CNRS-Rdmp: Complex Numeric Representational System Programme Full Programme Scope: All Four Problems

This document records the full scope of the Complex Numeric Representational System (CNRS) programme across all four problems, the current state of each, and the precise open questions remaining. Version 6 supersedes all prior roadmaps and reflects the programme state as of May 2026, following further working sessions (Sessions 5–6) in which seven new documents were produced (Papers 16–19 and MMS, plus Papers 14–15 finalised). Changes from v5. Paper 13 Steps 3–4 complete (Papers 16–17): The complex zs conjecture is a quantum conjecture — classical field equations force ϕ = 0. Discriminating prediction: interference phase shift ∆φ = ϕ∆s/L near compact objects. Both papers deposited to Zenodo. Paper 13 Step 5 substantially closed (Paper 19): CNRS Layers 1–3 satisfy all three representational sub-requirements. Triangulation identity 1 + 2/L = 1 + 1/2 ln(1 + e^(4/L) − 1) proved exact at all orders. Remaining open: e-base CNStheorem. Implementation 1 complete (Paper 18): Gierer-Meinhardt system via CNRS-H digit strings. Key finding: two-threshold Turing structure across biological hierarchy (absent at sub-cellular and organ scales; active at tissue-to-organism scales). Companion Python verified at machine precision. Schmidt response: Klaus Schmidt directed programme to Frougny (Paris IRIF), Berth´e (Paris IRIF), and Thuswaldner (Leoben). Outreach planned for May 2026.