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*) → C(X) reversing the grading, with block components Ψ_k: C_k(X*) → C_{n−1−k}(X), such that Ψ D^{X*} 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(Γ(X); {±1}) on the Hasse diagram Γ(X) of 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 nontrivial dual pairs of regular polytopes in dimension n ≤ 4: {cube, octahedron} (case n = 3), {icosahedron, dodecahedron}, {8-cell, 16-cell} (case n = 4), and {600, 120} treated in the companion article (DOI: 10.5281/zenodo.24040098); (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 are exact integer identities, reproducible from the supplementary scripts. The note is purely algebraic and combinatorial and makes no physical claims.