CNRS Schmidt: The CNRS Programme and Schmidt's Symbolic Cover Framework: A Complete Mathematical Mapping

We establish a precise correspondence between the Complex Numeric Repesentational System (CNRS) programme and the symbolic cover theory of Schmidt 2, 1 and Lindenstrauss–Schmidt 3, 4. The CNRS-A arithmetic system (base z0 =−2+i, digit alphabet D= 0, 1, 2, 3, 4) is the explicit symbolic cover and arithmetic transducer for the polynomial h (u) = u² + 4u+ 5 (the minimal polynomial of−2 + i over Z) within Schmidt’s framework. Every major result in the CNRS Problem 3 (arithmetic closure) programme is an explicit instantiation of a theorem in 1. The Problem 1 ( (−β) -expansion) programme is the symbolic cover theory for the class of hyperbolic polynomials h with a single small root, i. e. Pisot and Salem beta-shift polynomials (Schmidt’s Examples 7. 1–7. 5). The invariance gap of Problem 1 is the obstruction that separates the Pisot (hyperbolic, Theorem 3. 1) from the Salem (nonhyperbolic, Corollary 4. 3) regimes in Schmidt’s classification. This mapping has three immediate consequences: (1) the CNRS mathematics papers can be framed as the explicit physical and operational instantiation of Schmidt’s programme for h (u) = u² + 4u+ 5; (2) Schmidt’s open problems (Problems 6. 1 and 6. 8) acquire a concrete new instance; (3) the collaborator outreach to Schmidt is the highest-priority action in the programme, superseding other outreach.