Algebraic Friction in Mixed Characteristic: A Computational Exploration of Witt Deformations, Period Rings, and Adic Topologies

This paper investigates the generalization of algebraic tri-operate fields---specifically governed by the power-logarithmic star operator $x \\star y = x^{\\log(y)}$---to infinite characteristic 2 domains. While finite constructions are strictly permissible under Mersenne prime restrictions ($2^p - 1$) establishing cyclic isomorphisms $(\\mathbb{F}_{2^p} \\setminus \\{0,1\\}, \\star) \\cong (\\mathbb{F}_{M_p}^\\times, \\cdot)$ and preserving Frobenius Galois actions, lifting these structures to infinite fields introduces profound geometric incompatibilities. We attempt to construct an infinite tri-operate functor utilizing perfectoid tilting mechanisms, Fontaine's period ring $\\mathbb{A}_{\\text{inf}}$, and the associated adic space $\\operatorname{Spa}(K, \\mathcal{O}_K)$. By examining the Fargues-Fontaine curve $X_{FF}$ under a quadratic Harder-Narasimhan slope dilation, we demonstrate a fatal structural torsion collapse. Ultimately, this paper proves a No-Go Theorem: the topological and non-Archimedean rigidity required by perfectoid tilting obstructs the requisite tri-operate symmetries, establishing a fundamental impossibility for the existence of infinite tri-operate fields within this geometric framework.