KOS as the Osterwalder–Schrader Boundary Operator of the Standard Model Spectral Triple

This paper identifies the structural boundary operator underlying the coupled Dirac–Lambda framework as KOS, the Dirichlet-to-Neumann map of the squared Dirac operator on a collar boundary, restricted to the internal noncommutative geometry of the Standard Model spectral triple. We prove that the nonzero internal spectrum of KOS coincides with the singular-value spectrum of the Yukawa mass matrices. The nine Standard Model fermion masses appear with fixed representation-theoretic multiplicities {4, 4, 4, 12, 12, 12, 12, 12, 12}, independent of CKM and PMNS mixing. The total dissipative channel count is 84, and massless neutrinos reside in the kernel as record modes of the boundary semigroup. From this identified spectrum we compute: • Spectral determinant and zeta invariants• Internal power sums and boundary budget scaling• Gauge coupling normalization ratios• Higgs quartic boundary condition in the sigma-extended sector• One-loop Higgs mass locus and implied seesaw scale• Cosmological constant cap scaling proportional to H0 squared• Structural exclusion of the strong CP phase via OS positivity The Dirichlet-to-Neumann construction provides a positive selfadjoint, determinant-class boundary operator consistent with Osterwalder–Schrader reconstruction. Within the capacity-constrained Dirac–Lambda scheme, this operator is not optional: OS-reconstructibility, determinant-class admissibility, and the global capacity inequality uniquely identify KOS up to a single normalization constant fixed by the UV anchor. This identification places noncommutative spectral geometry and OS boundary reconstruction within a common operator-theoretic framework and establishes the boundary operator as the central structural object of the closure program. The framework is formulated in Euclidean signature and uses fixed global spectral and determinant conventions throughout.