How is free-will formally defined as distinct from determinism, randomness and determinism-randomness hybrid to support moral responsibility?

Usually free-will is assumed by most faith traditions as a prerequisite for moral responsibility in order to justify eternal punishment. The argument goes as "you are truly responsible for your immoral deeds because you were perfectly capable of acting morally, but chose not to, and therefore you are guilty and deserve punishment". In other words, there is an assumption of a self capable of having chosen otherwise. But this definition sounds kind of handwavy to me. What do they really mean by "capable of having chosen otherwise"? Does this mean that if we *somehow* go back in time and press the *"univere's play button"* again, a different decision would have come out of the same self? But how would that be different from randomness? I would appreciate a **formal** (hopefully, **mathematical** if you can) definition of free-will that **clearly** differentiates it from **determinism**, **randomness** and any kind of **determinism-randomness hybrid**. Furthermore, I would love to read a logical discussion of how the formal definition provided would support the existence of **true** moral responsibility. Let me explain why. If the only options available were *determinism*, *randomness* and *determinism-randomness hybrid*, I cannot see any hope for **true** moral responsibility: 1. **Determinism**: your actions, either moral or immoral, are just the ripple effect of past events, you cannot help but do what the laws of Physics make you do, you had no choice, it's not your fault, therefore there is no true moral responsibility on your part. 2. **Randomness**: your actions are random, chaotic, there is no control, you are lucky/unlucky that you behave morally/immorally by random chance (e.g. because of weird quantum randomness in the brain, for example). In other words, it's just a matter of luck, a matter of winning the morality lottery. If you behave immorally, it's not really your fault, it's just your bad luck. Therefore there is no true moral responsibility on your part either. 3. **Determinism-Randomness Hybrid**: your actions are the result of a combination of deterministic rules applied to past events combined with random quantum noise or something along those lines. Depending on how strong are the causal ripple effects from past events and how strong are the random noises altering them, you end up acting either morally or immorally. In part you are unlucky, in part you have no choice. Anyway, it's not your fault. Therefore there is no true moral responsibility on your part either. So, *somehow*, **free-will** is supposed to have a mysterious **formal definition** that distinguishes it from *determinism*, *randomness* and *determinism-randomness hybrid* that allows for **true moral responsibility** in a way that these other concepts can't. I would really appreciate such a formal definition.