u/Bbrhuft

Published AI Math solutions vs independently verified AI Math solutions (Aug 2025 – Aug 2026)

These include AI discovered, AI co-developed and AI assisted math proofs of conjectures, hypotheses etc.

Data is from VibeMathed, a catalogue of open mathematical problems solved or advanced with AI (n=509, CC BY 4.0, snapshot 6 Aug 2026). Chart was generated using Claude, which accessed VibeMathed's API.

The three lines apply progressively stricter standards of proof:

- All tracked entries (506) — every recorded claim, regardless of status: unreviewed announcements, partial results, candidates awaiting review.

- Resolved + site-confirmed or better (129) — the problem is fully settled, by either via independent reviewer via hand or Lean, or the site reproduced the proof.

- Resolved + expert- or Lean-verified (89) — the problem is full settled settled, checked by an independent expert by hand (11) or via Lean (78): a Lean proof is a formal statement confirming a solution is correct.

Lean is an interactive theorem prover and programming language used to write and check formal mathematical proofs. It allows mathematicians to translate human written proofs into computer code so that a software can verify every logical step with absolute certainty.

The shaded area is the gap between all tracked entries and confirmed proofs:

380 entries are recorded but not yet independently checked. Although there's a delay between a new AI solution announcement and its verification, verified solutions appears quite linear, this may indicate AI solutions are outpacing the verification process. That said, authors generally include a Lean proof themselves, though this chart is limited to confirmation by independent peer review.

Vertical dashed lines mark OpenAI (blue) and Anthropic (orange) model releases. I added those lines as I wanted to see if there's an up-tic in solutions following model releases. There isn't a clean correlation likely because there's a several week delay between finding solution and publishing it. Also, the chart is likely showing AI's growing adoption by mathematicans and not just increasing model capability.

There were only 3 retractions in the dataset (not included on chart).

Data from: https://vibemathed.com/

Summary

I challenged an AI to survey the vast breath of mathematics and devise a useful unifying mathematical object, then used another AI as an adversarial referee; after three rounds of criticism, we abandoned the original “new number” claim and developed the Valuative Branch Scalar (VBS), a computational scalar abstraction designed to represent and track complex algebraic quantities near singularites, where traditional floating-point values, Taylor series, or uncorrelated root sets fail.

By combining dynamic algebra (the* D5 principle), Henselian branch decomposition, local rational Puiseux expansions, and logarithmic differentials ($dz/z$), VBS automates singular algebraic sensitivity while controlling combinatorial expression swell.

Definition

A based VBS over K is an algebraic element z∈Ω together with a selected extension ṽ of a current centered valuation v to K(z), and with provenance identifying joint occurrences of algebraic generators. Its unbased form is the finite Galois/monodromy orbit of z. Equality is equality in Ω for based values and isomorphism of the corresponding finite K-algebras with distinguished elements for presentations.

Key Features

* Branch-Aware Scalar State: Tracks local valuations, idempotents, ramification indices, and monodromy exchange across parameters.

* Regular-to-Ramified Transitions: Automatically refines chart valuations when leading coefficients vanish on residue discriminants.

* Anti-Swell Dynamic Evaluation: Delays algebraic splitting field construction until an exact zero test or branch predicate requires it.

* Exact Degree Conservation: Enforces ∑ e𝑖f𝑖 = [L:K] assertions across all factor splits and composita in characteristic zero.

Prototype benchmarks (Python) found that VBS (Lazy) preserved near-machine precision through catastrophic cancellation and, on a synthetic nested-radical test, was up to 18,000 times faster while using 51,000 times less memory* than eager branch expansion.

*Benchmark run in Pydroid on my Android tablet

The result is a mathematically specified, falsifiable proposal, with a paper, algorithms and benchmarks (on Github, see below), for making singular algebraic calculations substantially faster, more reliable and tractable.

The most natural physics applications for VBS are problems in which an observable is defined implicitly by an eigenvalue or dispersion equation and becomes multivalued or singular as physical parameters change.

Critical review and feedback is welcome.

Github: Valuative Branch Scalar (VBS)

ELI5: An ordinary calculator stores an answer as one number, but some physics problems have answers that split into several connected paths, like a road splitting at a complicated junction. VBS keeps a compact map of those paths, how they meet, which one you are following and how quickly they change, without calculating every possible route in advance. This could make calculations near critical points, known as singularites, much faster and less likely to give the wrong answer.