Video Summary - Can $500 of AIs Solve an Open Maths Conjecture?

Scientific concepts / discoveries / phenomena mentioned

Reports indicate that large language models (LLMs) can prove open mathematical conjectures / open theorems at an accelerated rate following releases such as:
- Anthropic’s Fable
- OpenAI’s GPT-5.6
The video contrasts:
- impressive problem-solving
- with hallucinations—plausible-sounding but incorrect outputs.

The central math topic is random matrix theory, specifically:
- the limiting distribution of eigenvalues
- in the large (N) limit
The key phenomenon under investigation:
- whether the limiting distribution shows “two bumps” (i.e., bimodality)
- meaning two local maxima in the limiting eigenvalue density/distribution
The goal is a proof that the limiting distribution has exactly two bumps.
- Existing work is said to establish at least two bumps, but not a fully general proof of exactly two.

Earlier attempt (Codex):
- used high-precision numerical evaluation to support empirical behavior (e.g., at least two bumps)
- but this does not automatically generalize into a rigorous proof of exactly two
Another reported component involves a “residual projector” style decomposition/identity with terms resembling (I_3), and an equation structure of the form:
- “(A = B + C + A - B - C)”
- leading to a residual projector argument (described as part of understanding the earlier proof)

The workflow uses coding agents to:
- explore strategies
- generate documents (such as PDFs)
- collaborate through a file/folder-based messaging system
  - using markdown files stored in a Git-like folder structure

A repository/folder is created with structured notes and directory paths representing explored strategies.
Multiple coding agents are employed and coordinated via shared folders and file-based messaging.

Each model produces a short PDF (≈ three pages) containing:
- intuitive diagrams
- explanations of methods explored for proving bimodality in the large (N) limit

After agents review each other’s work, they produce a research plan aimed at replacing the earlier machine-assisted component with something more:
- human-readable
- analytically interesting
Three separate folders are created:
- Strategy A (MiniMax assigned)
- Strategy B (GPT-5.6-SolMax assigned)
- Strategy C (Codex assigned; “folder C” referenced)
Each agent works independently within its assigned folder.

Agents then review each other’s strategy folders, adding suggestions as markdown files.
The cycle is repeated multiple times over about a week to converge on a more rigorous analytic argument.

Strategy A (MiniMax):
- attempted to reduce the problem to a class of special functions
- claimed this could support conjectures and possibly full information
- ultimately did not yield the desired proof
Strategy B (simplified / toy model):
- removed eigenvalue interactions (treated eigenvalues as independent)
- observed that one hump persists
- argued the second bump depends on interaction effects (notably near the spectral edge)
- still did not establish the full claim of “exactly two bumps”
Strategy C (GPT-5.6-SolMax):
- tried various approaches
- ended with a counterexample showing why attempted approaches fail

Final conclusion:

The combined work of roughly ~500 AI models did not successfully complete the proof of the target statement (exactly two bumps / bimodality in the limiting distribution).
However, the process generated many new paths/ideas.