According to papers here https://users.cs.duke.edu/~reif/temp/RandomHardProblems/NovaFandina/succinctPermanent.pdf and here http://perso.ens-lyon.fr/natacha.portier/papers/PY86.pdf questions in $mathsf{NP}$ can be rephrased by questions in $mathsf{NEXP}$ by replacing inputs by succinct circuits if the $mathsf{NP}$-completeness reduction is a projection. In PY86.pdf paper it is mentioned that all known $mathsf{NP}$-completeness reductions (at time paper was written) are projections and for every reduction that is a projection the succinct version is also $mathsf{NEXP}$-hard. Is there an $mathsf{NP}$-completeness reduction that is not a projection?
My query then is when can arguments for $mathsf{NP}$ be replaced by $mathsf{NEXP}$? For instance does a proof contrasting $mathsf{NP}$ and $mathsf{P/Poly}$ be replaced by $mathsf{NEXP}$ and $mathsf{P/Poly}$.
Does $$mathsf{NP}subseteqmathsf{P/Poly}impliesmathsf{NEXP}subseteqmathsf{P/Poly}$$ follow?
This may not be true due to scaling issues.
(1) When would $$mathsf{NEXP}subseteqmathsf{P/Poly}$$ follow if $mathsf{NP}subseteqmathsf{P/Poly}$ holds?
(2) If scaling is only issue then does $$mathsf{NEXP}subseteqmathsf{P/Poly}implies mathsf{NP}subseteqmathsf{P/Log}$$ hold true?
It is clear converse $$mathsf{NP}notsubseteqmathsf{P/Poly}impliesmathsf{NEXP}notsubseteqmathsf{P/Poly}$$ follows because $mathsf{NP}subseteqmathsf{NEXP}$.