Boundary Moduli I:　Monster-Theta Enhancement of Boundary Moduli: Arithmetic Rigidity and Local Symmetry Refinement after Koizumi Descent

This paper is intended to be read as Part I of a two-paper sequence on exceptional symmetry for Clifford-valued boundary moduli after Koizumi descent. The earlier Koizumi paper established that exact local boundary equations live first on the algebraically visible Abelian image rather than on the ambient torus; the subsequent local symmetry paper introduced the Clifford-Koizumi local relative datum and formulated a Galois-type correspondence between local symmetry subgroupoids and intermediate visible-dark local structures. The present paper isolates a special arithmetic layer inside that general theory. We define theta enhancements of Koizumi-visible boundary germs, identify theta-admissible and arithmetic-rigid strata, and show that the first shell of the Leech lattice provides a distinguished exact averaging law for quartic overlap functionals. We then introduce Griess-type graded enhancements and their Monster-type specialisations as symmetry refinements of the visible theta sector. In the split rigid semisimple setting of Conjecture 7.1 of the paper [T. Matsuhisa. Local Visible-Dark Moduli and Clifford-Koizumi Symmetry. 2026. ⟨hal-05693872⟩], these enhancements select distinguished saturated subgroupoids on the symmetry side and hence sharpen the exact correspondence there without claiming to solve the entire visible-dark classification problem. Part I culminates in strict Monster enhancement and in a Monster isotropy theorem on a rigid one-object chart; the broader componentwise sporadic-groupoid programme is deferred to Part II [T. Matsuhisa,Boundary Moduli II: Sporadic Isotropy Groupoids after Monster--Theta Enhancement. Draft.Preprint (2026)]. Every definition, proposition, and theorem is bracketed by a model example and a Counter-example. Proof sketches are retained in the main text, whereas complete proofs and all necessary preliminaries are placed in the appendices.