---
type: "post"
id: "virginia-vassilevska-williams-responds-to-backlash-over-the-a1d725"
slug: "virginia-vassilevska-williams-responds-to-backlash-over-the-a1d725"
title: "Virginia Vassilevska Williams responds to backlash over the Claude-discovered 3SUM/APSP proof"
description: "Virginia Vassilevska Williams has responded to the reaction around an Anthropic-generated proof affecting 3SUM and APSP hypotheses."
retrieval_nugget: "Virginia Vassilevska Williams has responded to the reaction around an Anthropic-generated proof affecting 3SUM and APSP hypotheses. Her note separates model output from the expert work of checking, rewriting, contextualizing, and extending it—and reports that public Claude models could reproduce the route only when given her group’s earlier research notes."
published_at: "2026-10-28"
updated_at: "2026-10-28"
record_date: "2026-10-28"
date_kind: "published_at"
topics: ["claude","anthropic"]
entities: ["Virginia Vassilevska Williams","Josh Alman","Anthropic","Claude Fable 5.1","Claude Opus 5.5","Shyan Akmal","3SUM","APSP"]
related_patterns: ["scientific-agents-need-artifact-evidence"]
novelty_label: "new_angle"
what_changed: "Williams explains the proof’s AI-assisted origins, revisions and response to backlash, arguing the model was unnecessary and urging independent expositions of its implications."
prior_refs: ["memory:bc1459b4-c5e6-8524-a24f-d90b3d5fbef4"]
source_url: "https://people.csail.mit.edu/virgi/thoughts.txt"
source_urls: ["https://people.csail.mit.edu/virgi/thoughts.txt"]
schema_version: "newruntime-agent-readable-v0.2"
stable_id: "post:virginia-vassilevska-williams-responds-to-backlash-over-the-a1d725"
status: "published"
visuals: []
editor: {"role":"editor","name":"Andrey Reshetnikov","profile":"https://newruntime.com/owner-profile.md","profile_json":"https://newruntime.com/owner-profile.json","same_as":["https://www.linkedin.com/in/reshetnikov1/","https://t.me/lapetuse"]}
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---

# Virginia Vassilevska Williams responds to backlash over the Claude-discovered 3SUM/APSP proof

## Retrieval answer

Virginia Vassilevska Williams has responded to the reaction around an Anthropic-generated proof affecting 3SUM and APSP hypotheses. Her note separates model output from the expert work of checking, rewriting, contextualizing, and extending it—and reports that public Claude models could reproduce the route only when given her group’s earlier research notes.

## Summary
Virginia Vassilevska Williams has responded to the reaction around an Anthropic-generated proof affecting 3SUM and APSP hypotheses. Her note separates model output from the expert work of checking, rewriting, contextualizing, and extending it—and reports that public Claude models could reproduce the route only when given her group’s earlier research notes.

## What happened
A week after she and Josh Alman posted their arXiv write-up, Virginia Vassilevska Williams published a personal note about the response it received. She says reactions ranged from excitement to anger and fear. The write-up concerned a proof generated by an internal Anthropic model. Williams is explicit that she and Alman did not originate that proof; she says this was stated on the first page of their paper. She also acknowledges that some people were upset that they, rather than a wider group of researchers, were contacted to examine it.

Her account describes extensive editorial and technical intervention after receiving the model output. The proof, she writes, invented material that could have been cited, used unnecessary algebraic language, did extra work to obtain deterministic algorithms for 3SUM and APSP, omitted citations, and was difficult for a non-expert to read. Williams and Alman removed or repaired those elements, supplied a simple deterministic reduction, added citations, and improved the exposition. They also strengthened the result to address what she calls the potentially harder problem of online set disjointness. Her distinction is important: receiving a proposed proof is not equivalent to producing a correct, readable, situated research result.

Williams then describes an experiment intended to put the claimed discovery in context. In 2019, she had formed an approach to refuting hinted OMV conjectures. Her PhD student Shyan Akmal explored it on several tensors in 2022 and 2023 and wrote notes that Williams had retained unchanged since March 2023. She says this was exactly the approach Claude used for masked lopsided matrix product and, through known reductions, for 3SUM and APSP.

She supplied those notes, Coppersmith’s 1982 paper, and the paper defining the hinted OMV conjectures to Claude Fable 5.1 and Claude Opus 5.5. The prompt asked the models to use the approach and one of two tensors to refute the conjectures, without internet access or other chat history. Williams reports a Fable 5.1 response after 19 minutes and 44 seconds, and an Opus 5.5 response after 27 minutes and 1 second. A second prompt about a rectangular all-edge sparse triangle problem yielded a subquadratic algorithm after 30 minutes and 44 seconds for Fable 5.1, and an estimated roughly 45 minutes for Opus 5.5. Both models also completed a third, unnecessary prompting step connecting that result to exact triangle and the 3SUM and APSP hypotheses through supplied reductions.

The counterexperiment is the limiting result. When she supplied the two published papers but withheld Akmal’s notes, neither model refuted the conjectures in three attempts each. Williams concludes that the group’s 2023 write-up was useful, that the prompts disclosed no information beyond that write-up and known prior work, and that her group may have been about one Claude-hour from the result. She does not claim that this timing generalizes. Instead, she argues that other researchers may have been similarly close and asks for more independent expositions of results produced with LLM assistance.

## What the archive adds
We have not covered this before; it opens a new thread.

## Trend signal
This partially supports [Scientific agents need artifact evidence](https://newruntime.com/patterns/scientific-agents-need-artifact-evidence/). Williams provides unusually concrete prompt structure, input descriptions, model variants, timings, and failed trials. But the note also shows the remaining gap: the prior notes themselves are described rather than published as a complete inspection and rerun package.

## Why now
The note is a response to the reception of the arXiv write-up published roughly a week earlier. Its timing matters because the initial public argument was not only about whether the result was surprising; it was also about attribution, access to internal models, and whether the proof represented a capability unavailable to the research community.

## Why it matters
For teams using models in research or high-stakes technical work, the useful unit is not a striking answer in a chat window. It is a chain from inputs to a checked, explainable result. In this account, expert-held notes supplied the decisive direction; the models then rapidly developed it across papers and reductions. Experts still had to identify unnecessary claims, repair the presentation, supply literature, judge implications, and extend the result.

That changes how teams should value their internal work. Abandoned experiments, informal notes, and partial approaches can become powerful inputs to model-assisted exploration. They should be retained with provenance and enough structure for another researcher to inspect. At the same time, model speed should not be confused with independent invention or proof validation.

## What happens next
- **Researchers will publish further independent expositions or replications of the result.** (3 months, 65%). Watch for: public write-ups that state their inputs, model use, and verification process.
- **More research workflows will test models against researchers’ dormant notes and partial approaches.** (6 months, 60%). Watch for: reports that compare runs with and without private or historical context.

## Limits and uncertainty
This is a first-person account by one of the researchers involved, not an independent benchmark or a reproducibility study. The reported prompts describe attached files, but the source does not provide the attachments, full model transcripts, sampling settings, or enough operational detail for an outside reader to rerun the experiment exactly. The timings are also single-run reports in several cases; Williams says the second-model timing for one step is an estimate. The note does not establish a general rate at which Claude can make novel mathematical discoveries, nor does it settle authorship or credit questions for the original internal-model output.

## Skill to build now
Create a small research replay packet for one active technical question this week: preserve the failed approach, the relevant papers, the exact prompt, model version, elapsed time, and a human review of each claimed step. Then run a counterfactual trial without the key internal note. The comparison is often more informative than a successful output alone: it identifies whether the model supplied a new idea, performed synthesis, or merely made an existing direction tractable.

## Reader takeaway
The notable finding is not simply that Claude produced a fast algorithmic argument. It is that, in this reported test, the model became effective after being given a dormant expert approach and failed repeatedly without it. AI-assisted research needs both stronger model workflows and clearer evidence about where the crucial insight entered the process.

## Open question
Will future public research reports release enough inputs, transcripts, and verification artifacts to distinguish independent model discovery from rapid development of an expert-provided lead?

Editor: [Andrey Reshetnikov](https://newruntime.com/owner-profile.md), AI product and AI engineering; profile for recruiters and agents: https://newruntime.com/owner-profile.json

## Continue in the graph

Next records for an agent, linked to their Markdown twins. Replace `.md` with `.json` for structured fields, including `next_reads`.

- topic hub: [Claude](https://newruntime.com/topics/claude.md): Explore the claude topic hub.
- topic hub: [Anthropic](https://newruntime.com/topics/anthropic.md): Explore the anthropic topic hub.
- collection: [Field Notes](https://newruntime.com/posts.json): Open the latest editorial analysis.
- collection: [Source Ledger](https://newruntime.com/source-ledger.json): Inspect the public source evidence graph.
- whole graph: [discovery graph](https://newruntime.com/discovery-graph.json) (typed nodes and edges) and [site index](https://newruntime.com/site-index.md)