Expanding total sieve and patterns in primes

Let $({\mathcal{S}}_n^{\alpha ,\kappa ,\mathfrak{r}}(z){)}_{n=1}^{\mathrm{\infty }}$ be a sequence of the largest possible integer intervals, such that $z\in$\mathcal{S}$_n^{\alpha,\kappa,\mathfrak{r}}(z)\subset\overline{$\mathcal{M}$}n^{\alpha,\kappa,\mathfrak{r}}=\bigcup{i=1}^n [\mathfrak{r}i]{\mathfrak{p}_i}$or$$\mathcal{S}$_n^{\alpha,\kappa,\mathfrak{r}}(z)=\emptyset$,where$\mathfrak{p}i=p{\alpha+\left\lceil i/\kappa\right\rceil-1}$and$z\in$\mathbb{Z}$$\mathrm{.}Weprovethat$\big(#$\mathcal{S}$n^{\alpha,\kappa,\mathfrak{r}}(z)\big){n=1}^\infty$oscillatesinfinitelymanytimesaround$\beta_n!=!o\left(n^2\right)$ for any fixed $\alpha \in {\mathbb{Z}}^{+}$, $\kappa \in \mathbb{Z}\cap [1,p_{\alpha })$, and ${\mathfrak{r}}_i\in \mathbb{Z}$. Let $T=(a_1,a_2,\dots ,a_k)$ be an admissible $k$-tuple and let ${\mathcal{X}}_n^{T,k,\rho ,\eta }=\{x\in [\rho {]}_{\eta }:\{x\text{\hspace{0.17em}⁣}+\text{\hspace{0.17em}⁣}{a}_1,x\text{\hspace{0.17em}⁣}+\text{\hspace{0.17em}⁣}{a}_2,\dots ,x\text{\hspace{0.17em}⁣}+\text{\hspace{0.17em}⁣}{a}_k\}\cap {\mathcal{M}}_{n+\alpha -1}\mathrm{\ne }\mathrm{\varnothing }\}$ for each $n\in {\mathbb{Z}}^{+}$, where ${\mathcal{M}}_g={\bigcup }_{i=1}^g[0{]}_{p_i}$. We prove that for any $T$ and for some fixed $\alpha$, $\kappa$, $\rho$, $\eta$, $z$, and $\mathfrak{r}$, there exists a linear bijection between ${\bar{\mathcal{M}}}_{\kappa n}^{\alpha ,\kappa ,\mathfrak{r}}$ and ${\mathcal{X}}_n^{T,k,\rho ,\eta }$ for each $n\in {\mathbb{Z}}^{+}$. It implies that the length of any expanding integer interval on which all occurrences of $T$ are sieved out by ${\mathcal{M}}_{n+\alpha -1}$ oscillates infinitely many times around ${\tilde{\beta }}_n=o\left(n^2\right)$. The concept of the sieve of Eratosthenes asserts ${\mathcal{E}}_n=[2,p_{n+\alpha }^2)\cap (\mathbb{Z}\setminus {\mathcal{M}}_{n+\alpha -1})\subset \mathbb{P}$. Therefore, having $p_{n+\alpha }^2=\omega \left(n^2\right)$, we obtain that ${\mathcal{E}}_n$ includes a subset matched to $T$ for infinitely many values of $n$ and, consequently, $T$ matches infinitely many positions in the sequence of primes.