= Solution
\b[The scheme is not uniform in general.] A particular successful path has <probability> equal to the product of the reciprocals of its available-neighbour counts. Those counts depend on the path geometry.
For length four, the straight path $(0,0),(1,0),(2,0),(3,0),(4,0)$ has <probability> $1/(4\cdot3\cdot3\cdot3)=1/108$. The path $(0,0),(1,0),(1,1),(0,1),(-1,1)$ has only two available neighbours at its last choice, because the previous point and the origin block two directions. Its <probability> is $1/(4\cdot3\cdot3\cdot2)=1/72$. There is no trapping before four steps, so these are already the output probabilities. They differ, disproving uniformity on $S_4$.
At lengths where restarts occur, conditioning an attempt on success normalizes these path probabilities by a common success <probability>; it does not remove the unequal products.
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