Computing With Spin

Cohomology And Applications Of Holographic Satisfiability

Abstract

In [1], we presented an (holographic) algorithmic system for finding polynomial time proofs of both satisfiable and unsatisfiable 3 SAT problems and presenting them as differential varieties. In [2], the authors have identified holographic identities as spinor varieties. In this paper, we combine these insights and represent the solution manifold as a spinor space, that is, a spin manifold, along with varieties emerging as spin valuations using a form of complex arithmetic that we develop. Unsurprisingly, the spin forms that emerge can be associated with a type of algorithmic and relativistic, quantum gravity, which is not unlike the actual gravity associated with physical particles. We explore the implied gravitational field and also show that its structure is not only convenient for studying physical force fields, but also can shed proof theoretic insights on several deep mathematical problems by encoding their computations as variables associated with the spin cohomology introduced.

To obtain these results, we first examine the usual syntactical gadgetry invoked in other holography papers – variants of Pfaffian circuits - and find them to be inadequate to fully develop and associate the quite rigid notion of spin to definite numerical values in high dimensional differential space. We then introduce a fine-grained formalism that models the selection/deselection process as an algorithmization of a  Lie algebra type flow over a spin moduli stack in which the spins then arise as natural quotients of representations of the moduli group.