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 5 supersedes all prior roadmaps and reflects the programme state as of April 2026, following a further working session in which four new documents were produced. Changes from v4. Problem 1: Outreach letters sent (Mas´akov´a, Steiner, Komornik); awaiting responses. The β = (3 + √5)/2 computation is complete: both endpoints exit Iβ at step 1, confirming the invariance gap as the primary blocker. Conjugate contraction rate (≈0.382) is the fastest of the three cases examined. New working paper: CNRS problem1 beta golden v1.tex. Problem 2: Layers 1+2 standalone paper updated to v3 (CNRS layers12 standalone v3.tex), incorporating P3 completion. Submitted to Journal of Number Theory (primary) and Acta Arithmetica (secondary). Philosophical paper (philosophical paper v5.tex) submitted to both Philosophia Mathematica and the European Journal for Philosophy of Science; both rejected editorially. Resubmission pending new journal selection. Problem 4: Partial operational completeness paper written and complete (CNRS problem4 partial completeness v1.tex). Q3 (operational closure under addition and multiplication) is now formally proved. Triangulation: The correction F = 1 + 2/L is now proved to be exact to all orders within the static diagonal ansatz — not a leading-order approximation. The factor 12 in the triangulation function is derived independently from the vierbein (amplitude–metric distinction). All three derivation routes (area gap, vierbein, complex geodesic at ϕ = 0) are consistent. The triangulation is closed within the static framework. New working paper: CNRS tetrad subleading v1.tex.