We investigate how sorting algorithms efficiently overcome the exponential
size of the permutation space. Our main contribution is a new continuous-time
formulation of sorting as a gradient flow on the permutohedron, yielding an
independent proof of the classical $\Omega(n \log n)$ lower bound for
comparison-based sorting. This formulation reveals how exponential contraction
of disorder occurs under simple geometric dynamics. In support of this
Analyse, we present algebraic, combinatorial, and geometric perspectives,
including decision-tree arguments and linear constraints on the permutohedron.
The idea that efficient sorting arises from structure-guided logarithmic
reduction offers a unifying lens for how comparisons tame exponential spaces.
These observations connect to broader questions in theoretical computer
science, such as whether the existence of structure can explain why certain
computational problems permit efficient solutions.

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