Clotting reflects a balance between procoagulant and anticoagulant forces. Nephrotic syndrome disturbs that balance because the leaky glomerulus filters out plasma proteins by size.
Antithrombin III is roughly the size of albumin, so it passes into the urine and its blood level drops. Since antithrombin III normally inhibits thrombin and other clotting factors, losing it releases the brakes on coagulation. The body responds with extra hepatic output of large proteins like fibrinogen and with more reactive platelets, pushing the patient further into a clot-prone state, classically seen as renal vein thrombosis.
The wrong options each contradict the physiology. Fibrinogen actually rises, not falls. Vitamin K handling is not the central problem here. Protein C is itself an anticoagulant, so more of it would protect against clots rather than cause them.
The single best explanation for the prothrombotic tendency in nephrotic syndrome is the urinary depletion of the body's natural thrombin inhibitor.
\[\boxed{\text{Loss of antithrombin III}}\]