Prime-Index Isomorphic Arithmetic: Algebraic Structures, Asymptotic Dynamics, and ⊗-Irreducible Elements in the Prime Space

We introduce the Prime-Index Isomorphic Arithmetic (PIIA) framework, which constructs an ordered commutative rig (𝙋, ⊕, ⊗) isomorphic to (ℤ⁺, +, ×) via the bijection f(n) = pₙ. We formally prove that the ⊗-irreducible elements correspond exactly to the sequence of super-primes (primes with prime indices, OEIS A006450), and establish the Fundamental Theorem of PIIA: every prime admits a unique ⊗-factorization into super-primes. We derive a rigorous asymptotic expansion for the self-referential folding operator J(pₙ) = ppₙ, with an error bound of O(n ln ln n), validated empirically up to the 10¹² index scale with relative error monotonically decreasing to 0.17%. Source code, data, and interactive calculators are available at https://github.com/Ruqing1963/Prime-Universe-OS.