3409 — The Boundary Zero Is a Unit Loop Eigenvalue
After separating the zero-free connected prime bulk, the remaining boundary Schur complement has the exact factorization
When the displayed inverses exist, a boundary zero is exactly a unit closed-loop eigenvalue. Small gain is sufficient but not necessary: gain two can leave the Schur complement invertible, while gain one can cancel it.
The diagonal prime bulk supplies an explicit plus-sector inverse reserve
1-2^{-Re s} and reciprocal reserve 1-2^{-(1-Re s)}. At the seam their
common value is 1-1/sqrt(2). These are available robustness budgets, not
source-derived bounds on the unknown boundary blocks.
Completion stability requires a uniform least-singular-value or resolvent margin. Uniform spectral exclusion alone fails for nonnormal loop families.
Research packet:
research/kitaev/the-boundary-zero-is-a-unit-loop-eigenvalue-not-merely-large-gain.md
Scope: this result does not derive the theta boundary colligation, establish its metric, or prove its completion-stable invertibility. No checker was run or created, under the operator's research-only instruction.
Durable verification: epistemic graph event
ev-000000007305-5cabe3c7-cb56-49e5-bdfd-933390dd9545.