Can a Soulknife Harengon add a Psionic Energy Die to Initiative Checks?

Question

I've been searching forums and brainstorming about this idea. I am curious about how many holes can be put in this theory and how many DMs would allow this at their table.

Supporting Evidence

1. Every roll check in DnD is either an attack roll, saving throw, or ability check.
2. Initiative is not attack or saving, so it must be a DEX ability check.
3. Harengon gets proficiency to initiative checks.
4. Intent of initiative checks is to see if you go before your opponent.
5. Going after an opponent in initiative order means the check "failed".
6. Soulknife Rogue is allowed to add a Psionic Energy die to a failed ability check that they have proficiency with.
7. Soulknife Rogue only loses the Psionic Energy die if the new check succeeds.

Conclusion

1. Soulknife Harengon can use Psionic Energy die on "failed" initiative checks.
2. Soulknife Harengon doesn't lose the Psionic Energy die if new check still "fails" but gets to keep the new initiative.

Relevant Features

Hare-Trigger. You can add your proficiency bonus to your initiative rolls.

Psi-Bolstered Knack. When your nonpsionic training fails you, your psionic power can help: if you fail an ability check using a skill or tool with which you have proficiency, you can roll one Psionic Energy die and add the number rolled to the check, potentially turning failure into success. You expend the die only if the roll succeeds.

Answer 1

No, you are operating on false assumptions

The description of Psi-Bolstered Knack is very precise about when it can be used:

if you fail an ability check using a skill or tool with which you have proficiency.

An initiative roll is not an ability check using a skill or tool and Harengon's feature doesn't give you "proficiency in initiative rolls" because such proficiency doesn't exist. It merely allows you to add your proficiency bonus to your roll. There's also no way of failing or succeeding initiative roll, the roll merely determines the order in which participants get to take their turns.

Answer 2

Initiative checks do not succeed or fail

Success and failure for for ability checks is clearly defined in the rules:

To make an ability check, roll a d20 and add the relevant ability modifier. As with other d20 rolls, apply bonuses and penalties, and compare the total to the DC. If the total equals or exceeds the DC, the ability check is a success--the creature overcomes the challenge at hand. Otherwise, it's a failure, which means the character or monster makes no progress toward the objective or makes progress combined with a setback determined by the DM.

So for ability checks, a failure means rolling under the DC. However, unlike most ability checks, the dexterity check for initiative does not involve a DC:

Initiative determines the order of turns during combat. When combat starts, every participant makes a Dexterity check to determine their place in the initiative order. The DM makes one roll for an entire group of identical creatures, so each member of the group acts at the same time.

The DM ranks the combatants in order from the one with the highest Dexterity check total to the one with the lowest. This is the order (called the initiative order) in which they act during each round. The initiative order remains the same from round to round.

Hence, success and failure are not defined for initiative rolls, which means you cannot "fail" an initiative roll. This means that the trigger conditions for Psi-Bolstered Knack can never be met for initiative rolls.

Initaitive is not a contested check

You might argue that the initiative roll is a contested check, and therefore that rolling lower than the opponent counts as a failure. However, this is not the case. The rules for initiative do not reference contested checks, and initiative does not fit with the description of contested checks:

This can occur when both of them are trying to do the same thing and only one can succeed, such as attempting to snatch up a magic ring that has fallen on the floor.

Initiative does not determine which of two mutually exclusive actions succeeds, it determines turn order.

Initiative rolls do not involve a skill or tool

As pointed out by AnnaAG's answer, while an initiative roll is an ability check, it does not involve a skill or tool. This also makes it ineligible for Psi-Bolstered Knack, even if your DM rules that you can fail an initiative roll.

Answer 3

Ask your DM

This is honestly one of those corner cases that depend on low level details of the text. They show up sometimes.

The die can only be used on skill checks and tool checks for which you have proficiency. And initiative is neither.

Saying "attribute checks for which you have proficiency" is 99% the same, except there are some attribute checks where you can gain proficiency from something other than a skill or a tool.

These are initiative, counterspell and dispel magics, and remarkable athelete/jack of all trades bonuses (which are proficiency bonuses but not skill or tool bonuses).

Going this deep into the woods in 5e will result in either playing with legalese, or asking your DM.

There are 3 core types of rolls in 5e -- attribute check, attack roll and saving throw. The subtypes of attribute check -- "tool" and "skill" check -- are not usually used. You could take their use here as banning those 3 attribute checks. "Skill check" wasn't in the core rules of D&D at all.