This paper investigates the P vs NP problem within the framework of the standard computational model-the Turing machine. It is shown that the input space ^* is naturally endowed with a discrete Hausdorff topology d = P (^*), in which every string is an isolated point. In this topology, any function f: ^* ^* is continuous, and all cohomology groups Hⁿ (^*) = 0 (n 1) and homotopy groups ₙ (^*) = 0 are trivial, which indicates the absence of global topological obstacles to the existence of a polynomial-time algorithm. To analyze the computational speed, an algebraic framework is constructed. The SAT formula is transformed into a system of polynomial equations over F₂. The generating function F_ (z) = ₗ z^w (x) is defined, where w (x) is the number of satisfied disjunctions. Due to the complete disconnectedness of the discrete space, F_ (z) factors into a product of local factors, each of which corresponds to an independent component of the constraint graph. The coefficients of these factors are computed in polynomial time via a factorization into elementary fractions, which in discrete topology is an exact algebraic identity rather than an asymptotic approximation. To handle more complex structures, Ramanujan identities, the zeta function, and adeles are employed. It is shown that an exponential blow-up in complexity arises as a singularity when one attempts to glue the isolated points of ^* into a continuous asymptotic picture. An intervention by discretization removes these singularities, restoring the completely disconnected topology and preserving the polynomial nature of the local factors. Furthermore, the ratio of the running time of an optimal algorithm to that of any correct algorithm is bounded by a multiplicative constant, which depends only on the structural properties of the problem and not on the input size. It has been established that the runtime for each input x depends only on its local structure, and global topological invariants do not affect the computational process. Using the local-global principle (an analogue of Hasse's principle), it is proven that the total runtime is a polynomial function of the input size. The results obtained provide a justification for the polynomial speed of the optimal algorithm for NP-complete problems in the Turing model. Since exponential algorithms are already known for SAT, the source of complexity should be sought not in the structure of the formulas, but in the chosen computational model. The Turing machine proposed in this work, with an explicitly defined discrete Hausdorff topology on ^* and the property of closure, contains no artificial continuous obstacles, which allows the problem to be factored and guarantees polynomial runtime.
Berbet Tymur (Tue,) studied this question.