We present CLASSO, a path-following method for the LASSO subject to general linear equality and inequality constraints. The local direction is obtained from a quadratic programme on an augmented active set, which determines both the coefficients that leave zero and the weakly active inequalities that remain binding. For a positive-definite Hessian and a nonempty feasible set, this programme has a unique primal minimiser, even when the binding constraint rows are dependent, and gives the local derivative of the solution path. The construction recovers the LASSO-modified LARS direction without constraints, and over the full cone gives the segment leaving the origin. We prove that the primal path is piecewise affine with finitely many kinks, without a constraint qualification or unique multipliers. A finite path does not imply a finite traversal: re-certifying at each kink can fail to terminate. CLASSO instead follows the minimum-norm certificate along a lexicographic rate; we prove that this rule reaches $λ=0$ in exact arithmetic within an explicit bound and traverses the whole path. These results are machine-checked in Lean 4, and the floating-point implementation agrees with an exact rational one on all $210$ small test problems. We assess the implementation on a $39$-case synthetic suite and six constrained versions of the diabetes benchmark. On the $31$ nontrivial synthetic cases with $n\ge p$, sampled relative objective differences from numerical reference solves are below $10^{-9}$ in magnitude; unnormalised endpoint differences reach $1.5\times10^{-7}$ in magnitude. For $n<p$, matched differences range from $-3.6\times10^{-7}$ to $2\times10^{-9}$ across five ridge strengths. Across the synthetic and diabetes benchmarks, checks of all $580$ recorded segments found no missed interior event and all checked KKT residuals within the stated tolerances.