The sinking of the USS Indianapolis is certainly proof, with 900 men dead.
By the way, do you know where the USS Indianapolis was sunk? It was sunk between Tinian and the Philippines, about 1,000 miles from Japan. Do you know why it was sunk? Because it was traveling alone, since the Japanese naval threat was deemed to be so minimal that the U.S. Navy did not bother sending any ships to accompany the USS Indianapolis, even though it was a heavy cruiser. The ship was just incredibly unlucky that one of the few Japanese subs still patrolling in that area happened to come across her and saw that she was alone.
We could spend many pages detailing the evidence of Japan's prostrate condition by May 1945: her growing food shortages, her paucity of fuel, her virtually defenseless condition against air and naval attacks, etc., etc. In addition to the evidence on this point that I've already presented in this thread, I cite General Marshall's memo to Stimson, dated 15 June 1945, in which Marshall said, "
The Japanese know they are licked for this generation" (p. 2).
So not only were the Japanese "licked" by June 1945, but Marshall and Stimson, and most everyone else in the White House, knew it.
Another item from the mountain of evidence on this point comes from
Nuremberg prosecutor Telford Taylor's memoir. Taylor's comments give us some idea of how commonly it was known among top American officials that Japan was defeated by May 1945, that Japan’s civilian leaders knew it, and that we knew from decrypted Japanese cables that Japan wanted to make peace. Telford was a reserve colonel in Army Intelligence. In May 1945, he returned to the U.S. from Europe and was thinking about trying to get an assignment in the Pacific. He spoke with his superiors in Army Intelligence, especially Colonel Alfred McCormack, who was a good friend of Assistant Secretary of War John McCloy and the director of the Military Intelligence Service. Telford tells us what McCormack told him when he asked about the Pacific War:
I visited Jackson's staff headquarters and discussed the situation in the Pacific theater with my superiors in the intelligence division, particularly with Colonel Alfred McCormack, in peacetime a law partner of John J. McCloy, the Assistant Secretary of War. I knew that McCormack was as well informed and otherwise equipped as anyone to assess the prospects of the war against Japan. Whether or not he was in on the secret of the atom bomb I do not know, but he told me categorically that the Japanese military situation was hopeless, that the Emperor's advisers knew it, and that intercepted Japanese diplomatic messages revealed their anxiety to make peace. (p. xi)