Step 1: Sort each option by whether it signals immune collapse from HIV. Step 2: Lymphoma of the central nervous system and candida infection of the oesophagus are opportunistic, meaning they surface only when immunity is severely suppressed - both are AIDS-defining and HIV-associated. Step 3: Late-stage syphilis, by contrast, is a slow progression of the spirochete infection itself; whether it reaches the tertiary stage is governed by how long the syphilis went untreated, not by the presence of HIV. Step 4: The odd one out, with no HIV dependence, is tertiary syphilis. \[\boxed{\text{Tertiary syphilis}}\]