Active/Complement Transition Correspondence on Q5

This paper establishes that the distinguished-coordinate decomposition of Q₅ induces a canonical five-level complementary grading and a two-channel transition algebra that appears simultaneously in three forms: Gray edges ⟺ operator transitions ⟺ Hamiltonian commutator sectors Fixing coordinate i decomposes each vertex as \[ x = (xᵢ, x_comp) \] with complementary Hamming weight \[ w_comp(x) = Σⱼ≠ᵢ xⱼ \] and sector label \[ m(x) = 2 − w_comp(x) ∈ {−2,−1,0,1,2} \]. The operator \[ J_z^(i) = ½Σⱼ≠ᵢ Zⱼ \] encodes this grading exactly. Active-bit flips \[ (xᵢ via s±^(i)) \] preserve m; complementary-bit flips \[ (via J±^(i)) \] shift m by ±1. The spectral projectors Πₘ^(i) select the weight-(2−m) layer of the complementary Q₄ fibre. Under \[ H₀ = αΣZₖ + βΣZₖZₗ, \] the active commutator satisfies \[ [H₀, s±^(i)] = ±2(α + 2βm)s±^(i) \] and the complementary commutators are verified by explicit matrix computation with signs fixed to align with the Gray-edge orientation. All seven lemmas are solid. The \[ J±^(i) \] commutator sign convention was verified by explicit matrix computation (May 17, 2026): magnitude \[ 2(α + 2β) at m=1, z_i=+1 \] confirmed exact, with sign orientation fixed to align with the Gray-edge directionality of TA6–TA7.