A Summer with Claude and Relative Arbitrage, by Dmitriy Traytel

Dmitriy Traytel spent a summer vacation, interrupted by a week of FLoC, on semi-automatic formalization he would not check every hour. In lawrencecpaulson.github.io's report, dated 26 Oct 2026 on Machine Logic and tagged AI, Isabelle, and formalised mathematics, he describes a finished Isabelle/HOL proof. The case study sits in the Renaissance Philanthropy-funded project Copilots for Isabelle, which develops methods and tools so AI agents can interface with Isabelle. Kevin Kappelmann’s PIDE MCP component gives agents a live view of Isabelle, close to Isabelle/jEdit.

This piece is for Isabelle users and for readers following Claude through a long proof. It covers the paper he chose, the handoff from Claude Fable 5 to Claude Opus 5, Theorem 1.1, two corrections against that paper, and the tactic counts after a cut of about a third.

A Summer with Claude and Relative Arbitrage: what actually changed

Traytel’s second cousin Misha Shkolnikov spoke at Mathematics for the Real World 2026. Bound for Odense, he stopped in Copenhagen, and Traytel had a short tour and a surprise ready. Claude autoformalized, in Isabelle through PIDE MCP, the main result of Shkolnikov’s article with Jou-Hua Lai and H. Mete Soner. The article is not yet published but in the final stages, on the relative arbitrage problem under eigenvalue lower bounds. Traytel writes that he is quite clueless about financial mathematics, probability, and stochastic portfolio theory.

From the first sentence he learned that Fernholz introduced portfolio domination, now called relative arbitrage, to capture “beating the market.” That paper scored highly on the “recent work”–“short proof” curve among Shkolnikov’s recent arXiv papers. The prompt was “Formalize https://arxiv.org/abs/2512.17702 in Isabelle using PIDE MCP,” with encouragements to continue and to check the Archive of Formal Proofs before a new concept. Claude worked nearly non-stop for three weeks on Claude Max 5x, later replaced by the Claude Team plan for scientists, and finished an Isabelle/HOL proof of over 100,000 lines.

A Summer with Claude and Relative Arbitrage: how it works

Claude Fable 5 drafted until harsh token limits ran out. Claude Opus 5 resumed from the proof state, notes, and Fable’s plans. Traytel may have started with Opus 4.8. Theorem 1.1, as Isabelle HTML with comments hidden, fixes K in finite-dimensional real^'n, assumes compact K, k < CARD('n), 1 ≤ L, and 1 ≤ k, and sets v to enn2real of xval k L K. It bounds xval by ennreal B, adds a neighborhood condition on v, states visc_subsol_env2 and visc_supersol_env2 using interior K, v, and lsc_envK, and, when K is expandable, equates v with any bounded u under the same conditions.

Imports are HOL-Analysis, HOL-Probability, and six AFP entries: Disintegration, Kolmogorov_Chentsov, Levy_Prokhorov_Metric, Lower_Semicontinuous, Martingales, and Standard_Borel_Spaces. Traytel asked Claude to explain the added definitions and to compare the statement with the paper visually. He finds the page surprisingly readable next to raw Isabelle HTML, especially the nested tooltips he suggested. He includes the theorem display for completeness. He wants deterministic tools for that presentation. Isabelle/PIDE already has the markup, so the open question is presentation, and he does not judge the mathematics.

A Summer with Claude and Relative Arbitrage: why it matters now

Traytel believes Theorem 1.1 states the right result after eyeballing the paper and the definitions. A static PDF shown to Shkolnikov strengthened that belief. Claude’s first version used five lemmas with different assumptions. Boundedness said enn2real (xval k L K y) ≤ B for a real B. That misses infinity, because enn2real sends ∞::ennreal to 0::real, a totalization as with division. Claude proved xval is never ∞, then forgot it. Embedding B as ennreal on the right fixed the lemma.

Self-review caught L > 1 on the supersolution, against the paper’s L ≥ 1. L = 1 is Ambrosio–Soner co-dimension mean curvature flow, studied before by Ambrosio and Soner, the same Soner, though Shkolnikov mainly took L to be large. Claude traced L > 1 to an element of the open interval (1, L) asserted mid-proof, then restored L ≥ 1. Traytel does not understand the repair. The short account seemed to make sense to Shkolnikov, who said it will take a week to add to the paper. For Shkolnikov it was the most interesting outcome.

A Summer with Claude and Relative Arbitrage: who is affected

Traytel stayed off hourly checks because stochastic portfolio theory was new to him, and the surprise was ready for Shkolnikov’s Copenhagen stop on the way to Odense. Shkolnikov said the L ≥ 1 repair will take a week to enter the unpublished article he shares with Jou-Hua Lai and H. Mete Soner. Ambrosio and Soner had already studied that L = 1 case, even though Shkolnikov kept L large here. Fernholz’s portfolio domination is the first sentence’s rigorous form of “beating the market.”

Where proofs say “repeat the Larsson–Ruf argument word-by-word,” the agents read more than the 17-page paper, so the de Bruijn factor may be smaller than the line count suggests. They also built infrastructure the authors treat as given, beyond analysis and probability already in Isabelle and the AFP. Traytel had them drop lemmas unused_thms flagged on the dependencies. A few unused lemmas remained, but not many. His Claude Max 5x subscription from the summer has been replaced by the Claude Team plan for scientists.

A Summer with Claude and Relative Arbitrage: what to watch

The first complete proof ran past 100,000 lines, which Traytel doubted from the start. Restructuring cut about a third. He never pushed hard for concise proofs, and small steps remain. Leading methods are rule 6736, simp 6469, intro 1621, blast 1318, and auto 917. VeriMon, his 68,000-line Isabelle development, leads with auto 6962, then simp 2502, cases 1259, blast 909, and rule 890. He calls that a difference of style. Beside the volume of text, small steps are not a defect for him.

Shkolnikov said it will take a week to fold the L ≥ 1 repair, including Ambrosio–Soner at L = 1, into the article with Lai and Soner. Traytel writes that deterministic tools should turn Isabelle/PIDE markup into this kind of presentation. Nested tooltips he suggested to Claude already read more clearly than unprocessed Isabelle HTML, and the static PDF of that page is what Shkolnikov saw. Because Isabelle checked the scripts, he did not have to read every small step.