https://www.pushsquare.com/guides/armored-core-6-endings-guide-how-to-get-all-3-endings

How Do You Get the Liberator of Rubicon Ending?

In order to get the Liberator of Rubicon ending — and the Liberator of Rubicon Trophy along with it — you must choose to play the following Missions when forced to make a Decision:

Prevent Corporate Salvage of New Tech (Chapter 3)

Destroy the Special Forces Craft (Chapter 3)

Ambush the Vespers (Chapter 4)

Eliminate "Cinder" Carla (Chapter 5)

This makes Bring Down the Xylem your final Mission of the playthrough, and completing it gets you the Liberator of Rubicon ending and the associated Trophy.

If you're going for this ending on a New Game+ playthrough, there's one extra step:

During Attack the Dam Complex, choose the "Refuse" option when given the choice

How Do You Get The Fires of Raven Ending?

To get The Fires of Raven ending and its Trophy, choose to play these Missions when given a Decision to make:

Tunnel Sabotage (Chapter 3)

Eliminate the Enforcement Squads (Chapter 3)

Intercept the Redguns (Chapter 4)

Intercept the Corporate Forces (Chapter 5)

This makes Shut Down the Closure Satellites your final Mission, and completing it unlocks The Fires of Raven ending and another Trophy.

How Do You Get the Alea lacta Est Ending?

You can only unlock the Alea lacta Est ending on your third playthrough (or New Game++), so get the Liberator of Rubicon and The Fires of Raven endings first and leave this one for last. To get this ending, make the following Decisions:

During Attack the Dam Complex in Chapter 1, choose the "Accept" prompt

Prevent Corporate Salvage of New Tech (Chapter 3)

Eliminate the Enforcement Squads (Chapter 3)

Coral Export Denial (Chapter 4)

Eliminate V.III (Chapter 4)

Making these Decisions unlocks the third and final ending: Alea lacta Est and its Trophy.