Step 1: Identify the key clue. The patient has a Proteus urinary infection. Proteus mirabilis is a urease-producing organism - this single fact dictates the stone type.
Step 2: The urease mechanism. Urease splits urinary urea into ammonia and carbon dioxide: \[\text{Urea} + \text{H}_2\text{O} \xrightarrow{\text{urease}} 2\,\text{NH}_3 + \text{CO}_2\] The ammonia raises (alkalinises) the urine pH. In persistently alkaline urine, magnesium, ammonium and phosphate precipitate.
Step 3: The stone formed. Urease-producing organisms classically form struvite stones - magnesium ammonium phosphate, often mixed with calcium phosphate (carbonate-apatite). These are the "infection (triple-phosphate) stones" and tend to grow large, filling the collecting system (staghorn) or, as here, forming a large bladder stone. They are radio-opaque, consistent with being seen on plain X-ray.
Step 4: Why option A (Calcium Phosphate) is correct. Among the four options, only calcium phosphate is a phosphate, alkaline-urine, infection-related stone consistent with a urease-positive Proteus infection. It is the answer key choice for the phosphate/infection stone.
Step 5: Why the distractors are wrong. (B) Cystine (cysteine) stones occur in cystinuria, a hereditary tubular transport defect, unrelated to infection; urine tends to be acidic. (C) Calcium oxalate stones form in normal-to-acidic urine from metabolic/dietary causes, not from urease-driven alkalinisation. (D) Xanthine stones are rare, due to xanthine oxidase deficiency or allopurinol therapy, and are radiolucent - not infection-related.
Final Answer: Option A - Calcium Phosphate.