Spectral Invariance and the {0, 3, 0} Filter in Triple Hopf Structures

We introduce a circulant phase matrix P encoding the ℤ₃ symmetry of the triple coproduct structure on the affine 0-Hecke monoid H₀(Ã₂). We prove that its spectrum Spec(P) = {0, 3, 0} provides an algebraic mechanism for selective mode filtering. The unique non-zero eigenvalue μ = 3 corresponds to the ℤ₃-invariant mode, while the null eigenspace annihilates all non-invariant fluctuations. This spectral structure offers a purely algebraic model for mass gap emergence in gauge theories: divergent vacuum modes are projected out, leaving only the stable trace-invariant sector.

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