Spectral duality of Dirac operators on dual regular polytopes: exact conjugacy and cohomological obstruction

We establish the exact conjugacy of Dirac operators on the boundary chain complexes of dual regular convex polytopes. For a dual pair (X, X*) in dimension n, we prove the existence of a permutation-with-signs Ψ: C(X*) → Cn−1−•(X) with Ψ DX* Ψ⁻¹ = D^X as an exact identity of integer matrices, and identify the obstruction to finding such a Ψ as a pure permutation as a cohomological invariant [Ψ] ∈ H¹(L(X); {±1}) on the face lattice of X. The main results are: (i) a duality lemma relating cellular duality to operator conjugation; (ii) a proof that [Ψ] = 0 for the infinite family {n-cube, n-cross}, n ≥ 2; (iii) explicit constructions and exact computational verifications for all dual pairs of regular polytopes in dimension n ≤ 4: the family {8-cell, 16-cell} (case n = 4), the pair {icosahedron, dodecahedron}, and the pair {600, 120} treated in the companion article [(Fernández Paniagua, Zenodo 2026, DOI: 10.5281/zenodo.24040098)] (subsuming Theorem 10.2 of that work); (iv) an observed (G, det)-twisted equivariance of Ψ, verified by direct computation for the hyperoctahedral groups B_n (n = 2, 3, 4) and the icosahedral group H_3, and conjectured in general. All computations reported are exact integer identities, reproducible from the supplementary scripts. The note is purely algebraic and combinatorial and makes no physical claims.