It seems Samsung hasn't been totally forthcoming with us regarding the Galaxy Gear's compatibility with devices other than the Galaxy Note 3 and Galaxy Note 10.1 2014 Edition. The company has stated that compatibility of the Gear with the Galaxy S4 (and other devices) will arrive via a software update next month, but it turns out that the Gear is already compatible with the Galaxy S4.
The folks over at Planete Galaxy decided to do a test, and the Galaxy S4 managed to connect to the Galaxy Gear without a hiccup. Many functions work, such as changing the clock theme, controlling music playing on the phone from the watch, transferring pictures, and even reminder alerts. It was thought that the Gear would be compatible only with devices running on Android 4.3 (as that's the Android version which enables Bluetooth 4.0 LE, used by the Galaxy Gear to connect to devices), but as you can see in the video, that is clearly not the case.
However, at the moment, app support seems to be working only for the Galaxy Note 3, but hey, at least we know that if someone decides to fork out the cash for the Gear without purchasing the Note 3, they'll be able to use all the basic functions until the compatibility update arrives.