LISD Addendum A2: The Formal Derivation of R(t) — Regenerative Capacity as a Constrained Optimization

This addendum formally derives R(t), the regenerative capacity term in the Living Information Systems Dynamics (LISD) conservation law, using constrained optimization via Lagrange multipliers and the Karush-Kuhn-Tucker conditions. The correct optimization objective — maximizing R(t) directly subject to three constraints (viability, conservation law satisfaction, and a coherence floor) — is established and distinguished from entropy minimization and unconstrained maximization. Two modeling assumptions are stated explicitly: that regenerative effort increases structural order scaled by coherence (∂Σ/∂R = C(t)), and that coherence improvement follows a logistic form (∂C/∂R = C(t)·(1 − C(t))). The full derivation is shown including the bridge step connecting the first order condition to the final result via the active conservation constraint and KKT complementary slackness. The result R(t) = C(t)·(Σ(t) − S_min) passes all dimensional and range checks. The calibration gap Δ(t) = T(t) − R(t) is fully specified. A symmetry result is established and proven: T_dark raises the viability floor through S_eff(t) = S_min + D_KL(P_hist ‖ P_current), while R(t) scales the regenerative ceiling — opposite forces in the same mathematical register, bounding the Coherence Corridor from either side. This derivation closes the last formal gap in the LISD series. All terms in the conservation law are now derived from first principles. Part of the Living Information Systems Dynamics (LISD) preprint series. Extends the Coherence Corridor paper (DOI: 10.5281/zenodo.19352227).