We establish a spectral decomposition for the projection operator arising in Kosmoplex Theory. The OBMT (Octal Binomial--Modular Transform) operator $\mathcal{O}: \mathcal{H}_8 \to \mathcal{H}_4$ maps octonionic state vectors to spacetime observables. Its spectral structure reveals a fundamental duality: the projection operates through two distinct channels. The \emph{eigenvector channel}, organized by the Fano plane $\mathrm{PG}(2,2)$, yields 42 canonical glyphs with channel capacity $2\binom{8}{4} - 3 = 137$, corresponding to the fine-structure constant $\alpha^{-1}$. The \emph{eigenvalue channel}, organized by the Pascal 8-simplex, yields 80 primordial primes with channel capacity $\binom{9}{4} - 1 = 125$, corresponding to the Higgs vacuum expectation value (VEV) scale. The eigenvector channel carries directions (dimensionless) and loses 3 degrees of freedom to a triadic gauge redundancy; the eigenvalue channel carries magnitudes (dimensioned) and loses 1 to global scaling equivalence. The four coprime residue classes $\{1, 3, 5, 7\} \pmod{8}$ form the Klein four-group, mapping isomorphically onto the four normed division algebras. The eigenvalue channel exhibits symmetry breaking: it cycles through only three basins, with the Complex basin ($\C$) excluded, the geometric origin of electroweak symmetry breaking. The Hubble tension is conjectured to arise from the ratio of channel capacities: $137/125 = 1.096$.