TA34 derives the effective observable carrier dynamics after elimination of the residual sector established in TA31-TA33. The barrier scattering operator SB = [A, C, D, E] generates coupled carrier-residual evolution through the system \ (ψ̇M = AψM + CψR and ψ̇R = DψM + EψR \). Three lemmas establish the result progressively. Lemma 34. 1 (Residual Evolution) gives the formal solution \ (ψR (t) = e^tEψR (0) + ∫₀ᵗ e^ (t−s) EDψM (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) ~ e^λtψ \) to eliminate the residual sector, yielding the effective carrier generator \ (Hₑff = A + C (λI − E) ^−1D \), provided \ (λ ∉ σ (E) \). The effective generator Hₑff 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 \ (ψM (t) ≈ e^tHₑffψM (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) 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.
Craig Edwin Holdway (Sat,) studied this question.