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Closure-Parity Selection and Bi-Graded Structure on Q5

Craig Edwin Holdway

Published Apr 5, 2026• 0 citations
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Abstract

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.

DOI
10.5281/zenodo.19431284

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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.
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 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.
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,xcomp) x = (xᵢ, x_comp) x=(xi​,xc​omp) with complementary Hamming weight wcomp(x)=Σj≠ixj w_comp(x) = Σⱼ≠ᵢ xⱼ wc​omp(x)=Σj​=i​xj​ and sector label m(x)=2−wcomp(x)∈−2,−1,0,1,2 m(x) = 2 − w_comp(x) ∈ {−2,−1,0,1,2} m(x)=2−wc​omp(x)∈−2,−1,0,1,2. The operator Jz(i)=½Σj≠iZj J_z^(i) = ½Σⱼ≠ᵢ Zⱼ Jz(​i)=½Σj​=i​Zj​ encodes this grading exactly. Active-bit flips (xivias±(i)) (xᵢ via s±^(i)) (xi​vias±(i)) preserve m; complementary-bit flips (viaJ±(i)) (via J±^(i)) (viaJ±(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, H₀ = αΣZₖ + βΣZₖZₗ, H0​=αΣZk​+βΣZk​Zl​, the active commutator satisfies [H0,s±(i)]=±2(α+2βm)s±(i) [H₀, s±^(i)] = ±2(α + 2βm)s±^(i) [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) J±^(i) J±(i) commutator sign convention was verified by explicit matrix computation (May 17, 2026): magnitude 2(α+2β)atm=1,zi=+1 2(α + 2β) at m=1, z_i=+1 2(α+2β)atm=1,zi​=+1 confirmed exact, with sign orientation fixed to align with the Gray-edge directionality of TA6–TA7.
This Theorem establishes a complete transport-to-visible reduction theorem for a five-fibre system governed by Gray-structured transport on Q5. The cyclic reflected Gray code produces a unique surviving antisymmetric ordered source, the (5,1) closure class with ordered-pair count C₅₁ = 16. This reduces under the canonical projection–pairing–retention pipeline to a nearest-neighbour antisymmetric commutator operator K on a three-dimensional carrier space. Eliminating the mediator channel via Schur reduction forces a rank-one quadratic visible operator Q⊥ ~ vvᵀ, whose phase-bearing component is extracted by μ(Q) = ½Tr(QR). Path-class decomposition gives the single-sheet coefficient law μ = 2κ_adj + κ_nonadj; cross-sheet doubling gives the observable coefficient law: μ_obs = 4κ_adj + 2κ_nonadj Independently, Schur reduction of the five-fibre Hamiltonian gives h_z = v_L·v_R/Δ, bilinear in left and right couplings. Under the transport–Hamiltonian matching hypothesis in the near-symmetric regime, μ = 2h_z + O(δ²_eff). The bilinearity is irreducible (no subsystem supported on a single visible branch can reproduce a nonzero coefficient), establishing that the observable two-state dynamics are intrinsically non-factorizable. A geometric holonomy lemma establishes that μ_obs is a loop invariant, nontrivial despite vertex-level closure, vanishing on commutative transport classes. A Berry Phase Correspondence Criterion identifies the conditions under which γ_Berry = μ_obs + O(δ²), without asserting those conditions are satisfied. Status: Gray ordered-pair counts, uniqueness of (5,1) closure source, zero-pattern of K, rank-one quadratic emergence, phase extraction, cross-sheet doubling, bilinearity, and no-one-sided-support result all solid. Conditional on phase dressing assumption (full derivation from T17 kernel generators open. Conditional on path-class multiplicity), a 2:1 ratio requires explicit enumeration from T17 kernel geometry. Conditional on transport–Hamiltonian matching, bridge requires a near-symmetric regime hypothesis and second-order leg matching justification.
We present three theorems that complete the structural account of coherence and decoherence in the reduced transport sector of the Q5 framework. This mathematical system derives operator structures from the combinatorial geometry of the five-dimensional hypercube and its Gray code Hamiltonian cycle. Theorem 50 identifies coherence as the survival of the commutator-driven transverse mode under transport, and establishes a precise structural distinction between reversible phase scrambling and irreversible decoherence. No new operators are introduced; all objects are inherited from the prior theorem chain (T1–T49). Theorem 51 derives the structural form of the effective decoherence rate as a product of four independently motivated factors: a base loss rate, a quadratic coupling asymmetry term, a saturating orbit depth function, and a Lorentzian detuning suppression factor. Each factor is grounded in a specific prior result in the Q5 transport chain. Theorem 52 establishes that the coherence decay rate is the unique globally attracting fixed point of a self-consistent kernel-induced dynamical map. Existence and uniqueness are proved via the intermediate value theorem applied to a strictly decreasing map; global convergence follows from even/odd subsequence monotonicity. The exponential saturation law is derived as the asymptotic reduction of the fixed-point structure in the strong-suppression regime, rather than postulated. The primary remaining open step is the explicit derivation of the coupling parameter β from the commutator [M_n, D_n]Y geometry of Theorem 29. All results are conditional on the admissible linked-pair reduction of Theorem 17 and the retained-pairing assumption of Theorem 36. Epistemic classifications are provided for each result.
This paper establishes a structural and exact result for mediated three-sector systems, demonstrating asymmetry between transport and reconstruction channels in a closed finite-dimensional setting. A minimal three-sector model is introduced in which outer states are coupled only through a mediated intermediate state. Using second-order reduction, it is shown that outer-sector transport is governed by inverse detuning structure, while asymmetry between channel-resolved amplitudes is controlled by detuning imbalance. This produces suppression of reconstruction under restricted return operations, while exact reversibility remains preserved under full unitary evolution. An exact realization of this mechanism is then provided through a three-mode packet model derived from the Q₅ framework. In this setting, forward and return amplitudes are computed explicitly, yielding a coefficient hierarchy in which forward contributions are order-one while return contributions are suppressed by approximately three orders of magnitude. This leads to a rigorously defined coherence window in which forward amplitude strictly dominates return amplitude. The results establish that forward dominance is not an isolated numerical artifact, but arises from a structural asymmetry inherent in mediated multi-channel systems. Together, the structural theorem and exact realization provide a unified account of reconstruction suppression and transport dominance in a closed-system setting. This work forms part of a broader program investigating operator structure, grading behavior, and sector-resolved dynamics on the five-dimensional hypercube (Q₅), with related results available in accompanying white papers and operator decomposition studies.