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) = p䂸, 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.
Ruqing Chen (Thu,) studied this question.