Exact rook-pivoting growth factors in orders three and four

For Gaussian elimination with rook pivoting on real nonsingular matrices in
exact arithmetic, allowing every admissible choice of pivot, ties included, and
measuring growth over every entry of every active matrix, the manuscript proves
that the largest growth factor is $3$ in order three and $14/3$ in order four.
After normalisation the matrix is $A = LDR$ with positive pivots and
multipliers of modulus at most one. The order-three bound follows from a sign
argument on the entries of $A$. The order-four bound applies the order-three
result to a principal submatrix and combines three entry constraints with a
scalar inequality for two pivots and convexity in each of two variables.
Explicit rational matrices attain both values.