The archives catalog proceeds to greet us with 7 different screens, I have not found what all of them are but I've managed to decode the Verification code nonetheless. There are three known codes, two of them are provided by the developer on Discord and one is hidden in the game. There may be more codes but that is all I found.

The code is pretty simple, it is just an advanced version of the Ceasar cipher instead of a fixed coding number that changes the letter we are provided with PAWNCH decipher code. I won’t bother explaining how the code works because that is already explained in the game.

The developer provided us with the following Falcon codes on Discord:

WULFNRUWZMIBKIVLS
VHDVRQDOOZMMBQVOM

To decode those we only need the Pawnch code which we can get from MIRVIN. Here is how to do it step by step:
Press the „i” icon left of the settings button.
Choose „What or who made you?” option.
Choose „Who do you work for?” option.
Choose „The Soviet Union?” option.

Mirvin will glitch and he will give you for options, first three do nothing and the last one gives you the PAWNCH code which is CCCCCCCCHHHHHHHHT 8Cs and 8Hs and lone T at the end (17 letters).

Here are deciphered codes:
TRICKORTREATCANDY
SEASONALGREETINGS

In the C.T.T.W. we can get the following letters from the morse code: RXU

In the ♥♥♥♥, we can find the following letters if we press the Next button four times: PDWUB

In Jack we can see that Marylin Monroe is made out of the word: WAPSI

I found nothing else in Probopet and Tiler but what I've got was enough.

RXUPDWUBWAPSI

is 13 letters, which means it misses four additional letters. If we translate the first part anyway we get:

OURMATRYOSHKA

meaning there is some four letter word missing at the end of the code since if we would add anything in between or in the beginning it would change the code due to how the cipher works. I asked ChatGPT what 4 letter word he would add to Our Matryoshka and one suggestion was the correct answer. It is DOLL, makes sense right?

The final code is:

OURMATRYOSHKADOLL

I have no idea what it does though, but the verification screens confirms it is the correct code by saying ACCEPTED!

By encrypting the last four letters of the code DOLL we get LWTF. I found that one of Nixon's pets (Misha Bear) is named LWT so that could be it, but I still have no idea which player is meant to figure it out. Maybe somewhere in the game, there is a clue.

The last letter remains unfound F. It would make sense if it was hidden in the Tiler screen. I checked the definition of the word Tiler and it is (1. a person who lays tiles." a roof tiler" 2. the doorkeeper of a Freemasons' lodge, who prevents outsiders from entering.). That leads nowhere. However the phrase on the screen "In hoc signo vinces" translated into English as "In this sign, thou shalt conquer". also seems to not give a clue. Then as I was looking at the screen I realized the Eye has the letter F inside it, that was the last letter.