Elapsam semel occasionem...

The quotation on Phi's brooch is: Elapsam semel occasionem non ipse potest Iuppiter reprehendere

It seems to be based on a line from The Fables of Phaedrus. Phaedrus was a writer who rewrote the popular Greek fables of his day in Latin, with poetic meter. Perhaps coincidentally, the "Phae" of Phaedrus is pronounced the same as Phi.

The particular fable that is paraphrased in the game is from The Fables of Phaedrus, book 5, section 8, "Tempus".

Cursu volucri, pendens in novacula,
calvus, comosa fronte, nudo corpore,
quem si occuparis, teneas, elapsum semel
non ipse possit Iuppiter reprehendere,
occasionem rerum significat brevem.
Effectus impediret ne segnis mora,
finxere antiqui talem effigiem Temporis.

Here is one translation of it:

A Bald Man, balancing on a razor’s edge, fleet of foot, his forehead covered with hair, his body naked—if you have caught him, hold him fast; when he has once escaped, not Jupiter himself can overtake him: he is the emblem how shortlived is Opportunity. The ancients devised such a portraiture of Time, to signify that slothful delay should not hinder the execution of our purposes.

The above translation included a footnote:

From this figure of Time or Opportunity, Time came to be represented in the middle ages with a tuft of hair on his forehead; whence our common expression "To take time by the forelock," signifying to make the best of an opportunity.

This fable is about Caerus, the Greek personification of opportunity or luck. This comes from the Greek word kairos, which means the right moment. Caerus is said to have hair hanging down his forehead, but the back of his head is bald. He (an opportunity) can be seized by grabbing onto the hair of his forehead, but if he passes by, you can't catch him because the back of his head is bald, and not even Jupiter can catch him.

Plaudite! Acta est fabula!

"Acta est fabula, plaudite!" This is a Latin translation from Greek of what are said to have been the last words of Emperor Augustus Caesar. They mean "The play has ended. Applaud." This was traditionally said at the end of comedies in Greek theater. Suetonius wrote of Augustus's last words in The Twelve Caesars.

Dead Men Do Tell Tales

After inputting the ID and password on the two-headed lion screen in the Director's Office, seemingly random text is displayed on the screen. However, some of it is normal text. The text apparently comes from a CNN article about the recovery of an anchor from Blackbeard's flagship. The article begins with the famous saying "dead men tell no tales".

Voight-Kampff Test

Based on what is learned from the computer in the Director's Office, Phi wonders if the Nonary Game is a kind of a Voight-Kampff test. The Voight-Kampff Test is a fictional test to determine if someone is a human or an android, originally from the science fiction novel Do Androids Dream of Electric Sheep? which was later adapted into the film Blade Runner.

Knox's Ten Commandments

Sigma mentions Knox's Ten Commandments, which are ten rules that mystery stories should follow, according to author Ronald Knox.