Step 1: Recall the two ways cells obtain purine nucleotides: a costly de-novo route and an economical salvage route. Salvage of the purine bases hypoxanthine and guanine is done by one enzyme - HGPRT.
Step 2: Total loss of HGPRT removes the salvage option entirely, so the substrate PRPP that salvage would have used is spared and instead pushes the de-novo machinery into overdrive.
Step 3: The net result is excess purine turnover and a flood of uric acid, explaining the gout, renal stones, and hyperuricemia, while CNS involvement adds intellectual disability and compulsive self-injury.
Step 4: The distractors act on pyrimidine (orotate, uracil) or NAD precursor (quinolinate) handling, none of which links to this salvage-defect syndrome.
\[\boxed{\text{HGPRT}}\]