Paper 13 of the scale-space series identified a five-step programme to verify the complex zs conjecture — the conjecture that the scale coordinate of the (x, y, z, s) framework is intrinsically complex. Steps 1–4 addressed the complexified geometry; Step 5 — the “representational requirement” — was described as not a calculation but a mathematical research programme: thedevelopment of a positional numeric system in which complex numbers are single values, with e as the natural base and differentiation as a primitive operation. This paper demonstrates that the CNRS (Complex Numeric Representational System) programme, as developed through its three-layer architecture, constitutes a direct and substantially complete answer to that research programme. We verify the match against each of Step 5’s three sub-requirements, exhibit the quantitative triangulation between the physics metric correctionand the CNRS area gap, and identify the remaining open question: whether the Layer 3 system can be formulated with a bounded digit alphabet over a transcendental base. A positive resolution of this open question would complete Step 5.

The mathematical development in this paper was produced in dialogue with Claude.ai (Anthropic) in Spring 2026, directed by the author. Use of AI assistance is acknowledged in accordance with standard scholarly practice.