We establish two related structural results on the five-dimensional hypercube Q5. First, every symmetric three-sector Hamiltonian of mediated form arises as the unique compression of a Q5 operator family to the complement-symmetric block subspace K = span|L⟩, |M⟩, |R⟩, with diagonal energies generated by aI + bB̂ + cΠ̂ and off-diagonal couplings generated by the Gray-weighted transfer operator Aγ. Under this compression, the symmetric coupling condition vL = vR is emergent from complement symmetry rather than imposed, and the bulk asymmetry operator ηadj (PR − PL) is absorbed into the coefficients of I and B̂. Second, in any three-sector Hamiltonian with nonzero mediated couplings vL, vR ≠ 0 and nondegenerate outer energies EL ≠ ER, no eigenstate is localized in a single phase sector: every eigenstate has strictly nonzero amplitude in all three sectors. These results establish that the three-sector Hamiltonian underlying the mediated asymmetry framework is not a primitive model but a compressed object arising from Q5 geometry, and that eigenstates are intrinsically multi-phase distributed objects.
Craig Edwin Holdway (Fri,) studied this question.