Scoring a grid

Question

As a professor of Awesomeness at the prestigious Ad Hoc University (other questions in this series), I decided to assign my students this puzzle. Unfortunately, they were all unable to get it! I want to post it here to see if any puzzlers can get it.
Here we go:

Suppose I've defined an operation that takes in a 5x5 grid of booleans (or 1s and 0s) and outputs a number which represents its score. Here are some examples:
 = 5 + 7 = 12
 = 3 + 6 = 9
 = 3 + 0 = 3
 = $infty$
 = 6 + 4 = 10
 = $infty$
Your job is to tell me how I score my grids!

Note: all the information of the puzzle is in the blockquote; nothing outside the blockquote is relevant!

StephenTG · Accepted Answer

You score your grids by

The score is calculated from two pieces:

Steve · Answer

As the answer from @StephenTG states, the secret is to

Specifically,

Implementing the necessary calculations in Excel:

We can see that, as also stated in @StephenTG's answer,

Higher finite scores are possible. For example,

... and revisiting it a little later, some minor tweaks improve this:

Later, I finally got round to doing an exhaustive computer search for better solutions. The most relevant part of the output

Scoring a grid

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Scoring a grid​

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Scoring a grid