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 7 supersedes all prior roadmaps and reflects the programme state as of May 2026, incorporating corrections identified in three independent AI review rounds (Sessions 10–11).

Changes from v6 (corrections and status updates). Paper 18 Turing result corrected: v1–v3 of Paper 18 had an inverted Turing condition; corrected before BMB submission. The correct result—Turing active for s<sexit ≈0.52 nats (ℓ<17 µm, cell/sub-cellular scale), extinct at tissue scale and above—is now propagated throughout this document. A single extinction crossing is identified analytically from one computation.

Paper 18 status: At Scientific Reports (May 2026 submission, v6); 8 reviewers invited, awaiting reports.

Philosophical paper retired: After four desk rejections (EJPS, Synthese, Mathematical Intelligencer, Philosophia Mathematica), the standalone philosophical paper has been retired. The notation-necessity argument is being folded into the CNRS main paper.

References [24],[25] recredited: “Anonymous” replaced by Palmer (2026) working notes, consistent with the AI disclosure statement.

Document register updated: Zenodo concept DOI column added; Paper 18 and philosophical paper entries corrected; CNRS-Sum updated to v4.

Carried forward from v6. 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 CNS theorem.

Schmidt response: Klaus Schmidt directed programme to Frougny (Paris IRIF), Berth´e (Paris IRIF), and Thuswaldner (Leoben). Outreach initiated May 2026.

All papers 14–19 and MMS deposited to Zenodo with concept and version DOIs recorded.

Colour coding: green = proved; orange = active/waiting; blue = scoped/open; red = blocked on prerequisites.