Forced Residual Extraction Block

TA36 establishes the forced residual extraction block D: HM → HR 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 PR 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 \ (HR = spanBL, BR \) 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 \ (rD (ψM) = RD ψM ∈ Dₐ₅ \) (up to 32-dimensional) from the boundary-resolved residual defect, defined by two dual boundary probes \ (χL, χR ∈ Dₐ₅* \) determined by Q5 barrier geometry: \ δL (ψM) = ⟨χL, RD ψM⟩, δR (ψM) = ⟨χR, RD ψM⟩ \. The linear defect operator \ (RD = DY − D^adm \): \ (HM → Dₐ₅ \) is defined by upgrading the admissible target deposition to a linear map \ (D^adm: HM → Dₐ₅ \). The extraction block is then \ D ψM = δL (ψM) BL + δR (ψM) BR = PR RD ψM \, with boundary-resolved residual projection PR defined by the two probes. This construction forces rank (D) ≤ 2 automatically, consistent with the two-dimensionality of HR. Four lemmas establish: the rank constraint rank (D) ≤ 2 (elementary from codomain dimension) ; the forcing condition \ (εR (ψM) > 0 ⟹ DψM ≠ 0 \) (from linear independence of BL, BR) ; the vanishing condition \ (εR (ψM) = 0 ⟹ DψM = 0 \) ; and the norm identity \ (‖DψM‖ = εR (ψM) \) under the orthonormal normalization convention \ (‖BL‖ = ‖BR‖ = 1, ⟨BL, BR⟩ = 0 \) (basis-independent in the sense that any orthonormal basis for HR 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 (δL, δR) of the full deposition mismatch participate in extraction. The remaining directions of the mismatch do not enter HR 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 χL, χR 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.