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, authored by Sanna Foster. It is a certified polar contraction with κ = 0.85 and a conservative ℓ₂ ISS gain of 0.288. 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 public edition states the guarantees and verification status. Full proofs are in the restricted edition (10.5281/zenodo.20440589). Subject to patent application PRH2, filed with the Finnish Patent and Registration Office, April 1 2026.New in v1.1: module count corrected (three → four) — the new coupling module proves the signal the stabilizer reads is bounded and Lipschitz; details in the restricted edition. 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