Forced Residual Extraction Block

TA36 establishes the forced residual extraction block D: H_M → H_R as a geometric consequence of Q5 boundary structure, and corrects a linear-algebraic obstruction present in an earlier formulation. The earlier version claimed that the residual projection P_R could be faithful on the full deposition-defect subspace. This claim is obstructed by rank-nullity: a linear map into the two-dimensional residual sector \( H_R = span{B_L, B_R} \) cannot be injective on a subspace of dimension greater than 2. The corrected formulation explicitly identifies and resolves this obstruction. The repair distinguishes the full deposition defect \( r_D(ψ_M) = R_D ψ_M ∈ DQ5 \) (up to 32-dimensional) from the boundary-resolved residual defect, defined by two dual boundary probes \( χ_L, χ_R ∈ DQ5* \) determined by Q5 barrier geometry: \[ δ_L(ψ_M) = ⟨χ_L, R_D ψ_M⟩, δ_R(ψ_M) = ⟨χ_R, R_D ψ_M⟩ \]. The linear defect operator \( R_D = D_Y − Dᵃᵈᵐ \): \( H_M → DQ5 \) is defined by upgrading the admissible target deposition to a linear map \( Dᵃᵈᵐ: H_M → DQ5 \). The extraction block is then \[ D ψ_M = δ_L(ψ_M) B_L + δ_R(ψ_M) B_R = P_R R_D ψ_M \], with boundary-resolved residual projection P_R defined by the two probes. This construction forces rank(D) ≤ 2 automatically, consistent with the two-dimensionality of H_R. Four lemmas establish: the rank constraint rank(D) ≤ 2 (elementary from codomain dimension); the forcing condition \( ε_R(ψ_M) > 0 ⟹ Dψ_M ≠ 0 \) (from linear independence of B_L, B_R); the vanishing condition \( ε_R(ψ_M) = 0 ⟹ Dψ_M = 0 \); and the norm identity \( ‖Dψ_M‖ = ε_R(ψ_M) \) under the orthonormal normalization convention \( ‖B_L‖ = ‖B_R‖ = 1, ⟨B_L, B_R⟩ = 0 \) (basis-independent in the sense that any orthonormal basis for H_R yields the same norm). The key conceptual shift from TA32: TA32 established qualitatively that nonzero Hamming-deposition defect forces nonzero residual extraction. TA36 refines this; only the boundary-accessible moments (δ_L, δ_R) of the full deposition mismatch participate in extraction. The remaining directions of the mismatch do not enter H_R through D. This is physically more plausible: in reduced-channel transport systems, not every internal mismatch mode couples to every boundary channel. The explicit dual probes χ_L, χ_R are determined by Q5 boundary geometry and Möbius-access structure but are not computed explicitly here; their derivation is a remaining open computation addressed in TA37-TA38. Crucially, TA36 no longer depends on those computations for its structural validity; the rank constraint, forcing condition, and norm identity are all established without requiring the explicit probe values.