Let $S=\left(s_1, \ldots, s_n\right)$ be a finite sequence of integers. Then $S$ is a Gilbreath sequence of length $n$, $S\in\mathbb{G}_n$, iff $s_1$ is even or odd and $s_2, \ldots, s_n$ are respectively odd or even and $\min\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\leq s_{m+1}\leq\max\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\forall m\in\left[\left.1, n\right)\right.$. This, applied to the order sequence of prime number $P$, defines Gilbreath polynomials and two integer sequences A347924 \cite{oeisA347924} and A347925 \cite{oeisA347925} which are used to prove that Gilbreath conjecture $GC$ is implied by $p_n-2^{n-1}\leqslant\mathcal{P}_{n-1}\left(1\right)$ where $\mathcal{P}_{n-1}\left(1\right)$ is the $n-1$-th Gilbreath polynomial at 1.

Knowledge is not merely the characteristic of certain pieces of information being deemed true, believed and justifiable, but rather a process. The sentences that reaches the status of knowledge forms a universe that provides an interpretation of the reality one aims to describe. Language exchanges information through communication and the justification, if concludes successfully with respect to a sentence, it elevates the information to the status of knowledge. We provide a model where we define the fundamental elements of communication and grammar. As a result, knowledge must be \open\ (everyone must be able to acquire it), \free\ (everyone must be able to justifiy) and \tidy\ (everyone must be able to retrace the justification process in all its parts recursively). Ultimately, the development of knowledge processing, referred to as \emph{knowmatic}, represents the evolution of the information technologies (\it).