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Edit April 7th 2023: there may be a link between constructible reals and the statement "if $n$ has a Galois radius of level $2^{d}$ with positive $d$ then it has a Galois radius of level $2^{d-1}$". Indeed we may associate to $n$ such that $2n=p+q^{2^{d}}$ with $p$ and $q$ primes a constructible real number $x_{n}$ whose minimal polynomial $P$ has degree $2^{d}$ and whose Galois group expresses the indiscernability of the prime factors of $q^{2^{d}}$ bijectively associated to the roots of $P$. That way the (Grand) RHG2 conjecture would entail asymptotic Goldbach's conjecture.

Say a non negative $r$ is a Galois radius of $n$ of type $(a,b)$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime and positive $a$ and $b$. If $a\neq b$, say $r$ is "unbalanced" and say $s$ is an inversion shift of $r$ if either $r-s$ or $r+s$ is a Galois radius of $n$ of type $(b,a)$. Let $s_{0}(n,r)$ be the smallest positive inversion shift of $r$.

Does any unbalanced Galois radius of any large enough integer have an inversion shift? Does one have $s_{0}(n,r)\ll_{\varepsilon}r^{1+\varepsilon}$?

More precisely, is the following conjecture true?

Conjecture: for every couple of different positive integers $(a,b)$, there is an integer $N_{a,b}$ such that every integer $n$ greater than $N_{a,b}$ which has a Galois radius $r$ of type $(a,b)$ has a Galois radius $r'$ of type $(b,a)$.

Edit March 9th 2023: we may consider the special case where $a$ and $b$ have the same Hamming weight $H$ (as the number of $1$'s in the binary representation of an integer) and write them as a block of $0$'s and $1$'s of given length $L$. The set $\mathcal{P}_{L,H}$ of permutations of such integers forms a finite group $G_{L,H}$ under composition (if we also count balanced Galois radii hence allow $a=b$). Among its elements, those of order $2$ may correspond bijectively to the set of inversion shifts of Galois radii of type $(a,b)$ of a given set of integers $n$.

A particularly interesting subcase would be $L=2H$ and the involution we seek the flipping of each bit. That way we would have $a=(2^{L}-1)-b$ and the even more special subsubcase $L=2$ would be worth studying per se. Indeed in that case $G_{2,1}\cong C_{2}$, which may lead to the following conjecture:

Asymptotic (Grand) Riemann-Hamming-Goldbach-Galois (RHG2 for short) conjecture

Assume (Grand) RH. Then every large enough integer has a Galois radius of level at most 2.

We might be able to prove that in this case the level $l=ab$ of a Galois radius corresponds to the order of a subgroup of $C_{2}$ and that every large enough integer having a Galois radius of level $2$ has a Galois radius of level $1$ while the converse should not be true.

We may also hopefully establish a correspondence between "the" maximal L-rig and the rig (ring without negatives) of natural integers where self-dual L-functions correspond to integers having a Galois radius of level $2$ and non self dual L-functions to integers not having such a Galois radius.

A possible way to establish the desired correspondence would be to construct a rig morphism $h$ from the maximal L-rig $(\mathcal{M},. ,\otimes,s\mapsto 1,\zeta)$ to the rig of natural integers $(\mathbb{Z}_{\geq 0},+,\times,0,1)$ and require the following properties to hold:

1. $h(F.G)=h(F)+h(G)$
2. $h(F\otimes G)=h(F)\times h(G)$

Note that the degree of an L-function already fulfills these properties.

We may also want to identify precisely the integer $h(F)$ corresponding to an L-function $F$ by drawing a parallel between unique factorization of L-functions and the fundamental theorem of arithmetic on one hand and between primitive L-functions and prime numbers on the other hand.

Say a non negative $r$ is a Galois radius of $n$ of type $(a,b)$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime and positive $a$ and $b$. If $a\neq b$, say $r$ is "unbalanced" and say $s$ is an inversion shift of $r$ if either $r-s$ or $r+s$ is a Galois radius of $n$ of type $(b,a)$. Let $s_{0}(n,r)$ be the smallest positive inversion shift of $r$.

Does any unbalanced Galois radius of any large enough integer have an inversion shift? Does one have $s_{0}(n,r)\ll_{\varepsilon}r^{1+\varepsilon}$?

More precisely, is the following conjecture true?

Conjecture: for every couple of different positive integers $(a,b)$, there is an integer $N_{a,b}$ such that every integer $n$ greater than $N_{a,b}$ which has a Galois radius $r$ of type $(a,b)$ has a Galois radius $r'$ of type $(b,a)$.

Edit March 9th 2023: we may consider the special case where $a$ and $b$ have the same Hamming weight $H$ (as the number of $1$'s in the binary representation of an integer) and write them as a block of $0$'s and $1$'s of given length $L$. The set $\mathcal{P}_{L,H}$ of permutations of such integers forms a finite group $G_{L,H}$ under composition (if we also count balanced Galois radii hence allow $a=b$). Among its elements, those of order $2$ may correspond bijectively to the set of inversion shifts of Galois radii of type $(a,b)$ of a given set of integers $n$.

A particularly interesting subcase would be $L=2H$ and the involution we seek the flipping of each bit. That way we would have $a=(2^{L}-1)-b$ and the even more special subsubcase $L=2$ would be worth studying per se. Indeed in that case $G_{2,1}\cong C_{2}$, which may lead to the following conjecture:

Asymptotic (Grand) Riemann-Hamming-Goldbach-Galois (RHG2 for short) conjecture

Assume (Grand) RH. Then every large enough integer has a Galois radius of level at most 2.

We might be able to prove that in this case the level $l=ab$ of a Galois radius corresponds to the order of a subgroup of $C_{2}$ and that every large enough integer having a Galois radius of level $2$ has a Galois radius of level $1$ while the converse should not be true.

We may also hopefully establish a correspondence between "the" maximal L-rig and the rig (ring without negatives) of natural integers where self-dual L-functions correspond to integers having a Galois radius of level $2$ and non self dual L-functions to integers not having such a Galois radius.

A possible way to establish the desired correspondence would be to construct a rig morphism $h$ from the maximal L-rig $(\mathcal{M},. ,\otimes,s\mapsto 1,\zeta)$ to the rig of natural integers $(\mathbb{Z}_{\geq 0},+,\times,0,1)$ and require the following properties to hold:

1. $h(F.G)=h(F)+h(G)$
2. $h(F\otimes G)=h(F)\times h(G)$

Note that the degree of an L-function already fulfills these properties.

We may also want to identify precisely the integer $h(F)$ corresponding to an L-function $F$ by drawing a parallel between unique factorization of L-functions and the fundamental theorem of arithmetic on one hand and between primitive L-functions and prime numbers on the other hand.

Note also that we expect the number of Galois radii of $n$ to grow as $n$ grows. The degree of an L-function $F$ being a potential candidate for $h(F)$ we may relate the number of Galois radii thereof to the density of non trivial zeros of $F$ on the critical line.

Say a non negative $r$ is a Galois radius of $n$ of type $(a,b)$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime and positive $a$ and $b$. If $a\neq b$, say $r$ is "unbalanced" and say $s$ is an inversion shift of $r$ if either $r-s$ or $r+s$ is a Galois radius of $n$ of type $(b,a)$. Let $s_{0}(n,r)$ be the smallest positive inversion shift of $r$.

Does any unbalanced Galois radius of any large enough integer have an inversion shift? Does one have $s_{0}(n,r)\ll_{\varepsilon}r^{1+\varepsilon}$?

More precisely, is the following conjecture true?

Conjecture: for every couple of different positive integers $(a,b)$, there is an integer $N_{a,b}$ such that every integer $n$ greater than $N_{a,b}$ which has a Galois radius $r$ of type $(a,b)$ has a Galois radius $r'$ of type $(b,a)$.

Edit March 9th 2023: we may consider the special case where $a$ and $b$ have the same Hamming weight $H$ (as the number of $1$'s in the binary representation of an integer) and write them as a block of $0$'s and $1$'s of given length $L$. The set $\mathcal{P}_{L,H}$ of permutations of such integers forms a finite group $G_{L,H}$ under composition (if we also count balanced Galois radii hence allow $a=b$). Among its elements, those of order $2$ may correspond bijectively to the set of inversion shifts of Galois radii of type $(a,b)$ of a given set of integers $n$.

A particularly interesting subcase would be $L=2H$ and the involution we seek the flipping of each bit. That way we would have $a=(2^{L}-1)-b$ and the even more special subsubcase $L=2$ would be worth studying per se. Indeed in that case $G_{2,1}\cong C_{2}$, which may lead to the following conjecture:

Asymptotic (Grand) Riemann-Hamming-Goldbach-Galois (RHG2 for short) conjecture

Assume (Grand) RH. Then every large enough integer has a Galois radius of level at most 2.

We might be able to prove that in this case the level $l=ab$ of a Galois radius corresponds to the order of a subgroup of $C_{2}$ and that every large enough integer having a Galois radius of level $2$ has a Galois radius of level $1$ while the converse should not be true.

We may also hopefully establish a correspondence between "the" maximal L-rig and the rig (ring without negatives) of natural integers where self-dual L-functions correspond to integers having a Galois radius of level $2$ and non self dual L-functions to integers not having such a Galois radius.

A possible way to establish the desired correspondence would be to construct a rig morphism $h$ from the maximal L-rig $(\mathcal{M}, ,\otimes,s\mapsto 1,\zeta)$ to the rig of natural integers $(\mathbb{Z}_{\geq 0},+\times,0,1)$ and require the following properties to hold:

1. $h(F.G)=h(F)+h(G)$
2. $h(F\otimes G)=h(F)\times h(G)$

Note that the degree of an L-function already fulfills these properties.

We may also want to identify precisely the integer $h(F)$ corresponding to an L-function $F$ by drawing a parallel between unique factorization of L-functions and the fundamental theorem of arithmetic on one hand and between primitive L-functions and prime numbers on the other hand.

Say a non negative $r$ is a Galois radius of $n$ of type $(a,b)$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime and positive $a$ and $b$. If $a\neq b$, say $r$ is "unbalanced" and say $s$ is an inversion shift of $r$ if either $r-s$ or $r+s$ is a Galois radius of $n$ of type $(b,a)$. Let $s_{0}(n,r)$ be the smallest positive inversion shift of $r$.

Does any unbalanced Galois radius of any large enough integer have an inversion shift? Does one have $s_{0}(n,r)\ll_{\varepsilon}r^{1+\varepsilon}$?

More precisely, is the following conjecture true?

Conjecture: for every couple of different positive integers $(a,b)$, there is an integer $N_{a,b}$ such that every integer $n$ greater than $N_{a,b}$ which has a Galois radius $r$ of type $(a,b)$ has a Galois radius $r'$ of type $(b,a)$.

Edit March 9th 2023: we may consider the special case where $a$ and $b$ have the same Hamming weight $H$ (as the number of $1$'s in the binary representation of an integer) and write them as a block of $0$'s and $1$'s of given length $L$. The set $\mathcal{P}_{L,H}$ of permutations of such integers forms a finite group $G_{L,H}$ under composition (if we also count balanced Galois radii hence allow $a=b$). Among its elements, those of order $2$ may correspond bijectively to the set of inversion shifts of Galois radii of type $(a,b)$ of a given set of integers $n$.

A particularly interesting subcase would be $L=2H$ and the involution we seek the flipping of each bit. That way we would have $a=(2^{L}-1)-b$ and the even more special subsubcase $L=2$ would be worth studying per se. Indeed in that case $G_{2,1}\cong C_{2}$, which may lead to the following conjecture:

Asymptotic (Grand) Riemann-Hamming-Goldbach-Galois (RHG2 for short) conjecture

Assume (Grand) RH. Then every large enough integer has a Galois radius of level at most 2.

We might be able to prove that in this case the level $l=ab$ of a Galois radius corresponds to the order of a subgroup of $C_{2}$ and that every large enough integer having a Galois radius of level $2$ has a Galois radius of level $1$ while the converse should not be true.

We may also hopefully establish a correspondence between "the" maximal L-rig and the rig (ring without negatives) of natural integers where self-dual L-functions correspond to integers having a Galois radius of level $2$ and non self dual L-functions to integers not having such a Galois radius.

A possible way to establish the desired correspondence would be to construct a rig morphism $h$ from the maximal L-rig $(\mathcal{M},. ,\otimes,s\mapsto 1,\zeta)$ to the rig of natural integers $(\mathbb{Z}_{\geq 0},+,\times,0,1)$ and require the following properties to hold:

1. $h(F.G)=h(F)+h(G)$
2. $h(F\otimes G)=h(F)\times h(G)$

Note that the degree of an L-function already fulfills these properties.

We may also want to identify precisely the integer $h(F)$ corresponding to an L-function $F$ by drawing a parallel between unique factorization of L-functions and the fundamental theorem of arithmetic on one hand and between primitive L-functions and prime numbers on the other hand.