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, the reduced spectral determinant on the fermionic sector is approximately 2.58 x 10^-53, and massless neutrinos lie in the kernel of KOS as record modes of the dissipative boundary semigroup. From this identified spectrum the paper derives spectral determinant and zeta invariants, internal power sums and boundary budget scaling, gauge coupling normalization ratios, Higgs quartic boundary-condition data in the sigma-extended sector, a one-loop Higgs mass locus with an implied seesaw scale, cosmological-constant cap scaling proportional to H0 squared, and structural exclusion of the strong CP phase via Osterwalder-Schrader positivity. The Dirichlet-to-Neumann construction provides a positive self-adjoint, determinant-class boundary operator consistent with Osterwalder-Schrader reconstruction. Within the coupled Dirac-Lambda closure program, KOS is the boundary object through which dissipative admissibility, boundary budgeting, and internal spectral data are realized in a single operator-theoretic framework. This identification places noncommutative spectral geometry and Osterwalder-Schrader boundary reconstruction within a common operator-theoretic setting and establishes KOS 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.