What is the most standardized way to handle spells from a template?

I need help with a specific piece of a problem in a game I’m running, which generalizes to the question title.

Specifically, there is a big bad dracolich somewhere that the players want to be. They have a cunning plan to simply wish the dracolich away to a different place. They have access to wish through an essentially homebrewed acquired template that one of them has gained (a spiffier version of Smoking Eye Creature, from the Shackled City campaign path). The template is tied to a demiplane, and the wish effect works once/year, and the demi-plane eats the xp cost.

I have existing houserules that say wish can essentially only be used for the explicitly listed uses (it does those things without issue, and is actively malicious about perverting or failing any off-label wishes). However, this is one of the listed effects:

A wish can lift one creature per caster level from anywhere on any plane and place those creatures anywhere else on any plane regardless of local conditions. An unwilling target gets a Will save to negate the effect, and spell resistance (if any) applies.

This is where it breaks down: I haven’t specified a caster level or save value for the wish ability. What would be the most standardized way to determine those?

I’m really looking for a textual basis for this. I know there are not exact rules to cover this situation (are there?), so a good answer will probably explain how this matches some existing pattern, and have a clear method for finding caster level (for SR) and saving throw DC.

In D&D 5e, do most wands no longer require command words?

In the section on magic items in the DMG, under Activating a Magic Item, it states:

Activating some magic items requires a user to do something in particular, such as holding the item and uttering a command word, reading the item if it is a scroll, or drinking it if it is a potion. The description of each item category or individual item details how an item is activated.

Further down, under Spells in the same section, it states:

Some magic items allow the user to cast a spell from the item, often by expending charges from it. The spell is cast at the lowest possible spell and caster level, doesn’t expend any of the user’s spell slots, and requires no components [emphasis mine] unless the item’s description says otherwise.

Note this emphasized text does not say material components, just components. That would suggest verbal and somatic components as well as material.

The general description of wands says nothing about command words, either:

A magic wand is about 15 inches long and crafted of metal, bone, or wood. It is tipped with metal, crystal, stone, or some other material.

Further, some wand descriptions specifically mention a command word. For example, the wand of enemy detection says:

This wand has 7 charges. While holding it, you can use an action and expend 1 charge to speak its command word [emphasis mine]. For the next minute, you know the direction of the nearest creature hostile to you . . .

So going by the tenet that in 5e, the specific overrides the general, all this would suggest wands no longer need a command word to function, unless otherwise stated. But this seems like a really strange change to make from previous editions, and I’ve scoured both the rest of the rules and the web to see if I missed something.

Which magic item of very rare or lower rarity is most useful to protect a group of ordinary soldiers?


Background

I’m playing a mid-level artificer (artillerist) who’s a disgruntled veteran with a missing limb who, disillusioned by the leaders’ willingness to send soldiers to their deaths, has retired from the army and opened a shop. An adventure hook has people steal his work-in-progress masterpiece and now I need to find a fitting item he was trying to create.
Because of this background, the item he would be most interested in would be something that helps ordinary soldiers without magic powers survive the horrors of the battlefield. It might be something that protects a group of people from hostile spells or something that provides healing to them, similar to the artificer’s Protector cannon.

Criteria

  • I am trying to find an officially published item before resorting to homebrew (UA is probably fine, as is basic refluffing)
  • The DM has ruled that the item should be below legendary rank, so very rare at most
  • I probably won’t be held to strict prerequisites such as being able to cast every spell going into the items myself, but the item should still basically fit the artificer flavour
  • The item should be usable by someone who cannot cast spells
  • The item should be able to affect a group, not just the carrier
  • The item should be defensive in nature

My own research

I’ve gone through the "warding" and "healing" categories of magic items on D&D Beyond and found very little. There are almost no items that work on groups and those that do tend to be musical instruments or magic staves that need the user to be a spellcaster.
In general it seems that antimagic items aren’t really a thing in 5e. An item that can cast Antimagic Field on he regular would probably be in the legendary category and a Ring of spell Storing would again require a (powerful) spellcaster to be useful.
An ideal solution would be something like a banner of protection or an Eldritch Cannon: Protector that doesn’t need an artificer to be present. I’ve also considered something like a Ring of Regeneration, but that’s again a one-person item.

Finding the most frequent element, given that it’s Theta(n)-frequent?

We know [Ben-Or 1983] that deciding whether all elements in an array are distinct requires $ \Theta(n \log(n))$ time; and this problem reduces to finding the most frequent element, so it takes $ \Theta(n \log(n))$ time to find the most frequent element (assuming the domain of the array elements is not small).

But what happens when you know that there’s an element with frequency at least $ \alpha \cdot n$ ? Can you then decide the problem, or determine what the element is, in linear time (in $ n$ , not necessarily in $ 1/\alpha$ ) and deterministically?

Most efficient method for set intersection

Suppose I have two finite sets, $ A$ and $ B$ , with arbitrarily large cardinalities, the ordered integral elements of which are determined by unique (and well defined) polynomial generating functions $ f:\mathbb{N}\rightarrow\mathbb{Z}$ given by, say, $ f_1(x_i)$ and $ f_2(x_j)$ , respectively. Assume, also, that $ A\cap B$ is always a singleton set $ \{a\}$ such that $ a=f_1(x_i)=f_2(x_j)$ where I’ve proven that $ i\neq j$ .

Assuming you can even avoid the memory-dump problem, it seems the worst way to find $ \{a\}$ is to generate both sets and then check for the intersection. I wrote a simple code in Sagemath that does this, and, as I suspected, it doesn’t work well for sets with even moderately large cardinalities.

Is there a better way to (program a computer to) find the intersection of two sets, or is it just as hopeless (from a time-complexity perspective) as trying to solve $ f_1(x_i)=f_2(x_j)$ directly when the cardinalities are prohibitively large? Is there a parallel-computing possibility? If not, perhaps there’s a way to limit the atomistic search based on a range of values—i.e., each loop terminates the search after it finds the first $ i$ value such that $ f_1(x_i)>f_2(x_j)$ , knowing that $ f_1(x_{i+1}), f_1(x_{i+2}), f_1(x_{i+3}), \cdots, f_1(x_{i+n})>f_1(x_i)>f_2(x_j)$ .