Defect Survival Under Reduced Transport Compression

Within the Q5 Gray-code framework, this Theorem shows that the unique Gray-class closure defect survives the full reduced transport compression as a nonzero, non-removable remnant. Nine supporting lemmas establish the result in sequence. The distance-2 adjacency lattice of Q5 contains exactly 160 undirected edges (32 vertices × C (5, 2) = 10 neighbours / 2). The Gray-class closure traversal count is 161, giving closure excess τ = 1. The parity-optimal vector H₅ = −2, −2, −4, −8, 16 has normalized dominant residual aₘax = 16/16 = 1. These coincide: τ = aₘax = 1. The unique, unmatched defect bond is localized at Gray vertices g (15) = 8 and g (16) = 24, Hamming-weight type (1, 2). The oriented defect operator Ed = |8⟩⟨24| − |24⟩⟨8| is nonzero and skew-Hermitian. Under the retained-pairing reduction, all 160 matched closure classes cancel, and the reduced commutator remnant is Q = Ed. Projection to the A-supported terminal sector preserves the defect: RA = P·Q·P = Ed ≠ 0. Scalar collapse gives Πcol (RA) = ⟨8|Ed|24⟩ = 1. Main result: RA = ΠA·Πₛub·Πₚair·Πᵣet (M, D) ·ΠA = Ed ≠ 0 Πcol (RA) = τ = aₘax = 1 The same unit defect appears simultaneously in the combinatorial, spectral, and reduced-operator layers. RA is an operator lift of the unique unit counted combinatorially as the Gray closure excess. Status: Distance-2 count, closure excess, parity vector, defect-bond localization, concrete defect operator, and scalar readout are all solid by direct computation. Matched-class cancellation (Lemma 36. 7) and survival under A-sector projection (Lemma 36. 8) conditional on the retained-pairing assumption: full derivation from T17 kernel architecture first principles remains open. All results inherit T16/T17/T20 conditionality. Forward consequence: triple equality τ = aₘax = Πcol (RA) = 1 closes the T34 forward remark once T37 establishes two-dimensionality of the defect sector. Dependencies: T1, T2, T7, T14, T15, T16, T17, T18, T19, T20, T29, T30, T33, T34, T35.