Step 1: Anchor on the transporter. Cystinuria knocks out the renal and intestinal carrier for the dibasic amino acids, summarised by the mnemonic COLA: cystine, ornithine, lysine, arginine.
Step 2: With reabsorption gone, all four COLA amino acids spill into the urine. Cystine is the least soluble, so it crystallises into hexagonal crystals and forms recurrent stones, the clinical hallmark.
Step 3: The trap is cysteine versus cystine. The transporter and disease are about cystine, the disulfide-linked dimer, whereas free cysteine is a different single-residue molecule and is not the characteristically excreted species, so it is the exception.
Step 4: Confirm by elimination: arginine and ornithine are genuine COLA members and are excreted, leaving cysteine as the odd one out.
\[\boxed{\text{Cysteine}}\]