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This digest records the selected source statements and the paper-reported experiment in the arXiv v3 PDF, the copy read for the source card. The version is arXiv:2601.22401v3, submitted 2026-02-05; the PDF is dated February 6, 2026.

Experiment report

The authors report that Aletheia, a mathematics research agent built on Gemini Deep Think, was deployed from December 2 through December 9, 2025, on the 700 problems then marked open in Bloom's database (printed p. 3). Their pipeline returned 212 potentially correct responses, narrowed these to 27 solution responses for focused review, and ultimately classified 13 responses as meaningfully correct after human evaluation (printed pp. 3–4). Table 1 (printed p. 3) divides the 13 into two autonomous resolutions (E652, E1051), two partial AI solutions (E654, E1040), four independent rediscoveries (E397, E659, E935, E1089), and five literature identifications (E333, E591, E705, E992, E1105). The experiment dates, classifications, and model/human roles are the authors' report.

Authorial limitations (printed/physical pp. 5–7). The authors say that the longest and most difficult part of their evaluation was determining whether a response addressed the intended problem and whether its argument was already in the literature. They describe the novelty classifications as upper bounds that remain subject to revision. For independent rediscoveries, they also say that checking the recorded reasoning trace cannot exclude indirect ingestion from training, a risk they call “subconscious plagiarism.” These are the paper's methodological cautions, not independent conclusions of this digest. The abstract also gives the authors' interpretation that the then-open labels in these cases reflected obscurity rather than difficulty.

Autonomous cases

E652 (problem statement, printed/physical p. 9). For points $x_1,\ldots,x_n\in\mathbb R^2$, define

$$R(x_i)=\#\{\,|x_j-x_i|:j\ne i\,\},$$

and order the points so that $R(x_1)\le\cdots\le R(x_n)$. Let $\alpha_k$ be the least value such that, for all sufficiently large $n$, some $n$-point set has $R(x_k)<\alpha_k n^{1/2}$. The question is whether $\alpha_k\to\infty$ as $k\to\infty$. The paper's solution (printed pp. 10–11) reports the stronger asymptotic lower bound $\alpha_k=\Omega(k^{1/4})$, with the incidence estimate from Pach–Sharir as the cited literature input.

E1051, Theorem 2 (statement printed/physical pp. 11–12). If $(a_n)_{n\ge1}$ is a strictly increasing sequence of positive integers with

$$\liminf_{n\to\infty}a_n^{1/2^n}>1,$$

then

$$S=\sum_{n=1}^{\infty}\frac1{a_na_{n+1}}$$

is irrational. The paper states that this is the affirmative answer to E1051. Its Remark 2.2 says that the original model output contains a minor error, taking strict inequalities in the proof of Lemma 2 (the proof as printed, p. 13, uses non-strict ones), and that the solution has been formalised in Lean 4 by Barreto.

Partial case statements

E654, Lemma 3 and Theorem 3 (printed/physical pp. 15–17). For an integer $m\ge10$, let $n=4m$, $K=\{10,11,\ldots,m+9\}$, and define

$$P=\{(0,y)\in\mathbb R^2:y\in\{3^k,-3^k:k\in K\}\},\qquad Q=\{(x,0)\in\mathbb R^2:x\in\{2^j,-2^j:j\in K\}\},$$

$S=P\cup Q$. The source proves that no four points of $S$ lie on a circle (Lemma 3). If

$$D(u)=\{\,|u-v|:v\in S,\ v\ne u\,\},$$

then Theorem 3 states that $|D(u)|<3n/4$ for every $u\in S$. Thus this construction answers the displayed no-four-concyclic question negatively. The paper counts the result as a partial case because earlier sources, as Bloom's problem page notes, pose a weaker question with the additional hypothesis that no three points are on a line; Aletheia's answer to that case was incorrect and is omitted (Remark 3.1, p. 15).

E1040, first question (statement and construction printed/physical pp. 18–19). For a closed infinite $F\subseteq\mathbb C$, let $\mu(F)$ be the infimum of the planar areas

$$\bigl|\{z\in\mathbb C:|f(z)|<1\}\bigr|$$

over monic polynomials $f(z)=\prod_j(z-z_j)$ with every $z_j\in F$. The paper answers the first question negatively by taking

$$F_1=\{0\}\cup\{1/n:n\ge1\},\qquad F_2=\{0,R\}\cup\{1/n:n\ge1\}\cup\{R+1/n:n\ge1\},$$

with $R>4$. Both sets are countable compact sets with transfinite diameter zero. The paper gives $\mu(F_1)\ge\pi/4$ and, using $f(z)=z(z-R)$, $\mu(F_2)\le2\pi/(R^2-4)$, so $R$ can be chosen with $\mu(F_2)<\pi/4$. Its p. 19 footnote is a human annotation on the claim that every countable compact set has transfinite diameter zero: the model output did not justify or cite that claim. It points to Ransford [Ran95, Corollary 3.2.5] and the equivalence of transfinite diameter and logarithmic capacity, and notes that direct verification of the two constructed sets is easy. Its Remark 3.2 says that the second question was omitted because the model's reduction was incorrect.

Independent rediscovery

E935, second question (problem statement printed/physical p. 21; construction and conclusion pp. 22–23). If $n=\prod_p p^{k_p}$, define the powerful part

$$Q_2(n)=\prod_{k_p\ge2}p^{k_p}.$$

For every fixed integer $\ell\ge2$, the paper reports

$$\limsup_{n\to\infty} \frac{Q_2(n(n+1)\cdots(n+\ell))}{n^2}=\infty.$$

Its proof reduces to $\ell=2$: for Pell solutions $x_k+y_k\sqrt8=(3+\sqrt8)^k$, set $n_k=8y_k^2$, so $n_k+1=x_k^2$. For every prime $p\equiv5\pmod8$, the paper's Lemma 5 produces $k$ with $p^2\mid n_k+2$, and hence the normalized powerful part is at least $Q_2(n_k+2)\ge p^2$. The unbounded primes $p\equiv5\pmod8$ give the stated limsup. The other two E935 questions are not asserted here.

The paper's Addendum 4.1 (printed/physical p. 23) records that Wouter van Doorn identified an almost identical question in #367, and that the construction is the same as in van Doorn's comment there dated 2025-11-20. The authors report checking Aletheia's thinking logs and confirming that it did not access that page, and add that the comment postdated the base model's knowledge cutoff, so it was not in the training data; in light of this, they reclassified the case as an independent rediscovery. The E367 link is provenance/context only; no E367 mathematical result is credited.

E1089, Theorem 5 (statement and result printed/physical pp. 26–28). For positive integers $d,n$, let $g_d(n)$ be the least integer such that every set of $g_d(n)$ distinct points in $\mathbb R^d$ determines at least $n$ distinct nonzero distances. The source states

$$g_d(1)=2,\qquad \lim_{d\to\infty}\frac{g_d(n)}{d^{\,n-1}} =\frac1{(n-1)!}\quad(n\ge2).$$

The paper says human auditors traced the result to Bannai–Bannai, Remark 3(ii); the classification is therefore an independent rediscovery report rather than a new literature claim.

E397 and E659 are listed in Table 1 as independent rediscoveries. The paper's selected pages identify prior work for those cases; no additional theorem is reproduced in this digest.

Literature identifications

The paper reports the following existing-literature pointers (printed pp. 29–33): E333 to Erdős–Newman [EN77, Theorem 2], E591 to Schipperus and the Darby/Larson results, E705 to O'Donnell [O'D99/O'D00], E992 to Berkes–Philipp [BP94], and E1105 to Montellano-Ballesteros–Neumann-Lara [MBNL05] for cycles and the unpublished Yuan [Yua21] result for paths. These are source pointers; they do not by themselves replace review of those older primary sources.

Corrections and verification layers

Corrections to the inherited filing text. The long result list has been replaced by the exact statements above. In particular, E1040 is restricted to the first question because the paper calls the second reduction incorrect; E935 is restricted to the second question; E652's corrected incidence exponents are retained; and E1089's prior Bannai–Bannai attribution is stated. The five literature cases are recorded as citations, not new solutions.

Formalization report. The paper reports a Lean 4 formalization of the E1051 argument by Kevin Barreto. It supplies no pinned formal file, version, or local build result for this filing.

Reported verification. The taxonomy, dates, model runs, human evaluation, and claims about prior literature are all reported by the authors. They note that some technically correct responses contained minor inaccuracies or omissions, and that the first E1051 sweep had a disputed evaluation before a later ablation run was used in the paper.

Local verification. The PDF bytes and SHA-256 were checked. Rendered physical pp. 1, 3–7, 9, 11–12, 15–19, 21–28, and 29–33 were inspected for the taxonomy, statements, selected results, citations, and disclosure; the corresponding extracted text was compared while drafting this digest. No code or Lean proof was run.