The week was rough. In two or three days everything happened, mostly with clients. I won’t go into detail, but the blows landed.
The first, oddly the lighter one, was a client who decided not to renew their contract. We parted on very good terms, with lovely words about us and the chance that they might get back in touch in six months, if some of the technology changes. Still, it hit me hard, partly because these days I’m looking for liquidity. Someone I work with, for the first time, sent me a message that might have looked harsh from the outside. It was a lashing, like the ones an icy wind gives you, and it helped me enormously.
The next day another client, almost out of nowhere, gave me the worst news of the week. Devastating. But oddly enough I handled it better precisely because of that lashing the day before.
The topic isn’t the struggle
I won’t dwell on the difficulties. In this complicated period (you know how long it has been going on), and in a crazy week like the last one, the best part is the growth and the results I’m getting. I’m happy. The things I wrote in recent weeks work, and it’s in weeks like this that you can see it. I won’t tell you the rest of the good news, about work and beyond. I’ll tell you about a discovery in math that I barely knew, the invariant, and the strange road it came from.
How I got here
I had started testing Limite 1B Violetto, an LLM focused on mathematics from Paradigma, a lab in Rome. On competition math it did great, one problem after another. But I couldn’t find a practical use for my work. Talking with Claude, the concept of the invariant came up. I tried it on Limite with the classics, the chocolate bar, the grids, the coins, the chessboard, and it worked. So I started studying it with Claude, then testing it on Limite with those examples.
What it is
Among the math concepts I love most are the ones tied to structure, pattern recognition, problem solving. Not mere calculation. Equating math with calculation is often a mistake, and it distracts from its power. That’s why, along with philosophy, I love it.
An invariant is what doesn’t change while everything else does. To be precise, what doesn’t change with respect to certain moves: if the moves change, it can change too. Put like that it sounds trivial, but it isn’t.
The chessboard and that two
Take a chessboard, eight by eight, with light and dark squares alternating: next to a light square there is always a dark one, in every direction. Remove two opposite corners. They have the same color, say both dark, so 32 light squares and 30 dark squares remain. You’re asked whether you can cover the 62 remaining squares with dominoes, each covering two neighboring squares, without overlapping and without going off the board. You could spend an afternoon trying.
Or you ask what a domino does, always. Two neighboring squares have different colors, because the colors alternate, so a domino can never cover two squares of the same color. It covers one light and one dark, wherever I put it, horizontally or vertically. Ten dominoes cover ten light and ten dark squares, thirty-one cover thirty-one and thirty-one. That’s the invariant: the dominoes change place, but the light squares covered and the dark squares covered stay equal.
Now the count. To cover everything you’d need 31 dominoes, so 31 light and 31 dark squares, but the board has 32 and 30. You’d be left with two light squares, and no domino can take them together, because two light squares are never neighbors. That two doesn’t depend on how you place the dominoes, it’s already written in the board, and it’s enough to say no. You haven’t tried anything and you already know it can’t be done.
If you remove two corners on the same side instead, they have different colors, the count goes back to 31 and 31, and the covering can be done. That two is exactly what breaks everything.
Double-entry bookkeeping
A closer example for work. In a company money moves between accounts, departments, line items, but the sum of debits and credits stays the same. When it doesn’t add up, before you even look you already know there’s an error. The invariant doesn’t tell you where it is, it tells you it exists.
Rereading my clients
At some point I tried it on my own work cases, with Claude, and it was out of this world. I started looking at my clients through this lens, beginning with the new ones. I trained on past work cases, which I understood better only afterwards, and then on a personal one. Thinking it through, I understood how it works and which signals lead to the right questions.
I used it with people close to me too. In that moment it gave me one missing piece about something I didn’t understand.
Where Limite stops
I went back to Limite with the same work cases. If the problem was well structured, it understood it and answered. Otherwise it struggled. Claude told me that makes sense, because it’s a small model, and up to a point that’s fine. I haven’t verified that explanation myself.
Without planning it, I did something Popperian: I had a hypothesis, that Limite could be useful for my work, and I tested it until it gave way. I’m not saying the model is wrong, it was built for something else. I’m saying that for my use and my problems it doesn’t hold. They were two intense days and I consider them extremely useful. I understood Limite, I understood my own math better, I understood the invariant and a way of reasoning that unblocks me on certain problems.
Peace of mind, not a solution
That missing piece about the people close to me didn’t give me the solution, it gave me peace of mind. The problem is still there, but now I know something I didn’t know before, and that changes how I stand in it.
I carry four questions with me. What is changing, and what stays the same? Which moves can I make, and which can’t I? Is there a quantity or a property those moves don’t touch? If this thing really changed, what signal would I have seen earlier? The second matters more than it seems, because an invariant exists only with respect to certain moves: on the chessboard the move is placing a whole domino, and if I could break it in half the two would no longer weigh anything. With a client it means understanding what I can really move and what I can’t. The last one is the one I take to work: if the invariant breaks, the break is a signal, and sometimes it could have been seen earlier. I use them like this, one at a time, on the cases that come my way.