Schur-Reduced Effective Carrier Dynamics

TA34 derives the effective observable carrier dynamics after elimination of the residual sector established in TA31-TA33. The barrier scattering operator S_B = [[A, C], [D, E]] generates coupled carrier-residual evolution through the system \( ψ̇_M = Aψ_M + Cψ_R and ψ̇_R = Dψ_M + Eψ_R \). Three lemmas establish the result progressively. Lemma 34.1 (Residual Evolution) gives the formal solution \( ψ_R(t) = etEψ_R(0) + ∫₀ᵗ e(t−s)EDψ_M(s)ds \) via the variation-of-constants formula. Lemma 34.2 (Carrier Feedback) substitutes this into the carrier equation, showing the carrier inherits feedback, memory, and possible amplification or damping effects from residual circulation through the convolution kernel \( Ce(t−s)ED \). Lemma 34.3 (Schur Reduction) applies the stationary-frequency ansatz \( ψ(t) ~ eλtψ \) to eliminate the residual sector, yielding the effective carrier generator \( H_eff = A + C(λI − E)⁻¹D \), provided \( λ ∉ σ(E) \). The effective generator H_eff is frequency-dependent through the resolvent \( (λI − E)⁻¹ \) and represents an effective reduced generator at a given frequency, not a globally closed semigroup generator. The approximation \( ψ_M(t) ≈ etH_effψ_M(0) \) is valid in the stationary-frequency regime and should not be interpreted as literal globally autonomous evolution. Three corollaries establish: effective carrier memory (nonlocal temporal dependence through the convolution kernel); a stability criterion classifying damped (ω(E) < 0), conservative (ω(E) = 0, semisimple), and rupture (ω(E) > 0) regimes via the spectral abscissa consistent with TA33; and approximate closed evolution when residual circulation is weak \( (‖C(λI − E)⁻¹D‖ ≪ ‖A‖) \), explaining how effectively unitary carrier evolution can emerge from a larger residual-scattering system. The resolvent \( (λI − E)⁻¹ \) is a standard functional-analytic object. The residual ecology language used throughout TA30--TA34 is a phenomenological interpretation of the dynamical content of this resolvent and the convolution kernel, not a separate physical claim beyond the operator structure. The ecological interpretation becomes geometrically grounded when block entries A, C, D, E are derived explicitly from Q5 geometry in TA35-TA38. The central result: observable carrier dynamics arise from the elimination of hidden residual degrees of freedom rather than from intrinsically isolated evolution.