string dynamics of prime factorization: homotopy of thermodynamics and quantum complexity

We try to penetrate the puzzle of the distribution of primes, and prime factorization, and give from a mathematical point of view a positive answer to the question: is there exists a mathematics based on the prime factorization? \\Our theory based in the first paths on the Boltzmann entropy, then takes tow far ways, both in the mathematical side, and in the theoretical computer science side. Unfortunately, our progress blocked completely from this last side. \\Anyway, we prove that our conjecture is true if and only if it is true for all primes.So, in this paper we will present the program behind the crown of our theory:$$\Delta \Lambda^{N}(\alpha,\beta)+\langle \nabla^{\xi\eta}_{\star} + \nabla^{\phi\psi}_{\star} \mid P_N\rangle+ \Delta_{N,T}^{\ast} (\alpha,\beta)\log\frac{9}{8}=\eta(\alpha,\beta)$$but, since the above equation has enormous difficulty to explain it now, we say that it is equivalent under an assumption we will site it later to the following equation:$$J^{\star}(N,N+1)\prod_{p_i=2}^{p_{\pi(N)}}\left(\frac{p_i}{p_i-1}\right)^{\Xi_{p_i}(N+1)}=\prod_{p_i=2}^{{p_{\pi(N)}}}\left(\frac{p_i}{p_i-1}\right)^{\Xi_{p_i}(N)}$$Where $N \in \mathbb{Z}^+$, $p_i$, $1 \leq i \leq \pi(N)$, are the $i^{th}$ prime, $\pi(N)$ is the prime counting function, $\Xi_{p_i}(N)$, $\Xi_{p_i}(N+1)$, are integers based on the prime factorization, and:$$J^{\star}(N,N+1)=\begin{cases}\frac{2(N+1)}{3N} ~ ~ ~ \text {if $N=2l ~ and ~ \pi(N+1)=\pi(N)$}\\\frac{2}{3} ~ ~ ~ \text {if $N=2l ~ and ~ \pi(N+1)=\pi(N)+1$}\\\frac{3(N+1)}{4N} ~ ~ ~ \text {if $N=2l+1$}\end{cases}$$