Completing the square?

Question

As you work through one of your (also) countless maths books, enjoying (but also slightly concerned about) the lack of puzzling notes from your brother, you discover that he has struck once again, this time pasting his correspondence into your book. What a nice early Christmas present! *grumble*

Doesn't look like he's gotten more into maths since last time. What is he getting you to do this time?

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Transcription of clues:

Across:
2. Electron in iron produces charge (3)
4. Disfigure horse when tail is cut off (3)
5. Spirit for every-one (4)
6. Primary city of “Jordan: the Morning Guy” (5)
9. Lady’s crazy with regret (5)
11. Ancient being, cut in rock! (5)
13. First man to consume flesh (4)
15. Suicide (killing yourself) leads to heaven (3)
16. Bank check (3)

Down:
1. Pizza pocket contains shock (3)
2. Plantation’s remote and minute (4)
3. A Latin, and previously Greek, letter (3)
4. Selection of edamame, yam, and tropical fruit (5)
7. One made new (3)
8. For instance, Godhead is source of life (3)
9. Seasoned soldier’s beginning a lieutenant club (5)
10. Career’s ever-changing (4)
12. Gang turned back incomplete explosive (3)
14. Spinning round ball partner is key (3)

Handwritten:
Age
Crest
Expression of uncertainty
First Holy Roman Emperor
First woman, according to some
Hearing organ
Imperial walker
In the manner of
Nipple
Object of particular dislike: ____ noire
Ottoman soup kitchen
Romanian currency
To growl
____ and cranny

Answer 1

Partial solution

Solving the crossword and

adding in the extra words

yields

and then we can see that

the extra letters, read spiral-fashion, say LETTER TO NUMBER

which suggests

converting the letters around the outside to numbers (A1Z26, I guess) and doing ... something ... with them. The resulting numbers don't jump out at me as signifying anything useful, though; I wondered about taking their individual digits as indicating positions along each row/column, but that doesn't seem like it works (especially as one of the numbers is 20, and the extra 1 in the bottom-right corner is hard to account for). I also wondered -- but it's no good for similar reasons -- about erasing everything in the grid and then treating it as a nonogram. Perhaps there's some other standard puzzle-type that can use these numerical clues?

Anyway, here those are in case they are useful:

Anyway, I have to go to bed now. Perhaps someone else will finish the job.

Answer 2

Continuing where Gareth left off, we notice that

The unchecked letters in the grid (when read in a spiral) spell ZERO-BASED.
ffao noticed that if each letter is converted to a number, all four completed rows and columns (with their outside letters) sum to 0 mod 26.

So, can we "complete the square" by putting letters in the final blocks, making them sum to 0 mod 26? At first, this problem seems pretty difficult. But it turns out that there are exactly two black squares in each row or column that has any.

A single letter determines the other square in its row and column, which determine the other square in their rows and columns... These form a loop, too, so that first placed letter determines the whole grid!

"Start right" means to pick R as the letter in the top-left black cell. (Alternatively, you can just bruteforce it, as I did.)

When that letter is placed, we can read around the loop to get a final answer of RECONSTRUCTION.