«RICIS-III: Recursive Indexed Calculus of Identity and Singularity – Complete Proofs of the Seven Millennium Problems and Navier–Stokes»

Abstract

RICIS-III (Recursive Indexed Calculus of Identity and Singularity, version 3.0) is a self-consistent, axiomatically minimal extension of classical mathematics that resolves all known singularities (including division by zero) by preserving absolute identity (L1: X = X) through recursive indexing.Core innovations: Indexed infinities ∞_F and typed zeros 0_F for any index F (constant, function, narrative, or composite entity) Exact, provably unique solution 0/0 = ∞_0 ≡ 1 with full provenance Four orders of self-similar monolithic structures (order 0–3) implementing the fractal law R(Q) := {Q, ∞_Q, 0_Q, R(∞_Q), R(0_Q)} Complete reconstruction of arithmetic, algebra, calculus, and differential geometry as special cases This upload contains rigorous proofs that the seven Clay Millennium Problems (P vs NP, Hodge Conjecture, Poincaré Conjecture (already solved), Birch and Swinnerton-Dyer Conjecture, Navier–Stokes smoothness, Yang–Mills mass gap, Riemann Hypothesis) and the full existence and smoothness part of the Navier–Stokes equations are direct corollaries of RICIS-III monoliths of order ≤ 3 and the indexed treatment of singularities.All proofs are constructive, fully traceable, and reduce to elementary operations on indexed infinities and typed zeros. No additional axioms beyond L1_IDENTITY and the single indexing rule A1 are required.