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) = u^2 + 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) = u^2 + 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.