Is the formula for Occupied Size in the Size and Refinement Summary on page 110 of Covenants correct? Because Iâ€™ve been trying to get this to make sense and itâ€™s not.

Page 110 says â€œOccupied Size: current Virtue points minus Flaw points â€“ Refinement.â€ And then the Base Safety is â€œRefinement â€“ occupied Size (if greater than 0)â€ which makes no sense as it has refinement in two parts of the equation.

On page 109 the text says, â€œOccupied Size is equal to the Size minus the number of free points available for new Virtues. (This is exactly the same as the number of points of Virtues minus Flaws, minus the Refinement.)â€

So, if Iâ€™m moving into a brand new Size 2 lab how does this work? It hasnâ€™t been refined. No Virtues have been installed:
Size 2, Refinement 0, Current Virtue Points 0, Current Flaw Points 0.
Occupied Size: Size (2) - Current Virtue Points (0) â€“ Flaw Points (0) = 2.
Limit of Virtue Points Minus Flaw Points: Size (2) + Refinement (0) = 2.
Base Safety of Lab: Refinement (0) â€“ occupied Size 2 = -2.

This much makes sense to me. Now, letâ€™s spend a season refining the lab. Weâ€™ve still not installed any Virtues of Flaws:
Size 2, Refinement 1, Current Virtue Points 0, Current Flaw Points 0.
Occupied Size: Size (2) - Current Virtue Points (0) â€“ Flaw Points (0) = 2.
Limit of Virtue Points Minus Flaw Points: Size (2) + Refinement (1) = 3.
Base Safety of Lab: Refinement (1) â€“ occupied Size 2 = -1.

Okay, this is still good. Now, Iâ€™m going to spend two seasons and add a major virtue:
Size 2, Refinement 1, Current Virtue Points 3, Current Flaw Points 0.
Occupied Size: Size (2) - Current Virtue Points (3) â€“ Flaw Points (0) = -1.
Limit of Virtue Points Minus Flaw Points: Size (2) + Refinement (1) = 3.
Base Safety of Lab: Refinement (1) â€“ occupied Size 0* = 1.

Do I have occupied Size right for this last step? Since the occupied Size is less than zero itâ€™s taken out of the Base Safety, right? Leaving a safety of +1?

Does this make any sense at all?

If Iâ€™m missing something, please feel free to go step by step and treat me like an idiot, because I donâ€™t feel like Iâ€™m getting this.

You got the Occupied Size formula wrong. It doesn't represent the "free space" (i.e. effective size of the lab while it is occupied), but the "occupied space" (the space taken up by those virtues that are uncompensated by flaws or refinement). It is independent of the total size of the lab, as is the Base Safety.

Size 2, Refinement 0, Current Virtue Points 0, Current Flaw Points 0. Occupied Size: Current Virtue Points (0) â€“ Flaw Points (0) â€“ Refinement (0) = 0.
Limit of Virtue Points Minus Flaw Points: Size (2) + Refinement (0) = 2.
Base Safety of Lab: Refinement (0) â€“ occupied Size (0) = 0.

Size 2, Refinement 1, Current Virtue Points 0, Current Flaw Points 0. Occupied Size: Current Virtue Points (0) â€“ Flaw Points (0) â€“ Refinement (1) = -1.
Limit of Virtue Points Minus Flaw Points: Size (2) + Refinement (1) = 3.
Base Safety of Lab: Refinement (1) â€“ occupied Size (0) = 1. [because occupied size is less than zero]

Size 2, Refinement 1, Current Virtue Points 3, Current Flaw Points 0. Occupied Size: Current Virtue Points (3) â€“ Flaw Points (0) â€“ Refinement (1) = 2.
Limit of Virtue Points Minus Flaw Points: Size (2) + Refinement (1) = 3.
Base Safety of Lab: Refinement (1) â€“ occupied Size (2) = -1.

RAR -
Re-read the description at the end of p 109. The Occupied size is still capped by the size - you simply can't (normally) occupy more room than there is.

"It is sometime necessary to use the occupied Size, which might be less than the Size. More often than not, the laboratory is filled to capacity (there is no more room for extra points of Virtues), in which case the occupied Size is simply equal to the Size..."

But not always. If Size exceeds Occupied Size by too much, you get the Empty Flaw, which is not generally desirable.

The important thing to remember is that Virtues and Flaws do not have numerical values (+/-) unless they either take up space or provide space. So, if you have magical lighting (which takes up no space), or if your lab is simply way too hot (likewise), it's not space-related. If you put a Feature in your lab, some nifty gizmo that helps you do whatever it is that you do so well, that will take up space.

(So, here, whenever I say "Virtues/Flaws" here, I'm talking about ones that cost/provide points, not the "free" ones - because those never affect size or occupied size.)

"...However, if the lab space is not fully used, then the occupied Size is equal to the Size minus the number of free points available for new Virtues...", which is what you have, above.

Refinement is "compacting" or "organizing" - taking your "stuff" and making more room. So -2 Refinement can take +2 worth of "space-taking" stuff, that was filling that size 2 lab, and open it up enough for more Virtues. You don't have to have "stuff" before you can refine it - there is some basic "stuff" in your Lab to start, and that can just keep getting more and more organized. (But unless you already have Virtues, refinement only helps Safety.)

Now, wait - that's not the formula you quoted at the very top, which is the correct one. What you said was:"Occupied Size: current Virtue points minus Flaw points â€“ Refinement.â€So, OS = 0 V pts - 0 flaw - 0 refinement = 0 occupied size*

(Note that with Size 2, but 0 occupied size, this lab qualifies for the "empty" flaw, noted above! Good thing you put some stuff in there!)

If you had a simple +1 Virtue (a lesser Feature, perhaps), it would be:

OS = 1 V pts -0 Flaw pt - 0 refinement = 1 occupied space, as Virtues (with points) occupy space (and the lab is no longer so "empty"). Once you have 2 virtues in a size 2 lab, you need to Refine, or take a Flaw.

Think of it this way - you start with empty space - the basic lab & equipment, very bare. You occupy some of that with Flaws and/or Virtues, adding nifty or annoying stuff, and then Refine it to compact that stuff to create less occupied Space, and thus have more space for new Virtues (without needing more Flaws).

"Limit of Virtue pts - Flaw pts = Size + Refinement" (formula summary, sidebar, p 110).

So... Size 2, Refinement 1... you have equivalent of 3 space for virtues; your lab has

Occupied Size: 3 Virtue points - 0 Flaw points â€“ 1 Refinement = OS of 2. Perfect.

Limit of Virtue Points Minus Flaw Points: Size (2) + Refinement (1) = 3 (max Virtues)
Base Safety of Lab: Refinement (1) â€“ occupied Size (-1)* = 1 (* IF POSITIVE, so we do not subtract the -1.)

Let's do this - let's add a flaw to the Lab.Occupied Size 2

"Limit of Virtue pts - Flaw pts = Size + Refinement"
(or Limit = Size + Refinement + Flaws)
L = 2 + 1 +1 - We've got room for 1 more Virtue Point!So, let's design it with an "auspicious shape" - we'll say the shape is both auspicious for magic and awkward to work in - a hexagram with 6 pointy corners - urgh.Flaws:
-1 Awkward Shape

Virtues:
+3 Major Feature
+1 Auspicious Shape

Limit of Virtue Points = 2 size + 1 refinement + 1 flaw = +4 pts, so we're good!

Occupied Size = Current Virtue Points (4) â€“ Flaw Points (1) - refinement (1) = 2... ...which is the same as our base "Size", which also tells us we've crammed as much as we can for now. It's full. So, we Refine what we have one more season...

Occupied Size = current Virtue points minus Flaw points â€“ Refinement.â€
OS = 4 Virtue -1 Flaw -2 Refinement = OS 1 - and we have room for 1 more virtue in our Size 2 Lab.

Two seasons later, we have the same Size 2 lab, and one more Greater Virtue (+3 - more astronomical gear, can't get enough!). Nice organizing!

(Note that Auspicious Shape, Awkward Shape and Greater Feature will add to Characteristics and Specializations, and may ("will", in the case of this Flaw) also affect Safety.)