Proof

The Proofs are sets of key items which appear in Kingdom Hearts II Final Mix and Kingdom Hearts χ, which are obtained after completing specific side-missions. In all appearances, they manifest as a crown or head adornment.

Kingdom Hearts II Final Mix
New costume modifications, known as Crowns, are unlocked by obtaining Proofs. These are equipped on Sora automatically and are present at all times, except during cutscenes and while Sora is in a world-specific costume. There are several types of crowns; possessing one Proof grants Sora a Copper Crown, two a Silver Crown, and all three a Gold Crown. The Proof of Connection is obtained by defeating the Lingering Will, the Proof of Nonexistence is rewarded after defeating all of Organization XIII's Replica Data, and the Proof of Peace is given after appeasing the entire Mushroom XIII.

Kingdom Hearts χ
In Kingdom Hearts χ crown accessory items are monthly ranking rewards for events and the Dark Corridor Trials, they are named after the. Four variants are available: Gold, Silver, Bronze and finally "Charm", which is actually a hair decoration and not a hat like the rest.

Kingdom Hearts 0.2 Birth by Sleep -A fragmentary passage-
Bronze, silver, and gold tiaras can be earned as part of the system.