Mythos is good but not infallible

While models like Mythos undeniably have very good mathematical ability they are still not infallible and it’s important humans still critically evaluate and verify certain findings. Mythos’s recent writeup on the Erdős unit-distance problem shortly after OpenAI’s proof is a good example of why this matters. The following note specifically concerns only Proposition 7.1 as written and expands on a short tweet about the issue I wrote last week. It could be wrong if Mythos has a hidden upper-bound argument elsewhere, if $D(T)$ is intended differently from how it is defined and used, or if a corrected version supplies the missing estimates. I am not claiming that the statement $u(P)=n^{1+o(1)}$ is necessarily false for Mythos’s constructed point sets. I am saying that Proposition 7.1 does not prove it, and its written proof is mathematically invalid. It’s very possible that it is indeed an artifact that happened during the rewrite by Opus 4.7. We will dissect sometimes maybe even too fine-grained for the well versed mathematician but I want this to be very understandable. That said feel free to skip certain sections if obvious to you.

Proposition 7.1 claims that for the constructed point sets

$$ P=P_{K_m,d_m^A}, $$

one has

$$ u(P)=n^{1+o(1)}. $$

Equivalently, it needs to prove

$$ \frac{\log(u(P)/n)}{\log n}\to 0. $$

That requires an upper bound on $\log(u(P)/n)$. But the previous sections only prove lower bounds for $u(P)$.

In Proposition 7.1 Mythos writes

$$ \log(u(P)/n)\le \log D(T)\le r_2\log(2T)+O(d). $$

That’s weird huh. This is not supported by the earlier results. Proposition 4.2 proves the opposite-direction estimate

$$ \frac{u(P)}{n}\ge \frac12D(T)e^{-C_2d}, $$

so

$$ \log(u(P)/n)\ge \log D(T)-C_2d-\log2. $$

Thus $D(T)$ is used earlier to give a lower bound for the number of unit distances not an upper bound. The construction gives an injection from the counted algebraic directions into the set of unit-distance pairs. That proves there are at least this many unit distances and it does not prove there are at most this many unit distances! Mythos never proves that all unit-distance pairs in $P$ come from the bounded-height directions counted by this specific $D(T)$.

You can skip the following since it’s a minor algebraic issue. An arbitrary unit-distance pair in $P$ is not unrelated to the setup. If

$$ \begin{aligned} p &= \left(\frac{\sigma(x)}2,\frac{\sigma(y)}2\right), \ q &= \left(\frac{\sigma(x’)}{2},\frac{\sigma(y’)}{2}\right). \end{aligned} $$

are at unit distance, then with

$$ \begin{aligned} a &= x’-x, \ b &= y’-y. \end{aligned} $$

one gets

$$ \sigma(a^2+b^2)=4. $$

Since $\sigma$ is injective,

$$ a^2+b^2=4. $$

So the displacement corresponds to some $\zeta\in U^{(1)}$. But Mythos does not prove that this $\zeta$ lies in the same bounded-height set counted by

$$ D(T),\qquad T=\log(M/4). $$

For arbitrary points of $P$, the displacement comes from differences of elements of $B_M$, so one only gets a larger height window, for instance $|L(\zeta)|_\infty\le \log(2M)$, not $T=\log(M/4)$. Even a possible upper-counting argument would need a different count, together with a genuine quantitative upper bound for that larger count. Mythos does not provide that.

Lemma 6.1 also confirms the lower bound. It gives

$$ \log D(T)\ge c_\star d\log\log d-O(d) $$

and

$$ \log(u(P)/n)\ge c_\star d\log\log d-O(d). $$

Both are lower bounds. Lemma 6.1 does not give

$$ \log(u(P)/n)\le O(d\log\log d). $$

The second inequality in Proposition 7.1,

$$ \log D(T)\le r_2\log(2T)+O(d), $$

is also not established by the previous results. Earlier Mythos proves

$$ D(T)\ge \frac{T^{r_2}}{R^{(1)}}, $$

and then uses

$$ R^{(1)}\le e^{C_1d} $$

to make this lower bound large. That does not imply an upper bound for $D(T)$. Finiteness of $D(T)$ is not enough to justify a quantitative estimate of the form

$$ D(T)\le e^{O(d)}T^{r_2}. $$

It makes the same type of mistake after “even more simply” argument. From

$$ \log(u(P)/n)\ge \Omega(d\log\log d) $$

and

$$ \log n=\Theta(d\log d), $$

one obtains the lower bound

$$ \begin{aligned} \frac{\log(u(P)/n)}{\log n} &\ge \Omega\left(\frac{\log\log d}{\log d}\right), \end{aligned} $$

not the upper bound

$$ \begin{aligned} \frac{\log(u(P)/n)}{\log n} &= O\left(\frac{\log\log d}{\log d}\right). \end{aligned} $$

The fallback estimate in Section 7 is just the trivial binomial-pair upper bound

$$ u(P)\le \binom{n}{2}. $$

That gives only

$$ \frac{\log(u(P)/n)}{\log n}\le 1+o(1), $$

which is far too weak to imply

$$ u(P)=n^{1+o(1)}. $$

Even the stronger general Spencer-Szemeredi-Trotter upper bound

$$ u(n)=O(n^{4/3}) $$

would only give

$$ u(P)\le O(n^{4/3}), $$

hence

$$ \frac{\log(u(P)/n)}{\log n}\le \frac13+o(1), $$

still not

$$ o(1). $$

So neither the trivial binomial-pair bound nor the Spencer-Szemeredi-Trotter bound proves Proposition 7.1. Now what does this mean gum give us a sum up. Basically Proposition 7.1 tries to prove an upper bound using inputs that only establish lower bounds. This does not by itself invalidate Mythos’s lower-bound theorem but it invalidates Proposition 7.1 and the claim that the constructed sets are known from this proof to satisfy

$$ u(P)=n^{1+o(1)}. $$

The reason or at least why I think Mythos or Opus 4.7 decided to keep Proposition 7.1 in is it kinda functions like a Spencer-Szemeredi-Trotter consistency check. Mythos wants to reassure the reader that its construction does not threaten the known general upper bound which is understandable since

$$ u(n)=O(n^{4/3}). $$

Mythos introduction explicitly references Spencer-Szemeredi-Trotter bound as the best known upper bound and says the produced exponent is $1+o(1)$. It blurs the lower bound exponent guaranteed by the construction with the actual exponent of the constructed point sets. Mythos does prove a guaranteed lower-bound exponent tending to $1$. It does not prove that the constructed sets have no additional unit distances beyond those counted by the construction. The global Spencer-Szemeredi-Trotter theorem already prevents any contradiction with

$$ u(n)=O(n^{4/3}). $$

What Proposition 7.1 tries, and fails, to prove is the stronger self-contained statement

$$ u(P)=n^{1+o(1)}. $$

Even though the result of Mythos is weaker than the result of OpenAI’s version I felt the need for completeness to point this out since at least I haven’t seen anyone point this exact issue out in the comments of the original Mythos proof which is understandable since it doesn’t invalidate the main finding of it and mathematicians likely already used OpenAI’s published proof as their main reading source so this is very easy to overlook. Besides that for completeness and keeping up good research principles it is important to establish critical feedback loops which will become even more important with increasing capabilities of these models.

References and acknowledgements

• Anthropic. Mythos / Opus 4.7 writeup.
• OpenAI. Unit distance proof.
• OpenAI. Model disproves discrete geometry conjecture.
• gum. Follow-up tweet on Proposition 7.1.

@online{gum2026mythos,
  author = {gum},
  title  = {Mythos is good but not infallible},
  year   = {2026},
  month  = {06},
  day    = {07},
  url    = {/post/mythos-is-good-but-not-infallible/},
}