Proving the Grothendieck--Teichmüller Conjecture for Profinite Spaces & The Galois Grothendieck Path Integral

We establish that the Grothendieck-Teichmuller conjecture, which predicts an isomorphism between the Grothendieck-Teichmuller group GT and the absolute Galois group of rational numbers Gal, holds in the setting of profinite spaces. To access arithmetic information within this framework, we introduce a generalization of the notion of path integrals, defined simultaneously for GT and Gal. This construction reveals new arithmetic invariants for both groups, capturing, in particular, the rationality of periods associated with GT and the rational structure of the coefficients in the Drinfeld associator. Moreover, this perspective provides a mechanism to detect the descent of cohomology classes to Q under the action of Gal, and more broadly, a method to compute ranks of arithmetic objects. Furthermore, we introduce an algorithm, which we call the Cubic Matrioshka, encoding paths in both GT and Gal as unique infinite binary sequences. The compatibility of these binary encodings reflects the underlying homeomorphism and ensures that symmetries of GT are faithfully mirrored in the corresponding binary representation within Gal.