From Divergent Series to Transfinite Power Towers: A Generalized Averaging Method Based on "Computable Infinity" and "Continuous Numbers"

In classical mathematics, the infinite power tower ^∞ x = x^{x^{x⋯}} converges only for x ∈ [e⁻ᵉ, e1/e] , with range [e⁻¹, e] . When the target value exceeds e (e.g., ^∞ x = 3 ), standard real analysis declares the equation unsolvable. This paper does not reject that conclusion; rather, it constructs a self-consistent generalized solution method for such equations beyond that boundary, by introducing two new concepts — "Computable Infinity" Ω and "Continuous Numbers." The core idea is: decompose the transfinite target value (e.g., 3) via a scale parameter k as 3 = k · s , where the kernel s is constrained to the classical convergence interval [e⁻¹, e] ; then use the classical identity s = x^s to derive the base function x(k) = (3/k)k/3 . Due to the nonlinear mapping between k and x , two natural perspectives — scale-uniform and variable-uniform — yield distinct means x̄_k and x̄_x . This paper further proposes the concept of "Continuous Numbers," acknowledging that these two results together constitute a principled interval; then, following the discrete compromise idea of Grandi's series 1-1+1-1+⋯ = 1/2 , it applies a second equal-probability averaging over this interval, finally obtaining the unique generalized point solution Xfinal = (x̄_k + x̄_x)/2 . This method does not rely on external regularization rules (such as Cesàro or Abel summation). Instead, within a nonstandard analysis framework, it uses "statistical averaging over infinitely many parallel instances" as its operational foundation, while maintaining full respect for classical mathematics. The paper concludes with a candid risk statement pointing out the particular-solution nature and theoretical limitations of the method, and looks ahead to its potential applications in high-dimensional infinity problems (such as continuous path integrals and geometric probability measure selection).