Could the precursors of perfectoids emerge in TGD?The work of Peter Stolze based on the notion of perfectoid has raised a lot of interest in the community of algebraic geometers. One application of the notion relates to the attempt to generalize algebraic geometry by replacing polynomials with analytic functions satisfying suitable restrictions. Also in TGD this kind of generalization might be needed at the level of M^{4}× CP_{2} whereas at the level of M^{8} algebraic geometry might be enough. The notion of perfectoid as an extension of padic numbers Q_{p} allowing all p:th roots of padic prime p is central and provides a powerful technical tool when combined with its dual, which is function field with characteristic p. Could perfectoids have a role in TGD? The infinitedimensionality of perfectoid is in conflict with the vision about finiteness of cognition. For other padic number fields Q_{q}, q≠ p the extension containing p:th roots of p would be however finitedimensional even in the case of perfectoid. Furthermore, one has an entire hierarchy of almostperfectoids allowing powers of p^{m}:th roots of padic numbers. The larger the value of m, the larger the number of points in the extension of rationals used, and the larger the number of points in cognitive representations consisting of points with coordinates in the extension of rationals. The emergence of almostperfectoids could be seen in the adelic physics framework as an outcome of evolution forcing the emergence of increasingly complex extensions of rationals. See the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the article Could the precursors of perfectoids emerge in TGD?.
