Microscopic Origin of the Weyl Evolution Equationfrom SU(2) Chern-Simons Boundary Theory

Third paper in the holographic equilibrium series. Paper I (PRD, submitted) derived a long-mode lapse perturbation dΨ/d ln a = -(1+q)f_ss from the perturbed Clausius relation on the cosmological apparent horizon, resolving the H₀ and S₈ tensions with Δχ² = -58.4. Paper II (CQG, submitted) derived the interaction modulation, screening function, and coupling β = 1/12 from SU(2) Chern-Simons theory on the dynamical horizon. This paper closes the remaining gap: the Ψ ODE is derived from the Weyl Ward identity of the sourced SU(2)_k Wess-Zumino-Witten boundary theory. Main results: (1) The non-adiabatic excess fraction f_ss = 1 - 1/S is derived from the Kubo linear-response decomposition, with the adiabatic threshold set by the per-puncture Pendry bound. (2) The bulk-to-boundary normalisation N(k) = 1 is established by two independent routes: the 2D spatial stress trace of the Brown-York tensor at the Israel-Darmois pullback (geometric), and the trace anomaly of the unique isotropic monopole on S² with the Kodama-flow normalisation (gauge-theoretic). Both give N = 1 exactly for all k. (3) Metric inheritance — the statement that new punctures inherit the local lapse — is shown to be a consequence of the Israel-Darmois junction conditions on the perturbed horizon, not an additional assumption. (4) The holographic self-consistency condition σ_max = Ṡ_BH (maximum boundary entropy production equals Bekenstein-Hawking entropy growth) holds if and only if q = -1/2 (Ω_m = 1/3), selecting the late-time attractor as the unique holographically self-consistent state with eigenvalue -2/3 and global basin of attraction (0, 1). The H₀ resolution and the coincidence problem are unified as perturbation-level and background-level consequences of the same SU(2)_k boundary theory.