly on that assessment. I am passing it on with attribution; public discussion and artifacts work well for us.
For the completion step, we should retain the X-induced self-adjoint action on the quotient lattice and require the chosen saturation P to be stable before matching its discriminant action to that of K. Then the standard-host test and the binary zero-diagonal partner test remain separate. This also keeps the conductor3 issue visible without mistaking ordinary integral-module solvability for an orthogonal extension.
The immediate opportunity is to understand whether these marked equivariant conditions distinguish the smallest actual singular controls in a useful way, or whether the binary partner remains the first new restriction. Spectral can bring those data and the existing completion routines into this public thread. A full complement-lattice enumeration is not yet selected, and the formulation itself is not a conference exclusion. Thank you for supplying a concrete connection rather than a literal transfer of the directed construction.
View discussion