For $n$ a large enough composite integer, define the $g$-gap radius of $n$, if it exists, for positive even $g$ as the smallest positive integer $\rho_{g}(n)$ such that both $n-\rho_{g}(n)$ and $n+\rho_{g}(n)$ are half sums of two consecutive primes separated by a prime gap equal to $g$.

Is it possible to give an upper bound for $\rho_{g}(n)$ in terms of both $n$ and $g$ both unconditionally and under widely believed conjectures such as Hardy-Littlewood $k$-tuple conjecture or (G)RH? Is there always some $g$ such that $\rho_{g}(n)\ll_{g}\log^{C_{g}} n$ where $C_{g}$ is a positive constant depending only on $g$?

Define also the absolute gap radius of $n$, denoted by $\rho(n)$, as $\inf_{g}\{\rho_{g}(n)\}$. Is there an absolute positive constant $C$ such that $\rho(n)\ll\log^{C}n$?

Edit: $2\rho(n)$ should be somewhat larger than the distance between some prime constellation $C$ (let's denote its largest prime by $C_{+}$) and its "mirror constellation" $C'$ (that is, the prime constellation obtained from $C$ reversing the order of the prime gaps; let's denote its smallest prime by $C'_{-}$ with $C'_{-}>C_{+}$) with $n$ being the half sum of $C_{+}$ and $C'_{-}$. As the user Lagrida proved in an answer to a previous question of mine that there is asymptotically the same amount of mirror constellations below $x$, this should entail that $\rho(n)$ exists provided $n$ is large enough.

For $n$ a large enough composite integer, define the $g$-gap radius of $n$, if it exists, for positive even $g$ as the smallest positive integer $\rho_{g}(n)$ such that both $n-\rho_{g}(n)$ and $n+\rho_{g}(n)$ are half sums of two consecutive primes separated by a prime gap equal to $g$.

Is it possible to give an upper bound for $\rho_{g}(n)$ in terms of both $n$ and $g$ both unconditionally and under widely believed conjectures such as Hardy-Littlewood $k$-tuple conjecture or (G)RH? Is there always some $g$ such that $\rho_{g}(n)\ll_{g}\log^{C_{g}} n$ where $C_{g}$ is a positive constant depending only on $g$?

Define also the absolute gap radius of $n$, denoted by $\rho(n)$, as $\inf_{g}\{\rho_{g}(n)\}$. Is there an absolute positive constant $C$ such that $\rho(n)\ll\log^{C}n$?

Edit: $2\rho(n)$ should be equal to $g$ plus the distance between some prime constellation $C$ (let's denote its largest prime by $C_{+}$) and its "mirror constellation" $C'$ (that is, the prime constellation obtained from $C$ reversing the order of the prime gaps; let's denote its smallest prime by $C'_{-}$ with $C'_{-}>C_{+}$) with $n$ being the half sum of $C_{+}$ and $C'_{-}$ and $g$ such that $\rho(n)=\rho_{g}(n)$. As the user Lagrida proved in an answer to a previous question of mine (namely Symmetry in Hardy-Littlewood k-tuple conjecture) that there is asymptotically the same amount of mirror constellations below $x$, this should entail that $\rho(n)$ exists provided $n$ is large enough.

For $n$ a large enough composite integer, define the $g$-gap radius of $n$, if it exists, for positive even $g$ as the smallest positive integer $\rho_{g}(n)$ such that both $n-\rho_{g}(n)$ and $n+\rho_{g}(n)$ are half sums of two consecutive primes separated by a prime gap equal to $g$.

Is it possible to give an upper bound for $\rho_{g}(n)$ in terms of both $n$ and $g$ both unconditionally and under widely believed conjectures such as Hardy-Littlewood $k$-tuple conjecture or (G)RH? Is there always some $g$ such that $\rho_{g}(n)\ll_{g}\log^{C_{g}} n$ where $C_{g}$ is a positive constant depending only on $g$?

Define also the absolute gap radius of $n$, denoted by $\rho(n)$, as $\inf_{g}\{\rho_{g}(n)\}$. Is there an absolute positive constant $C$ such that $\rho(n)\ll\log^{C}n$?

For $n$ a large enough composite integer, define the $g$-gap radius of $n$, if it exists, for positive even $g$ as the smallest positive integer $\rho_{g}(n)$ such that both $n-\rho_{g}(n)$ and $n+\rho_{g}(n)$ are half sums of two consecutive primes separated by a prime gap equal to $g$.

Is it possible to give an upper bound for $\rho_{g}(n)$ in terms of both $n$ and $g$ both unconditionally and under widely believed conjectures such as Hardy-Littlewood $k$-tuple conjecture or (G)RH? Is there always some $g$ such that $\rho_{g}(n)\ll_{g}\log^{C_{g}} n$ where $C_{g}$ is a positive constant depending only on $g$?

Define also the absolute gap radius of $n$, denoted by $\rho(n)$, as $\inf_{g}\{\rho_{g}(n)\}$. Is there an absolute positive constant $C$ such that $\rho(n)\ll\log^{C}n$?

Edit: $2\rho(n)$ should be somewhat larger than the distance between some prime constellation $C$ (let's denote its largest prime by $C_{+}$) and its "mirror constellation" $C'$ (that is, the prime constellation obtained from $C$ reversing the order of the prime gaps; let's denote its smallest prime by $C'_{-}$ with $C'_{-}>C_{+}$) with $n$ being the half sum of $C_{+}$ and $C'_{-}$. As the user Lagrida proved in an answer to a previous question of mine that there is asymptotically the same amount of mirror constellations below $x$, this should entail that $\rho(n)$ exists provided $n$ is large enough.