Forced Residual Circulation Block

TA37 derives the forced residual circulation block \( E_B: H_R → H_R \) from the helical leakage adjacency structure of T127 via Schur elimination of the mediating middle mode u_3. This is the second theorem of the forced block trilogy (TA36 for D_B, TA37 for E_B, TA38 for C_B). The helical leakage generator \( B = i*A_L \) acts on \( span{u_2, u_3, u_4} \) with the chain u_2 ↔ u_3 ↔ u_4 and no direct u_2 ↔ u_4 coupling (T127). The residual sector \( H_R = span{B_L, B_R} \) corresponds to the outer modes u_2 and u_4. Partitioning B into outer-outer \( (B_oo = 0) \), outer-middle (B_om), middle-outer (B_mo), and middle-middle (B_mm = 0) blocks, the stationary-frequency Schur elimination of u_3 gives \[ E_B(λ) = B_oo + (1/λ) B_om B_mo \]. Three lemmas establish the result. Lemma 1 performs the Schur elimination. Lemma 2 computes the four scalar coupling products: \( B[u2,u3]·B[u3,u2] = -1/80, B[u2,u3]·B[u3,u4] = -i/80, B[u4,u3]·B[u3,u2] = i/80, B[u4,u3]·B[u3,u4] = -1/80 \) , using \( i² = -1 and eiπ/4·eiπ/4 = i \). Lemma 3 assembles these into \[ B_om B_mo = (1/80) * [[-1,-i],[i,-1]] = (1/80)(σ_y − I) \], giving the main result: \[ E_B(λ) = (1/80λ)(σ_y − I) \]. An earlier theorem summary recorded \( E_B(λ) = (1/80λ)σ_y \). Explicit Schur elimination shows this is missing the diagonal self-energy term −I/80λ arising from the round-trip contributions B[u2,u3]·B[u3,u2] and B[u4,u3]·B[u3,u4], each equal to −1/80. This is standard Schur/Feshbach behaviour: second-order mediated processes generically produce both off-diagonal coupling and diagonal self-energy shifts. The correction is documented transparently within the theorem. The corrected operator has eigenvalues 0 and −1/40λ (for positive real λ), rather than the symmetric pair ±1/80λ from the pure σ_y form. This yields one conserved circulation mode and one damped corrective mode. The spectral abscissa \( ω(E_B) = 0 \), placing the residual ecology at the conservative/stable boundary of the TA33 classification. This is structurally more natural than purely oscillatory behaviour. E_B is not a free parameter. It is determined entirely by A_L from T127 through the Schur elimination. The forced structure confirms that all outer-mode coupling passes through the mandatory middle-mode mediator u_3, and the off-diagonal structure of E_B is entirely second-order in this mediation.