The Hyperoperation Continuum: Existence, Uniform Construction, and Complete Classification of Continuous Extensions

Abstract

We address the continuous extension problem for the hyperoperation hierarchy: given a base g>0, extend the discrete height index of exponentiation, tetration, pentation, hexation, and beyond to a real parameter. A single strip construction S_F[s](x)=F^{\lfloor x\rfloor}(s(\{x\})), applied recursively along the height ladder, produces a continuous extension H_n at every height n\ge1, satisfying H_n(x+1)=H_{n-1}(H_n(x)) with H_n(0)=1. The construction is exact: continuity and the functional equation hold identically, not approximately. At height n=1 the extension admits the elementary closed form \mathfrak P^{(1)}_\lambda(x)=((1+\gamma x)^\lambda-1)/\gamma. The solution set at each height is completely classified: it is a torsor under the gauge group \mathcal H\cong\mathrm{Homeo}_+([0,1]), and is therefore infinite-dimensional. Consequently, no uniqueness statement can hold without conditions that collapse \mathcal H to the identity. Numerical verification for g=\sqrt2 through height 7 confirms functional-equation residuals at double-precision roundoff (\le2.3\times10^{-16}), gauge recovery to 4.4\times10^{-16}, and solution-manifold rank equal to the number of independent gauge directions up to k=8. The paper provides a complete, verifiable, and uniform resolution of the hyperoperation continuum problem.