Relational Field Stabilizer: A Machine-Checked BIBO Bounded Memory Accumulator with Asymptotic Decay

The Relational Field Stabilizer (RFS) is the bounded memory centre of the REAL E3 Stability Triad. It maintains a scalar field memory B_t bounded in [0,1] for all inputs (BIBO certified), and decays asymptotically to zero under recovery. The BIBO bound, memory contraction, mutual exclusivity of load and decay, and asymptotic decay are machine-checked in Lean 4 with zero sorries and no added axioms across four lemmas. This restricted edition contains the full memory dynamics, adaptive decay mechanism, and coupling coefficients. Subject to patent application PRH2, filed with the Finnish Patent and Registration Office, April 1 2026. New in version 1.2. Three remarks concerning Lemma 2c, none of which changes a claim. The standing per-turn load the memory carries when the presence layer sits at its origin is now stated as an explicit design cost rather than left implicit. A model-fidelity note records that the development leaves one projection unclipped where the implementation clips it, so the development is strictly more permissive and the boundedness result is unaffected either way. And the coupled counterpart of the asymptotic-decay theorem is recorded, of which the published base theorem is the bias-free special case. Every claim published in earlier editions is unchanged and still exactly true. 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