*This article is a readable write-up of episode 22 of the weekly podcast hosted by 服部陽良, founder of coiai Inc.
Below is the paper 橋本勇 submitted! Please take a look!
https://arxiv.org/abs/2603.26043
Hello, this is 服部陽良, founder of coiai Inc.
coiai is an IT company centered on making things, doing system development and product development. Every Monday on this show, I, the founder, share the reality of running a startup.
This time we're changing things up with a guest episode. 橋本勇 (Hashimoto Yu), a coiai team member who also does mathematics research, recently submitted a paper, so I sat down to hear all about it in depth. But first, let's start with this week's usual AI news.
This Week's AI News: OpenAI Unveils "GPT-5.6" — Sol / Terra / Luna
The thing that caught my attention most this week was OpenAI's release of "GPT-5.6."
GPT-5.6 comes in three models:
- Sol … the strongest flagship model
- Terra … the balanced option
- Luna … fast and low-cost
Sun (Sol), Earth (Terra), Moon (Luna) — that's a great naming scheme.
One of Sol's strong points is that it's said to be very strong on security. Another is that conversation has become remarkably natural. When I actually talked to it by voice, it was genuinely impressive. On the show, I had it improvise a reading of a poem from the Hyakunin Isshu, and after reciting "Ama no hara furisake mireba Kasuga naru Mikasa no yama ni ideshi tsuki kamo," it added its own commentary:
"This poem overlays the vastness of the sky with a gaze fixed on the moon, leaving behind a quiet lingering resonance."
The texture of its speech felt about as natural as a Japanese person who had lived abroad for a long time.
I haven't had a chance to really put it through its paces for coding yet, so I can't speak to that, but on the voice side it feels very strong.
Guest Segment: 橋本勇's "Unusual Background"
Now for the main segment. Our guest is 橋本勇. He was involved with the show back in its early days, so it's been a while since he's been on.
橋本 has done all sorts of things over the years — professionally, a few years in the IT industry, and on the research side, mainly linguistics and mathematics. It's a pretty rare combination. Think of him as someone with a genuinely unusual background as you listen.
What Is a "Disjoint Covering System"?
The topic of 橋本's paper is the Disjoint Covering System. We started with the obvious question: "What would you even call this in Japanese?"
A Tiny, Tiny Niche Field with No Settled Translation
As it turns out, there's essentially no settled Japanese term for this. Almost no one researches it in Japan, and there's almost no Japanese-language literature on it. The latter half, "Covering System," is translated on Wikipedia and elsewhere as "hifuku-kei" (covering system), but it's not widely known in Japan. If you had to translate the whole thing, it would come out to something like "non-intersecting covering system."
First, "Covering System"
Let's walk through 橋本's explanation step by step.
The foundation here is the arithmetic progression. For example, the set of even numbers — "0, 2, 4, 6, 8 …" — keeps going at a constant interval forever. Now, gather up a handful of these arithmetic progressions.
When that collection covers every single integer (meaning any integer at all falls into at least one of them), it's called a Covering System. Apparently it's a fairly fundamental concept that shows up across many areas of mathematics.
Adding "Disjoint"
Disjoint means "having no overlap." The clearest example is the set of even numbers and the set of odd numbers.
- Evens and odds share no elements at all (they don't overlap)
- But together, the two sets cover every single integer exactly
In other words, "the set of evens plus the set of odds" divides up the integers with no overlap and no gaps — a Disjoint Covering System. That's the simplest possible example; there are many others, and apparently once you try to consider all of them, there's a huge amount that's still unknown — that's the kind of field this is.
Inside the Paper: Proving "Classifiability" in General
So what exactly did 橋本's paper advance within this topic?
Within Disjoint Covering Systems, there's a special class (framework) that satisfies a certain condition. What 橋本 did was classify that class.
"Classification" here means being able to say "everything that satisfies this condition is exactly this set, and nothing more." In other words, showing that there are only finitely many such objects (i.e., they can all be listed out).
Until now, the classification had only been settled for small cases; it wasn't known whether the number stayed finite once things scaled up. This time, 橋本 proved it in general. Rather than knocking down individual small cases one by one, he raised the level of abstraction by one notch and proved it as a general theorem — that's the result of the paper.
It's currently in the review stage. But this field is so niche that "maybe three people in the world truly understand it fully," so it's a world where the reviewer isn't necessarily even a specialist in exactly the same area.
What's Next: Digging Up Old Research and Connecting It to the Present
What 橋本 is aiming at next is the more general Covering System. Much of the existing literature on the Disjoint side is old, whereas the general Covering System is the hot topic right now, with new papers still coming out.
This time around, there was an element of "focusing on a problem I could solve relatively easily." Next, he wants to do something a bit more challenging. One approach he finds interesting is digging up old, forgotten research and connecting it to modern results.
People in the past reached their results under constraints we can barely imagine today — no internet, for one. And sometimes those results hold up just fine against modern ones, yet they're buried and forgotten. Pull them back out and connect them to the present, and you sometimes see something genuinely interesting.
I thought this applies exactly the same way in business. On the show, we talked about Echigoya (the predecessor of Mitsui). Back when bolts of cloth were only sold in full, uncut lengths ("hagire"), Echigoya introduced the idea of cutting and selling exactly enough fabric for one garment, and even something like a credit / deferred-payment system. Even from ages ago, there were people doing genuinely remarkable things.
I used to think of scholarship as something that only ever builds up in one direction, but really, "whatever we're thinking about now, people in the past have mostly already thought about it." It just reappears in a different form; the essence has often already been grasped by people long before us. Never losing respect for the people who came before us matters a great deal — that was the takeaway.
Math × AI: Humans Provide the Ideas, AI Handles Simplification and Implementation
Just as engineers use AI every day, mathematics research is also being fundamentally reshaped by AI, says 橋本.
What does he actually use?
Apparently a lot of people in the math community use ChatGPT. He doesn't see many people using Claude yet, but after recently trying it together on the show, he's come around to thinking "this is genuinely useful."
How do you get an AI to read mathematical notation?
Mathematics is full of unfamiliar symbols. So how does he get the AI to read them? Either photographing handwritten equations and running them through OCR, or just typing them in directly.
As for Claude's strength in handling huge amounts of context, in 橋本's own workflow that particular strength doesn't necessarily come into play. Rather than feeding it a pile of papers all at once, his style is to closely read one to a few papers and reconstruct the argument from there. It's less "find the one needle in a huge haystack" and more "dig deeper into one specific spot." Some researchers work in a very data-driven way, feeding in large volumes, but that's not his style.
The ideas are his own; AI is a partner
The ideas in the paper are entirely his own. On top of that, he does get some partial collaboration from AI — getting pointers like "this part of the argument can be simplified this way," and then fixing it accordingly.
Once the idea is solid, everything after that goes incredibly fast. Put another way, if you're willing to put in the time, you could ultimately do it all yourself.
This felt exactly like the feeling in programming. Coming up with what to build is something we have to do ourselves, but the implementation itself is fast with AI. Once you can see the whole picture and know "this is how it can be done," you can knock down the problems that come up one by one — math and software turned out to be continuous with each other here.
For Anyone Starting Out in Math: Begin by "Solving a Problem"
Finally, I asked for advice for anyone who wants to get started in mathematics.
橋本's recommendation: start by "solving a problem" rather than starting with studying.
Traditional mathematics is often described as "cumulative — you can't move forward without understanding each step." That's true to some extent, but now that AI exists, it's become much easier to work backward from the problem you actually want to solve and gather what you need along the way.
Start with something you want to build or a problem you want to solve, then gather the information you need for it — that's an engineering mindset, a hacker mindset. Mathematics is drifting in that direction too.
"Find a problem you want to solve, and just start" — that was the message. Hearing this made me want to go try something myself.
Closing
This week's guest was 橋本勇. I really hope he'll come back again (next time, we agreed it should be "an episode digging into 服部's own depths").
coiai Inc. works on web development / XR development / core business system development / on-premises AI adoption support.
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