Elliptic Meta-Operational Mathematics: A Complete and Rigorous Extension of Meta-Operational Mathematics to Elliptic Functionsand Their Inverses

This work presents a completely rigorous and self-contained extension of the full apparatus of Meta-Operational Mathematics to the specific and important class of elliptic functions and their inverses. The central philosophical principle -- that operations upon operations constitute meta-operations -- is established with complete mathematical precision through a hierarchical framework: Level 0 (elements of a base space), Level 1 (operations as mappings on the base space), Level 2 (meta-operations as mappings on operations), and Level 3 (meta-meta-operations acting on meta-operations). Within this framework, the Weierstrass elliptic functions , ', ζW, σW and the Jacobi elliptic functions sn, cn, dn together with their inverses arcep, arcsn, arccn, arcdn are shown to admit canonical lifts to meta-operations via composition, and these meta-operations interact with one another through composition, pointwise addition, pointwise multiplication, differentiation, exponentiation, and logarithm in arbitrarily many iterations---integer, fractional, real,and complex. A fundamental distinction from the hyperbolic case is established: the double periodicity of elliptic functions leads to the Elliptic Duality Axiom (Axiom 2.19), replacing the translation group (C,+) with the quotient group C/Λ. This crucial modification permeates the entire theory.