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

We establish a precise correspondence between the Complex Numeric Representational 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 beta-shift polynomials (Schmidt’s Examples 7.1–7.5).

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 a high-priority action in the programme.

Changes in version 2. Section 3 has been corrected and substantially rewritten following correspondence with W. Steiner (IRIF, Paris). Version 1 used an incorrect interval [−2/(β + 1), 1/(β + 1)) and incorrect admissibility blocks (01)^ω/(20)^ω for the two proved Pisot instances. The correct interval is the Ito–Sadahiro interval [−β/(β + 1), 1/(β + 1)); the correct blocks are
(2, 1^ω)/(0, 2, 1^ω) for β = 1 + √2 and (2, 0^ω)/(0, 2, 0^ω) for β = 1 + √3. The invariance gap analysis of v1 has been removed; no invariance gap exists in the corrected framework. The uniform admissibility conjecture has been retracted. The Schmidt correspondence for the Pisot results is updated accordingly. Two rows of the correspondence table (Section 6) have been corrected.