Self-Referential Decimal Power Towers and Integer Roots: Algebraic Reductions, Combinatorial Variants, and Certified Computational Search

This paper investigates generalized self-referential decimal power tower identities of the form √[k]{T(n)} = n, extending the classic 262144 square-root puzzle to higher root indices k > 2. By leveraging prime valuations and algebraic reductions, we show that non-trivial solutions reduce to power base representations, avoiding intractable tower evaluations. Under strict individual digit constraints, we prove the existence of exact non-trivial solutions with colossal indices exceeding 10¹⁰⁰ (such as n = 4294967296 and n = 3433683820292512484657849089281) and establish negative search bounds up to 10,000 digits. Full Python search scripts and data verification are included.