A few years ago, a graduate student and I got stuck.
We were chasing a conjecture we were both convinced was true; the numerical evidence was overwhelming, the kind that doesn’t leave much room for doubt. We’d handled the low-dimensional case cleanly and knocked out a good many of the sub-cases one dimension up. And then we hit a wall: a fact about polynomials we needed and simply didn’t have. Not a wall we could climb with more effort, one that needed a tool that didn’t exist, at least not in any form we could find. So we did what you do. We wrote up what we had, and the general case went into the drawer, that mental drawer every researcher keeps, the one that starts filling with your first unfinished problem and never empties, full of things that are true, believed, and unfinished. Mine had been accumulating for years. This week I learned what one entry in it was quietly worth.
About a week ago, a paper appeared on the arXiv claiming a formula about exactly the kind of polynomials we’d been stuck on.
Here’s the small habit that mattered: I checked it against the drawer. This is Feynman’s old trick, keep a dozen of your favorite unsolved problems alive in the back of your mind, and every time you learn a new technique, run it against the whole list. Most of the time nothing happens. This time the formula looked like it might be the missing fact. But confirming that by hand would have been daunting, exactly the kind of computation that had stopped us years earlier.
So I didn’t do it by hand. I handed AI, the tool, the new formula, my old computations, my files, and the relevant literature, and asked whether the case we couldn’t finish could now be finished.
Two afternoons later, it appears, and I want to stress appears, because a great deal still needs to be checked, that the case is done. Not only that: the same approach suggests a cluster of other cases might now fall, and the conjecture we’d shelved two years ago is suddenly, plausibly, within reach.
I’ve written before about watching an execution moat drain, a problem that used to take weeks collapsing into a single session, and the uneasy feeling that came with it. This is the same drainage. The same collapse of execution time. But this time it didn’t feel like loss. It felt like someone had handed me the key to a drawer I’d given up on. Same shift in the world; opposite valence. And the gap between the two experiences taught me more than either did alone.
The difference was what I brought to the encounter.
In the story where it felt like loss, the tool executed the idea faster, at machine speed. In this one, the tool executed nothing on its own, it needed the stuck problem, the two years of computation, the stored files, and the judgment to recognize that this formula might fit that wall. I supplied all of that. The tool supplied the afternoons I no longer had to spend. Which points at something I only half-understood before: the tool’s value to you is almost exactly proportional to what you bring to it. Bring it a blank page and it gives you something generic and replaceable. Bring it a stocked drawer and a specific question, and it gives you back years.
I’ll name the uncomfortable part, because anyone who supervises will notice it: the case that cracked in two afternoons is the kind of case that might have been a satisfying chunk of a student’s thesis. The tool didn’t only save me time, it quietly removed a training-sized piece of work from the pipeline. I don’t have a clean answer for that yet. It’s the shadow over this whole story, and I’d rather point at it than pretend it isn’t there.
But set that beside the problem I actually want to work through, because solving one stuck case didn’t simplify my life, it complicated it, the way abundance always does. The same afternoon that case cracked open, it suggested a dozen further directions. And separately, I’d just come back from a conference having learned about an area of harmonic analysis I didn’t know well, where I could see, clearly, not vaguely, connections to problems I do know, and techniques worth developing. So now I had two live, genuinely valuable directions in front of me and no way to tell from the outside which was worth more. You can only see the value of a research direction from the other side of it. That’s not a failure of planning; it’s the nature of the thing. If you could see the payoff in advance, it wouldn’t be research.
So the question isn’t “which is more valuable?,” that’s unanswerable up front. It’s “how do I spend my attention so I find out as cheaply as possible?” And when I looked honestly at how I’d actually been deciding, I found I’d been using rules I’d never articulated.
The first was time-to-signal. Without naming it, I’d defaulted to the direction that would tell me fastest whether it worked. The stuck conjecture was days from a yes-or-no; the new area was a longer, fuzzier bet. I front-loaded the one that would resolve quickest, not because it was more valuable, but because finishing it removes that uncertainty and frees my judgment for everything else. Resolve the fast questions first.
But there was a second rule underneath, and this is the one I hadn’t seen at all until I stopped to look. I’d told myself I was choosing by speed. What I was actually protecting was an edge no one else had. The approach we’d built on that conjecture wasn’t one other groups were running, it was ours. We had the data, the computations, the specific idea, and now a formula that appears to have turned the lock. I’d been thinking of those two shelved years as sunk cost, the thing you’re supposed to ignore. They weren’t sunk cost. They were accumulated positional advantage, and it looks identical to sunk cost from the outside, which is exactly why it’s so easy to walk away from at precisely the wrong moment.
That distinction matters more than any other rule I’ve named. Sunk cost is spent effort that shouldn’t influence you. Positional advantage is altitude, you’re near the summit not because you’re throwing good work after bad, but because you’ve already done the climbing, and nobody else is on this face of the mountain. In mathematics, where nearly all the value lives discontinuously at finished, a 90%-proved conjecture is worth almost nothing until it’s 100%, converting a near-complete result you’re uniquely positioned to close will, more often than not, beat opening a fresh ascent on something new. Not always. But the default should lean toward finishing what only you can finish. The half-finished project where you hold altitude no one else has is worth more than the newer, more fashionable one where you’re starting from the valley floor, a sorting rule that matters most precisely when a career clock makes the choice feel urgent.
Seen that way, the triage wasn’t close. The new area waits, not because it’s less valuable (I still can’t see its value from the outside), but because it resolves slower and I hold no altitude there yet, while on the conjecture I hold years of it and a key that just turned. That’s discipline, not leisure. The new area doesn’t wait because I can afford to dawdle; it waits because spending scarce judgment on the slower, un-advantaged climb while a nearly-finished, uniquely-held result sits open would be the wrong allocation.
I still don’t fully trust the result. Lots needs to be checked, I’ve written that phrase to myself a dozen times this week, and I mean it as more than caution. Verification used to be free. It came bundled with doing the computation yourself; by the time you’d ground through it by hand, you understood it well enough to trust it. Now the computation arrives pre-done and plausible-looking, and the understanding doesn’t come bundled anymore. It has to be added back, deliberately, as its own act. The faculty that now matters most, telling a real result from one that merely looks right, is exactly the one the tool doesn’t exercise for me. So I exercise it on purpose, or I lose it right when it has become the whole game.
There’s a version of this new world where you drown, where cheap exploration floods you with more plausible directions and plausible derivations than you can wade through, and the abundance that was supposed to carry you pulls you under instead. What keeps you afloat isn’t the tool. It’s the drawer, the taste to know which of its contents are worth a key, and the discipline to check what the key appears to open.
I may have gotten back a few years of work this week; if the checking holds. But only because I’d worked previously to put something in the drawer worth unlocking.