The phrase "actually specified" is correct and usable in written English.
It can be used when clarifying or emphasizing that something has been clearly defined or detailed.
Example: "The requirements for the project were actually specified in the initial meeting, so there should be no confusion."
Alternatives: "clearly defined" or "explicitly stated".
That day's production plan actually specified rainfall, for the scene of Pollock and Krasner's arrival at their new home on a November evening in 1945.
The kiddies are then hauled off to a boot camp, the location and identification of which are also unclear in some cases; although I caught one show wherein it was actually specified as a facility of the Colorado County Juvenile Boot Camp, located in Texas.
When scientists were asked to rate the level of endorsement of their own papers, in the 237 papers that actually specified the proportion of human-caused global warming, over 96% agreed that humans have caused more than half of the recent global warming.
Then, whenever ω 2 m is given, the auction mechanism is actually specified.
The ASCII code is actually specified as a 7-bit code with 128 characters plus an extra bit designed as a parity bit that makes it an 8-bit code.
However, as Rowling herself stressed, Hermione's race and skin color were never actually specified in the books.
(They didn't actually specify this, but it was heavily implied).
So to actually specify an amount for the budget would be a bit of a stretch.
The law does not actually specify how much the bags should cost, but retailers have been told to charge at least 5p.
When phoneticians describe vowels as high or low, they probably are actually specifying the inverse of the frequency of the first formant.
They have monikers that sound like exotic birds — cleeks, spring-neck brassies, long-nose baffing spoons — but actually specify types of antique golf clubs, a sampling of the nearly 800 in the collection of Jeffery B. Ellis, which Sotheby's is to sell in New York on Sept. 27 and 28.
