Representation Meta-Operational Mathematics: A Unified Framework for Iterations, Renormalization, and Noncommutative Geometry

We develop a comprehensive theory of representation-operational meta-mathematics, extending the earlier meta-operational frameworks to the most general setting of representation-theoretic operations and their inverses. The central object is the space RepOp of representation morphisms, which we equip with a bornological structure and a system of twelve axioms. We construct the endomorphism operad PRep and prove that its unary part gRep carries a natural Lie algebra structure, with primitive elements classified as derivations on the representation category. A rigorous bornological convergence theory is established, including Mackey–Cauchy equivalence, completeness, and integral representations of fractional derivations. Exponential and logarithm meta-operations define fractional iterates f ◦t (t ∈ C), whose analytic continuation reveals logarithmic branch points and a natural boundary. The notion of non-idempotency degree is introduced, and a spectral criterion for collapse (loss of attractivity) in weighted one-parameter families is proved. A Hopf–representation operad structure (coproduct, counit, antipode) is constructed, and a morphism ΦRep to the Connes–Kreimer renormalization Hopf algebra is established, interpreting renormalized path integrals as the counit of this morphism. Applications to noncommutative geometry include representation-theoretic spectral triples, stability of the spectral triple property under bornological limits, and an index theorem for the noncommutative torus. Categorification yields a strict 2-category 2RepOp and an (∞, 1)-operad RepOp∞ via the dendroidal nerve. Classical representation-theoretic objects—Weyl reflections, Cartan matrices, characters, induced representations—are reinterpreted within the meta-operational framework, and a perfectoid correspondence for p-adic Lie algebras is proved. Numerical algorithms with optimal complexity and rigorous error bounds are provided. All conjectures from the original research program are resolved as theorems; a list of remaining open directions is given. This work provides a unified language for representation theory, quantum field theory, noncommutative geometry, and higher category theory.