TA37 derives the forced residual circulation block \ (EB: HR → HR \) from the helical leakage adjacency structure of T127 via Schur elimination of the mediating middle mode u₃. This is the second theorem of the forced block trilogy (TA36 for DB, TA37 for EB, TA38 for CB). The helical leakage generator \ (B = i*AL \) acts on \ (spanu₂, u₃, u₄ \) with the chain u₂ ↔ u₃ ↔ u₄ and no direct u₂ ↔ u₄ coupling (T127). The residual sector \ (HR = spanBL, BR \) corresponds to the outer modes u₂ and u₄. Partitioning B into outer-outer \ ( (Bₒo = 0) \), outer-middle (Bₒm), middle-outer (Bₘo), and middle-middle (Bₘm = 0) blocks, the stationary-frequency Schur elimination of u₃ gives \ EB (λ) = Bₒo + (1/λ) Bₒm Bₘo \. Three lemmas establish the result. Lemma 1 performs the Schur elimination. Lemma 2 computes the four scalar coupling products: \ (Bu2, u3·Bu3, u2 = -1/80, Bu2, u3·Bu3, u4 = -i/80, Bu4, u3·Bu3, u2 = i/80, Bu4, u3·Bu3, u4 = -1/80 \), using \ (i² = -1 and e^iπ/4·e^iπ/4 = i \). Lemma 3 assembles these into \ Bₒm Bₘo = (1/80) * [[-1, -i, i, -1] = (1/80) (σᵧ − I) \], giving the main result: \ EB (λ) = (1/80λ) (σᵧ − I) \. An earlier theorem summary recorded \ (EB (λ) = (1/80λ) σᵧ \). Explicit Schur elimination shows this is missing the diagonal self-energy term −I/80λ arising from the round-trip contributions Bu2, u3·Bu3, u2 and Bu4, u3·Bu3, 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 σᵧ form. This yields one conserved circulation mode and one damped corrective mode. The spectral abscissa \ (ω (EB) = 0 \), placing the residual ecology at the conservative/stable boundary of the TA33 classification. This is structurally more natural than purely oscillatory behaviour. EB is not a free parameter. It is determined entirely by AL from T127 through the Schur elimination. The forced structure confirms that all outer-mode coupling passes through the mandatory middle-mode mediator u₃, and the off-diagonal structure of EB is entirely second-order in this mediation.
Craig Edwin Holdway (Fri,) studied this question.