Can my Primal Companion roll initiative instead of my character?

In Tasha’s Primal Companion feature it doesn’t say that the beast doesn’t roll initiative, just that it acts during the player’s turn. Is that enough to mean that it’s the player that has to roll initiative?

I assume the answer is yes, but it would be nice if you were playing a strength build ranger, or to take advantage of the Primal Bond.

Does the Beast Master’s Primal Companion feature allows the beast to take two Actions in a single turn?

I watched a video from Dungeon Dudes on YouTube where they rated the Ranger subclasses. Around the 23:18 mark, they mentioned that the new Beast Master’s Primal Companion feature from Tasha’s Cauldron of Everything can allow the Ranger to forgo one of their attacks to allow the Beast to take the Attack action, on top of the action that they can make using the Ranger’s bonus action. The Primal Companion feature description says the following:

Primal Companion

[…]
In combat the beast acts during your turn. It can move and use its reaction on its own, but the only action it takes is the Dodge action, unless you take a bonus action on your turn to command it to take another action. That action can be one in its stat block or some other action. You can also sacrifice one of your attacks when you take the Attack action to command the beast to take the Attack action. If you are incapacitated, the beast can take any action of its choice, not just Dodge.
[…]

As far as I know, a creature can only take one action on their turn. Is this a specific ruling for the Primal Companion since it doesn’t have its own turn because "it acts on your [the Ranger’s] turn?" Also does this mean that the Ranger can break up their movement, actions, and bonus actions with the Beast’s on their turn, essentially controlling two characters freely? In the video, Monty mentioned that a level 11 Beast Master Ranger can make one attack and have the beast attack four times which is crazy in my opinion.

Can a spell cast outside a primal magic area affect its interior?

The question divides in two possibilities:

  1. A spell is cast outside a primal magic area but the target/place is inside of it. Does it work?

  2. A spell is cast outside a primal magic area but the target/place is outside of it, but the spell’s area of effect still affects part of the area. Does it work?

The example that came to my mind: a fireball cast to explode inside a primal magic area or outside of it, but the explosion still reaches people inside the area.

Does the Primal Awareness feature allow you to cast each spell once?

In the Unearthed Arcana – Class features article, Primal Awareness, a new variant feature for the ranger was introduced, replacing Primeval Awareness.

This feature lists a number spells as additional spells known that don’t count against the number of ranger spells you know. That part of the feature is fairly clear. However, it goes on to say:

You can cast each of these spells once without expending a spell slot. Once you cast a spell in this way, you can’t do so again until you finish a long rest.

To me, this section is unclear and has two possible interpretations:

  1. “You can cast each of these spells once…” – You can cast each of the spells lists 1/long rest without expending a spell slot. A total of 6 spells cast per day.
  2. You can cast any of these spells once. – Once you have cast one of these spells in this way, you cannot cast any of them until you finish a long rest. A total of 1 spell cast per day.

I can make a logical argument for both cases, one based on the first sentence, the other on the second.

Which of these interpretations is correct?

Is increasing the Primal Savagery spell’s damage die to 1d12 unbalanced/unnecessary?

The primal savagery spell states:

[…] Make a melee spell attack against one creature within 5 feet of you. On a hit, the target takes 1d10 acid damage…

The spell’s damage increases by 1d10 when you reach 5th level (2d10), 11th level (3d10), and 17th level (4d10).

Compare this to some other spells:

  1. Firebolt deals 1d10 fire damage from a range.
  2. Ray of frost deals 1d8 cold damage from a range and lowers the target’s speed by 10.
  3. Shocking grasp deals 1d8 lightning damage from melee and prevents the target from taking reactions, with advantage against targets wearing metal armor.
  4. Chill touch deals 1d8 necrotic damage from melee and prevents it from regaining hit points, also granting disadvantage to undead targets.

The primal savagery cantrip deals one die size additional damage (an average of +1 damage) and has no additional effects nor does it let you attack from a range. This feels particularly weak to me and so I wonder:
Would increasing its damage die to 1d12 (both the initial damage, and how it scales) be unbalancing or is the change simply unnecessary and the spell is already balanced against other cantrips?

Which spell level does the Primal Draconic Bloodline sorcerer’s Variant Bloodline Arcana feature refer to?

The Sorcerer’s Variant Bloodline Arcana for the Primal Dragon bloodline says

Variant Bloodline Arcana: Whenever you cast a spell with an energy descriptor that matches your draconic bloodline’s energy type, you can reroll a number of damage dice equal to half the spell’s level.

Is that base spell level, or spell level after applying metamagic feats?

a question about solving the primal problem via the dual

I have a question when reading the convex optimization by Boyd, it is about solving the primal problem via the dual(in page 248):

suppose we have strong duality and an optimal $ (\lambda^*,v^*)$ is known, and suppose that the minimizer of $ L(x,\lambda^*,v^*)$ is unique; then why “if the solution of $ minimize\quad L(x,\lambda^*,v^*)$ is not feasible, then no primal optimal point can exist”? What does “no primal optimal point can exist” mean, does it mean that the primal problem is unsolvable?

Please help me.