On a relation between uniform coding and problems of the form DTIMEF($\cal F$) =? DSPACEF($\cal F)$

Abstract.

If uniform coding (Gödelization) of potentially infinite sequences of numbers can be performed in PSPACEF, then PSPACE = EXPTIME $\neq$ EXPSPACE = 2-EXPTIME, and, for all p, we have $p-$ EXPSPACE = $p+1$-EXPTIME; if it can be performed in LINSPACEF, we also have LINSPACE = DTIME$(2^{cx})$; the proof fails, when relativized to oracle-TM's. A by-product of this research is that PTIMEF is not closed under number-theoretic, limited, course-of-values recursion.