Presence Layer: A Machine-Checked Polar ISS Contraction with Soft Relational Governance

The Presence Layer (PL) is the soft human-facing filter of the REAL E3 Stability Triad. It is a certified polar contraction with κ = 0.85 and a conservative ℓ₂ ISS gain of 0.288, authored by Sanna Foster. Unlike Field Guard, PL dampens adaptively rather than blocking, preserving signal meaning while maintaining stability. The contraction, ISS dissipation, and gain bound are machine-checked in Lean 4 with zero sorries and no added axioms. This restricted edition contains the full proof chain and input mappings. Subject to patent application PRH2, filed with the Finnish Patent and Registration Office, April 1 2026. 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: [email protected]