There are multiple books about ways to characterize the normal distribution. Eg, Bryc’s book starts with Herschel-Maxwell’s

Theorem: If $X$ and $Y$ are independent variables whose joint distribution is rotationally invariant, then $X$ and $Y$ are both normal.

He immediately notes that one can strengthen this to Polya’s:

Theorem: If $X$ and $Y$ are independent variables, and rotations of  $\pi/4$ and $\pi/2$ leave the distribution of $X$ invariant, then $X$ and $Y$ are both normal.

Perhaps somewhere in such books you’ll find a characterization with hypotheses that avoid moments but are number-theoretically tractable.

There are multiple books about ways to characterize the normal distribution. For instance, Bryc’s book starts with Herschel-Maxwell’s theorem:

If $X$ and $Y$ are independent variables whose joint distribution is rotationally invariant, then $X$ and $Y$ are both normal.

He immediately notes that one can strengthen this to Polya’s theorem:

If $X$ and $Y$ are independent variables, and rotations of  $\pi/4$ and $\pi/2$ leave the distribution of $X$ invariant, then $X$ and $Y$ are both normal.

Perhaps somewhere in such books you’ll find a characterization that avoids moments but is number-theoretically tractable.

There are multiple books about ways to characterize the normal distribution. Eg, Bryc’s book starts with Herschel-Maxwell’s

Theorem: If $X$ and $Y$ are independent variables whose joint distribution is rotationally invariant, then $X$ and $Y$ are both normal.

He immediately notes that one can weaken this to Polya’s:

Theorem: If $X$ and $Y$ are independent variables, and rotations of $\pi/4$ and $\pi/2$ leave the distribution of $X$ invariant, then $X$ and $Y$ are both normal.

Perhaps somewhere in those books you’ll find a characterization with hypotheses that avoid moments but are are number-theoretically tractable.

There are multiple books about ways to characterize the normal distribution. Eg, Bryc’s book starts with Herschel-Maxwell’s

Theorem: If $X$ and $Y$ are independent variables whose joint distribution is rotationally invariant, then $X$ and $Y$ are both normal.

He immediately notes that one can strengthen this to Polya’s:

Theorem: If $X$ and $Y$ are independent variables, and rotations of $\pi/4$ and $\pi/2$ leave the distribution of $X$ invariant, then $X$ and $Y$ are both normal.

Perhaps somewhere in such books you’ll find a characterization with hypotheses that avoid moments but are number-theoretically tractable.