Step 1: Hyponatremia is sorted by plasma osmolality into low (true hypotonic), normal (pseudo), and high (translocational). The question asks for the false statement.
Step 2: Pseudohyponatremia happens when high lipids or proteins take up plasma volume, so the measured sodium reads low even though the true sodium activity is fine. In this state plasma osmolality is normal, not low. Option A says low osmolality, which is false, so A is the answer.
Step 3: In hyperglycemia, high glucose pulls water out of cells into plasma, diluting sodium. The plasma osmolality is high because of the glucose load. So option B is true.
Step 4: In SIADH there is excess water retention but no edema and no volume depletion, so the patient is clinically euvolemic (normovolemic). So option C is true.
Step 5: NSAIDs block prostaglandins, which normally oppose vasopressin in the kidney. Removing that opposition lets vasopressin act more strongly, promoting water retention. So option D is true.
Step 6: The false statement is that pseudohyponatremia has low plasma osmolality; it is actually normal.
Correct option: A.