The REAL E3 Stability Triad A Compositional Stability Certificate

This paper certifies the compositional stability of the REAL E3 Stability Triad: Field Guard, the Presence Layer, and the Relational Field Stabilizer composed as a state-decoupled parallel system. The parallel boundedness, joint Lyapunov bound, state decoupling, cascade BIBO, and combined field load bound are machine-checked in Lean 4 with zero sorries and no added axioms in the REALE3.COMP module. The joint ISS gain is dominated by Field Guard at 39.5. This restricted edition contains the full composition argument, coupling parameters, and the complete Lean proof structure. Subject to patent application PRH2, filed with the Finnish Patent and Registration Office, April 1 2026.New in version 1.2. The trajectory-level joint gain, carried as pending in the Claims Ledger since the first edition, is now machine-checked and that row is discharged. Three theorems are added: a strict Sontag ISS estimate for the parallel triad, the coupled triad's trajectory estimate with its standing presence-deficit bias stated explicitly rather than absorbed into a constant, and the release theorem giving the level the coupled memory settles to under sustained recovery. Four scope remarks record where the development is more permissive than the deployed implementation. Zero sorries, no added axioms. RE3 are proprietary systems of REAL E3 Systems Oy, Finland. This document discloses the mathematical stability certificate for independent verification purposes only. The harness implementation code, state extraction methodology, and REAL-E3 internal architecture are proprietary and not disclosed. Publication does not constitute a licence to implement, reproduce, or deploy any component of the REAL-E3 architecture. This notice also applies to all prior publications by REAL E3 Systems Oy. Licensing enquiries: info@real-e3systems.fi