Paper 19: The Representational Requirement: The CNRS Programme as Paper 13 Step 5

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: the
development 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 correction
and 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.