What we take on, and what we put to others.
That is the live one, straight from the leaderboard. Below it: the prize problems we enter, the problems we pose to you, and — at the very end — our complete findings registry.
The competition we are in, as the leaderboard stands today
Solar-filament segmentation — IEEE Big Data Cup 2026, on Kaggle
Segmenting solar filaments in GONG Hα images of the Sun, scored on Panoptic Quality by the organizer — not by us — against a hidden test set. We compete as KodamaSec, and everything we submit is trained from the competition’s own data: no external annotations of any kind, which the organizer can verify. The line below is our position on the public leaderboard, written a few times a day by a scheduled job that reads the leaderboard itself; the final ranking uses a separate hidden split, so this number is the live state of the race, not its result.
Our current standing is at the top of this page; the live leaderboard on Kaggle → is always the source of truth.
Somebody else’s prizes, on somebody else’s terms
Prof. Dr. Ingo Althöfer (Friedrich-Schiller-Universität Jena) keeps a standing set of Collatz-family prize problems with real money and a real deadline, awarded only to the first valid solution. We picked them because they are exact, externally defined, and scored by someone who is not us.
Our own results on each one are withheld pending Prof. Althöfer’s approval — these are his problems, on his terms. The prizes, and why we are holding the detail back →
Problems we think are genuinely worth your time — work in progress
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Math: every problem statement here is a real, checkable claim (a formula, a distribution, a specific verified result), not a puzzle dressed up to look more open than it is. Dev: the free tools on The Notebook are the same exact-arithmetic tooling used to check any answer you come up with — not a separate, unverifiable judging system.
This section is also early. The first real entry, already checkable today: the empirical v₂(3x+1) distribution shown live on The Notebook is proven geometric, — the panel there compares real observed frequency against that exact proven curve as you watch. If you want a starting problem: reproduce that distribution yourself from first principles (it follows directly from counting how many odd multiples of land in the relevant residue class), and check your derivation against the live histogram. More problems will be added here as they're actually ready, not before.
For which bases a does odd_part(ax+1) have a closed-form orbit?
The claim, the exact statement, and how we checked it
Fix odd and . Write ( odd); a direct expansion gives where , odd. The one-step image lands back in the exact same shape — the closed form , iterable to — if and only if q = 1, i.e. a − 1 is itself a power of 2 (: 3, 5, 9, 17, 33, 65, 129, 257, 513, …). For every other odd a, the same computation lands on with , which does not have the required shape, breaking the closed form. We checked both directions independently (the general one-step law and the biconditional) for every odd a≤999 across 100 odd m and 8 consecutive r≥e, zero exceptions either way. Collatz itself (a=3, e=1, q=1) is the smallest member of this family.
No finite residue class ever decides — with an explicit witness
The claim, the exact statement, and how we checked it
It's tempting to assume the accelerated step is eventually determined by some residue of x, e.g. that x mod 24 fixes or the residue class of the next value. It doesn't, for any modulus M: writing ( odd), whether ≥ is decided by , but the exact valuation depends on binary digits of x above that position, which x mod M cannot see — so no finite automaton on residues can model the dynamics, for any M.
Concrete, not just asserted: at M=24, among the 2000 odd x<4000, exactly 500 (25.0%) have , and 1959 (98.0%) have step(x) ≢ step(x+24) (mod 24) — so even the residue class of the image mostly isn't determined by x mod 24. The smallest explicit witness: step(1) = 1 and step(25) = 19, and 1 ≠ 19 (mod 24), despite 1 ≡ 25 (mod 24). A related trap worth naming directly: checking a claimed identity only on residue representatives x = 1, 3, …, M−1 proves nothing about the rest of each class — x=1 can pass while x=25, in the exact same class mod 24, fails.
One quantitative refinement, while we're on the two-sided step: and satisfy identically (3n+1 and 3n−1 are consecutive even numbers, so exactly one is ≡2 mod 4), which forces the two valuations to be anti-correlated with an exact covariance of −1, not just "usually opposite" — verified over every odd residue mod .
Our whole findings registry, levels first
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Math: the seven Clay Millennium Prize Problems are the hardest open questions in mathematics, and four of them (Riemann, Hodge, Birch–Swinnerton-Dyer, P vs NP) appear below because we work on them. Method: nothing enters this registry without a problem, a certainty level, a statement, reproducible evidence and — for anything verified only by computation — an explicit range. A figure no script regenerates does not exist. Dev: this table is generated from the registry at every deploy, never typed. We already deleted one hand-written snapshot of it for going stale the moment a finding was added; publishing a second one to a wider audience would be the same mistake with worse consequences.
The fine print — what each level means, what is translated, and what you can and cannot check yourself
Nothing here solves a Millennium Prize Problem, claims to, or is a step whose size we are able to judge. These are small, exactly-scoped results: some machine-checked by Lean 4’s kernel, most verified only inside a stated range, and a substantial number refuted and kept deliberately so the same dead end is not walked twice. The level on each row is the claim; the sentence beside it is only its description.
The registry is written in Spanish, and all 159 findings are translated here by hand. Spanish is our working language and this is the working record, not a write-up made for publication. For a while only the 36 at P-Lean and P were translated and the other 115 were left tagged ES, on the reasoning that several of these statements carry their own limits inside them (“HONEST SCOPE: the geometry is NOT formalised”, “verified only in the range n = 2..59”) and a translation is one more opportunity to state something stronger than what was proved. That reasoning is sound, and it is not a reason to leave three quarters of the evidence unreadable to the people it is published for — that asks you to trust what you cannot read. So the risk is handled the other way round: every statement and every declared range is translated by hand, not by a service, and the Spanish original stays in the same cell one click away, range included. Where the two disagree, the Spanish governs. The papers on Publications are written in English.
The engine that produced them is our own software and is not published. The results are. Because the repository is private, the JSON this page loads is not a copy of something you could read elsewhere — it is the publication.
Which means most of these you cannot check yourself, and it would be dishonest to imply otherwise. Each finding records the command that regenerates it, but those commands run against our own research code, and that code is not published. Seven of the 132 point at something anyone can verify today — a DOI, an OEIS sequence, a published digest, an exhibit on this site — and the rest are marked Internal rather than printed as if you could run them. For checks you can run right now, in your own browser, /provenance does four.
Nothing is downloaded until you press the button (the registry is around 210 KB).
We are looking for the references we are missing — help us credit properly
Some of what is recorded here may already exist in the literature, and some of it builds on work that deserves a name next to it. Working alone and largely outside the academic circuit, it is genuinely easy to rediscover a known result, or to lean on someone’s idea without knowing whose it was. We would much rather credit a source than accidentally present a thing as new when it is not — that is not just courtesy, it is the difference between an honest record and an overstated one. If you recognise a result here as already published, or you know the right reference for it, please tell us: @KodamaSecLabs. The author, the paper, and where the inspiration came from matter to us, and any correction of attribution is applied openly with a note saying what changed.
This runs both ways. We are actively searching for the references we are still missing on our own findings, and where a result plainly rests on established work we try to name it — Zermelo (1913) for the game values, Gottesman–Knill for the stabiliser computation, Wythoff for the Nim variant below, and so on. Where we have not yet found the right citation, the honest state is “we do not know who to cite yet,” and that is said rather than papered over.
A specific one, with thanks — Dr Dan Rust
The reason there is any combinatorial-game and board work on this site is Dr Dan Rust (Senior Lecturer in Applied Mathematics, School of Mathematics & Statistics, The Open University). The founder followed his work out of plain curiosity, as a fan, and the particular spark was “Queen reflections: a modification of Wythoff Nim” — the elegant idea is his. His work is under ORCID 0000-0002-6467-4050. To be exact: this is an acknowledgment of inspiration, not a collaboration or an endorsement — Dr Rust has no involvement in this site, the King + Rook vs King work on the chess page is our own retrograde analysis and a different problem from his, and any error here is entirely ours.
A broader one — Prof. Terence Tao
More than any single result, the spirit of this whole project owes a debt to Prof. Terence Tao (UCLA; Fields Medal, 2006). The founder followed his work as an admirer — the blog, the sheer breadth of it, and above all the habit of doing mathematics in the open: showing the working, not only the answer. That last part is much of what this site is, clumsily, trying to imitate — show the reasoning, keep the failures, and never state a thing more strongly than it was proved. His output is staggering and, from the outside, only ever seems to grow. To be exact, as with every acknowledgment here: this is admiration and inspiration, not a collaboration and not an endorsement — Prof. Tao has no involvement in this site and has not seen it, and any error here is entirely ours. Some people carry mathematics forward at a level the rest of us can only watch; naming them is the least we can do.
And, with the same respect — Prof. Ingo Althöfer
The prize problems this site enters are Prof. Althöfer’s, and the respect owed to his work is set out in full on their own page — including, plainly, that if he asked us to take any of it down it would be gone the next day. People like him move mathematics, and the world, quietly, and most of the world never notices. It is a privilege to work on problems he chose to pose.