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.
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.
We establish that four previously independent structural results on the five-dimensional hypercube Q₅ maximal parity alignment in Hamiltonian cycle structure, parity-optimal cycle structure with its induced 2-adic tier organization, the active/complementary transition decomposition, and the graded adjoint operator decomposition under an Ising-type Hamiltonian, are equivalent realizations of a single graded operator system induced by a distinguished coordinate decomposition. Two lemmas establish the core compatibility: the complementary Hamming-weight grading coincides exactly with the spectrum of the grading operator J_z^(i), and the active/complementary transition partition matches the commutation structure of the transition operators. The unification theorem then identifies the global Hamiltonian cycle structure, the local transition decomposition, and the sector-resolved operator dynamics as consistent manifestations of one underlying finite combinatorial-operator system on Q₅. No new combinatorial or operator identities are introduced. The result makes explicit that the constituent theorems describe the same object.
We identify a distinguished quadratic diagonal operator on the five-dimensional hypercube Q₅, selected uniquely (up to scale) by a pair of anticommutation conditions on the dominant coordinates and commutation conditions on the subdominant coordinates. This operator, the closure-parity operator Ĉ = Z₁Z₂, defines a second Z₂ grading independent of global parity. The main theorem proves that Ĉ is the unique operator in the degree-≤2 diagonal sector satisfying these transition conditions. We show that Ĉ commutes with the diagonal Hamiltonian H₀, so closure parity is conserved under diagonal evolution, and derive the resulting bi-graded refinement of the H₀ eigenspaces with explicit sector dimensions (1,0), (3,2), (4,6), (4,6), (3,2), (1,0) across weights 0–5. A closed formula for closure parity along the reflected Gray cycle is derived. Interpretive sections describe the fibrewise three-level structure that follows from the bi-grading, outer states in the +1 sector, a doubly-degenerate intermediate bridge in the −1 sector, and the second-order mediated coupling between outer states arising through virtual transitions into the intermediate sector.
This Theorem closes the second open assumption of the T39 bridge lemma by deriving the leg-matching identity a = √2‖v_L‖ + O(δ²_eff) from Q5 carrier structure. Two independent Q5-native mechanisms force the leading-order identity. The complement involution ι: x ↦ 1 − x maps the Hamming-weight-1 subspace isometrically onto the Hamming-weight-2 subspace, intertwines the T24 ambient su(2) action by sign-reversing Ĵ_z and exchanging Ĵ_±, and satisfies ι(v_L) = v_R with ‖v_R‖ = ‖v_L‖. Within the spin-1 module V₁ carried by the Hamming-weight-1 sector, the raising operator Ĵ₊ acts with matrix element √2 between adjacent weight states, a standard spin-1 Clebsch-Gordan coefficient. Combining: at leading order a = √2‖v_L‖, where the √2 is the canonical spin-1 matrix element, not a normalization convention. The Y-transport correction is quadratic not linear: since θ = π/2 + O(δ_eff) from T43, the amplitude scales as sin θ = cos(O(δ_eff)) = 1 − O(δ²_eff), giving a = √2‖v_L‖ + O(δ²_eff). The leading-order result is stable to first order in the off-support leakage. Together with T43, both open assumptions in the T39 bridge lemma are now derived results. Status: Complement isometry solid: direct computation on the residual carrier basis. Spin-1 matrix element solid: standard result. Leading-order leg-matching conditional on identification of transport leg amplitude with Ĵ₊ matrix element on V₁ is structurally natural within the T24–T25 framework but not yet derived from explicit kernel-level transport generators. Quadratic correction inherits T43 parametric control conditionality. All results inherit T22–T25 and T39 conditionality. Dependencies: T22, T24, T25, T39, T43.
T96 provides the explicit matrix realization of the gate-admission and leakage operators that were introduced abstractly in T93–T95. The theorem establishes a canonical orthogonal decomposition of the reduced Q5 crossing sector HPQ=C2 into a Y-admissible crossing mode and its leakage complement. This converts the leakage architecture from an abstract projector framework into an explicitly computable model. The admissible crossing mode is defined as ∣Y⟩=21(1i), representing the quarter-turn phase relation associated with the Y-gate. The corresponding gate projector is ΠY=∣Y⟩⟨Y∣=21(1i−i1), and the leakage complement is K=I−ΠY=21(1−ii1). The theorem proves that both operators are orthogonal projectors satisfying \[\Pi_Y^2=\Pi_Y,\qquadK^2=K,\qquad\Pi_YK=K\Pi_Y=0,\] yielding the canonical decomposition HPQ=span{∣Y⟩}⊕ker(ΠY). Every reduced crossing state ψ=(PQ) admits the unique decomposition ψ=ΠYψ+Kψ, with norm identity ∥ψ∥2=∥ΠYψ∥2+∥Kψ∥2. The theorem further derives explicit formulas for gate admission and leakage strength. The crossing-mode amplitude is ⟨Y∣ψ⟩=21(P−iQ), and therefore the gate-admission strength is CY(ψ)=∥ΠYψ∥2=21∣P−iQ∣2. For normalized states, DY(ψ)=∥Kψ∥2=1−21∣P−iQ∣2, so that CY(ψ)+DY(ψ)=1. Gate coherence and leakage are therefore complementary quantities. Several structural consequences follow. A state crosses cleanly if and only if Kψ=0, equivalently ψ=c(1i) for some complex scalar c. Any deviation from this quarter-turn phase relation produces a nonzero leakage component. The theorem also provides the explicit realization of the abstract leakage operator appearing in T93–T95. Quantities such as the leakage norm ∥Kψ0∥2 and the short-time visibility decay derived in T95 become directly computable from the crossing amplitudes P and Q. The projector formalism, therefore, acquires a concrete matrix representation suitable for explicit calculations. T96 serves as the foundational algebraic theorem of the leakage program. It identifies the admissible crossing mode, constructs the associated projector and leakage complement, establishes the orthogonal decomposition of the reduced crossing sector, and converts the abstract coherence and leakage measures of T93–T95 into explicit computable formulas.