The finite-degree profile of essential dimension I: general theory and characteristic prime to $\mathrm{\mid }G\mathrm{\mid }$

Essential dimension counts the parameters needed to define a torsor, but it can drop sharply over a field extension, and it does not say how large an extension is needed. For a $G$-torsor $\tau$ over a field $F\supseteq k$ we study the \emph{finite-degree profile} $\text{edd}{d}_k(\tau )=\min \{{\text{ed}}_k(\tau {\otimes }_F{F}^{\mathrm{\prime }}):[F^{\mathrm{\prime }}:F]\le d\}$, a variant of invariants of Farb–Kisin–Wolfson and Farb–Wolfson, and its jump degrees $d_j(\tau )$, the least degree of an extension over which $\tau$ needs only $j$ parameters. The profile is geometric over every field: $\text{edd}{d}_k(\tau )\le j$ iff some generically free $G$-variety of dimension $\le j$ has a closed point of degree $\le d$ in the free locus of its twist by $\tau$. We prove a Sylow divisibility theorem for jump degrees and a valuation-theoretic rank–index theorem, $\text{edd}{d}_k(\text{tgen})\ge \min \{\text{rk}B:B\le A,[A:B]\le d\}$ for abelian $A$ of order prime to $\mathrm{char}k$, which determines the profile of every abelian group over a field with enough roots of unity, for instance $n-\lfloor {\log }_{2}d\rfloor$ for $(\mathbb{Z}\mathrm{/}2{)}^n$. For $\mathbb{Z}\mathrm{/}p$ in characteristic $0$ we show $\lceil m\mathrm{/}2\rceil \le d_1\le [k({\zeta }_p+{\zeta }_p^{-1}):k]$, $m=[k({\zeta }_p):k]$, with equality for $m$ even. For $S_5$ in characteristic $0$ we prove $d_1=4$, so Klein’s reduction of the generic quintic is optimal; the lower bound comes from Abel–Jacobi rigidity, not from cohomological invariants. For $n\ge 6$, $6\mid d_1(S_n)$.