What is the '4 bullets' thing all about?

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Just a random number between only 1 shot and the 6 bullets of a gun.

Is just a way to say

Yep, seemed pretty random. I liked how the 4 gunshots were synched up with 4 "film" shots. Not the first time it's been done I'm sure.

"Are birds free from the chains of the skyway?"

I think the significance was that earlier (25 years ago) Morales had told Benjamin something like "What use would it be to put 4 bullets into him?"
When Morales later tells Benjamin how he killed the murderer with 4 bullets (told visually as an "unreliable narrator" flashback), Benjamin doesn't immediately make the connection. Only after he leaves Morales' place there is a flashback to the earlier statement and Benjamin makes the connection and figures out that Morales didn't really shoot him. Also the flashback to Morales closing the curtain to hide the shed.

The number 4 doesn't have any special meaning. In Spanish if you had to pick a random number of shots, 3 or 4 would be the most common choice, neither too few nor too many to be special, just average.

Funny coincidence: 4 is also considered an unlucky number in Japanese, because its pronounced the same way as "death" (shi)

It may not have been intentional but I just realised that there were four people who had a "secret in their eyes" and were displaying it throughout the film. Irene and Esposito who were hiding a secret love for each other - even though they were not hiding it very well. Also Gomez and Morales were each hiding their own secret. Coincidence?

In Spanish you can say 4 as "some", like in "pegar 4 gritos", "4 gatos", etc.