P != NP and here’s a (not-so)-brief explanation of why:

Every [math]n[/math]-length sequence of states [math][\mathbf{x}]_n = \{(x_0,x_1,...,x_k)|k \in \mathbb{Z}_n\}[/math] a Turing Machine might take can be represented as the orbit of a dynamical system in the real numbers using Lagrange interpolation. To see this, let [math]\sigma[/math] be a Turing Machine’s transition function over a set of states [math]Q[/math] with [math]\Gamma[/math] the tape alphabet and [math]\sigma : Q \times \Gamma \to Q \times \Gamma \times \mathbb{Z}_2[/math]. A Differential Turing Machine is therefore defined as the oriented flag manifold [math]\mathfrak{M}[X_0][/math] defined by the bundle of [math]\sigma^k[/math] orbits of input data [math]\X_0[/math], for [math]k=1,2,...,n[/math]. Then for each [math]n[/math] the equivalence class [math][\mathbf{x}]_n[/math] defines a family of Lagrange polynomials over the finite field [math]\mathbb{Z}_n[/math]. In the inductive limit, the implementation space of an algorithm is defined by the transport [math]F[/math] of its combinatorial species, which is invariant under the [math]G[/math]-action of [math]S_n[/math], the symmetric group on [math]n[/math] letters. In other words, the computational complexity of a program will remain the same under permutation of symbols of the formal language’s alphabet. In this way, the combinatorics of a permutation [math]\pi : \Gamma \to \Gamma[/math] acting on a countable set determine the fiber bundles of orbits for a dynamical system’s evolution operator [math]\sigma[/math]. In the Almost Finite (AF) limit that [math]n \to \infty[/math], the space [math]\mathfrak{M}[X_0][/math] has an analytic representation through the one parameter evolution equation [math]\frac{d\pi(t)}{dt}=v(\pi(t))[/math], which by integration has the solution [math]\pi(t)=\sigma^t(\pi(0))[/math]. From this perspective, the dynamical system’s phase space is identified with the Turing Machine’s state space, and computation is interpreted as the flow of information across the corresponding phase manifold.

An element [math]\sigma \in S_n[/math] is isomorphic to the associated polynomial [math]\sigma \cong C_n = c_{0}+c_{1}x+\dots +c_{{n-1}}x^{{n-1}}[/math] of a circulant matrix, which itself is defined by the transpose of the Vandermonde matrix of [math]\sigma[/math] orbits. The normalized eigenvectors of the associated polynomial [math]C_n[/math] are given in terms of the [math]n[/math]th roots of unity, and furthermore, through the characteristic and associated polynomials of the circulant matrix defined by [math]C_n[/math], every permutation additionally defines a kernel of the circular convolution. Each Lagrange polynomial has a Vandermonde representation, generating a system of polynomial equations at the multiplicative identity [math]1 \in \mathbb{Z}_n[/math] which is solvable by the Chinese Remainder Theorem. This procedure can be inductively applied in the limit that [math]n \to \infty[/math]. By the Cayley-Hamilton Theorem, a matrix solves its characteristic equation, so the product of eigenvalues must equal [math]1 \operatorname{mod} n[/math], however by definition the [math]n[/math]th roots of unity are also solutions to the cyclotomic polynomial equation [math]\Phi_n = x^n-1[/math] with [math]\Phi_n = 0[/math], and therefore the eigenvalues of the Vandermonde matrix are necessarily algebraic numbers.

Every cyclotomic polynomial [math]\Phi_n[/math] has an equivalent representation as the tangent bundles of a Differentiable Turing Machine [math]\mathfrak{M}[/math] using truncated k-jets, and in the inductive limit, this process holds regardless of the indeterminate because the enveloping algebra forms a unitary representation of the cone of modules defined by the Turing Sequence [math][\mathbf{x}]_n[/math] under the action of an element of [math]S_n[/math]. Thus every effectively computable function has an embedded realization as an orbifold generated by a one parameter semi-group, and the orbits of [math]\sigma[/math] are expressible as a polynomial ring itself having an algebraic p-adic representation, which therefore enumerates the space of polynomial time solutions.

The complexity class of decision problems with an irrational p-adic Turing Sequence is called [math]\mathbf{HARD_{AF}}[/math], where by Dedekind cut construction, [math]\mathbf{HARD_{AF}}[/math] can be shown to be the algebraic completion of AF Turing Sequences (see the provided link for more information on rational power series). Remember that the set of algorithms with polynomial complexity [math]\mathbf{P}[/math] corresponds to the AF polynomial ring generated by rational Turing Sequences. Letting [math]\mathbf{HARD_{AF}}[/math] refer to both the complexity class and the set of solutions in this class, [math]\mathbf{HARD_{AF}}[/math] is the algebraic closure of the polynomially expressible algorithms, so necessarily [math]\mathbf{P}[/math] and [math]\mathbf{HARD_{AF}}[/math] are disjoint. The symmetric difference of disjoint sets equals their union, so [math]\mathbf{P}\ominus\mathbf{HARD_{AF}}=\mathbf{P}\cup\mathbf{HARD_{AF}}[/math]. Similarly, it is clear that [math]\mathbf{NP}[/math] and [math]\mathbf{HARD_{AF}}[/math] are not disjoint and the symmetric difference of two overlapping sets equals their intersection, thus [math]\mathbf{NP}\ominus\mathbf{HARD_{AF}}=\mathbf{NP}\cap\mathbf{HARD_{AF}}[/math]. As the symmetric difference is distributive over set intersection and the operation of intersecting sets is commutative and associative, it therefore follows that [math]\mathbf{P}\cap\mathbf{NP}=\emptyset[/math]. In other words, [math]\mathbf{P}\ne\mathbf{NP}[/math].