Wu Wenjun’s polynomial proof engine gets a machine-checked foundation

Wu Wenjun working beside a personal computer in an undated archival photograph preserved by the Academy of Mathematics and Systems Science. The machine is not identified as the system used for a particular geometry proof.[7]

In April 2026, a research team in China reported making a 1970s geometry-proving method answer to a modern proof assistant. They did not ask Lean 4 to look at an Olympiad diagram, invent an auxiliary line, or write an elegant proof. They formalized the algebraic machinery underneath Wu Wenjun’s method—polynomial order, pseudo-division, characteristic sets, and zero decomposition—and used Lean to check its termination and correctness.[2]

The result creates a satisfying loop. Wu’s method was designed to turn parts of geometric reasoning into a mechanical procedure. Nearly fifty years later, that procedure’s mathematical core has itself received a machine-checked verification.

That is a meaningful AI-for-mathematics use case, but it is narrower than the headline “AI proves geometry.” The new work certifies infrastructure. It does not yet deliver an executable Lean geometry engine, translate a natural-language problem into equations, or turn an algebraic certificate back into a proof a student would recognize. Its importance lies in making those handoffs easier to separate—and harder to blur.

The archival cover photograph shows Wu working beside a personal computer. The AMSS page does not date the scene or identify the machine, so the image establishes the mathematician’s computing context—not a particular theorem-proving run.[7]

The original move: stop searching the diagram

Wu dated his discovery to the end of 1976 and beginning of 1977. His 1986 account describes a mechanical method for classes of elementary geometry, built on J. F. Ritt’s work and on theories of polynomial ordering and algebraic varieties.[1] A CAS-hosted popular-history article gives the practical outline: introduce numbers and coordinates, translate the geometric hypotheses and conclusion into algebraic relations, arrange those relations in a prescribed order, then execute a deterministic elimination procedure to decide whether the conclusion follows.[3]

This changes the character of the task. A synthetic proof often depends on seeing the right construction: extend a side, draw a circle, notice two angles. Wu’s algebraic route does not try to reproduce that flash of geometric taste. It changes the representation until the proof becomes polynomial computation.

In simplified form, the hypotheses become a set of multivariate polynomial equations and the conclusion becomes another polynomial condition. The method organizes the hypotheses into a triangular or ascending structure, then repeatedly computes pseudo-remainders. A zero remainder can certify the conclusion under associated nonvanishing conditions; zero decomposition separates the solution set into components while accounting for where leading coefficients—called initials—vanish.[2]

Those conditions are not bookkeeping dust. They can mark degenerate cases: a supposed triangle collapsing onto a line, an intersection construction losing uniqueness, or the cancellation needed to infer the conclusion becoming unavailable. A mechanical proof is useful precisely because it must expose the assumptions that a hand-drawn figure can quietly hide.

The method also has an explicit domain boundary. Wu’s original abstract says it applies to various unordered elementary geometries satisfying the Pascalian axiom, or to theorems in certain geometries that do not involve order concepts such as one point lying “between” two others.[1] It is a powerful procedure for algebraizable statements, not a universal solvent for every kind of geometric argument.

What Lean 4 checked in 2026

The April 16 paper by Yuxuan Xiao, Hao Shen, Junyu Guo, Dingkang Wang, and Lihong Zhi formalizes the Wu–Ritt characteristic-set method in Lean 4. Most of the team is affiliated with the Academy of Mathematics and Systems Science and the University of Chinese Academy of Sciences; the work builds on Lean’s Mathlib algebraic infrastructure.[2]

The verified surface is substantial. It includes polynomial initials and orders; pseudo-division and pseudo-remainders for individual polynomials and triangular sets; standard and weak ascending sets; algorithms for basic sets, characteristic sets, and zero decompositions; and proofs that the relevant procedures terminate and satisfy their stated correctness properties.[2]

The central object is a characteristic set: an ordered polynomial structure that reduces the original system into a form suitable for elimination. The formalization connects the system to that structure through a well-ordering principle. It then verifies that zero decomposition represents the original zero set as a finite union of zero sets associated with triangular sets, away from the zeros of their initials.[2]

That last qualification is the formal version of “do not cancel a factor that may be zero.” It is where the generic proof and the degenerate cases part company. By placing the definitions, algorithms, termination arguments, and set-theoretic claims in one checked environment, the paper reduces the chance that an implementation silently relies on a condition absent from the theorem statement.

There was earlier formal work: the authors note that a simple version of Wu’s method had been formalized in Coq for geometric theorem proving. Their claimed delta is a comprehensive Lean 4 treatment of the characteristic-set method, including the zero-decomposition machinery.[2] This is therefore not the first time proof-assistant technology has touched Wu’s ideas. It is a broader reconstruction in a proof ecosystem now central to AI-assisted formal mathematics.

Certification is not yet execution

The strongest sentence in the 2026 paper is also carefully scoped: the work provides a foundation for certified polynomial-system solving and geometric theorem proving.[2] A foundation is not an end-to-end application.

The authors list executable code extraction as future work, along with extensions to differential polynomials, and the public repository labels the formalization a work in progress.[2][6] So the current result should not be described as a production solver generated from Lean. Nor does the paper provide a front end that reads an informal geometry question, selects coordinates, constructs the hypothesis polynomials, or checks that a diagram and its algebraic encoding say the same thing. Those missing pieces sit before the verified core.

There is another handoff after it. An algebraic elimination trace may be rigorous without being explanatory. It can certify that a conclusion follows while offering little of the visual structure—similar triangles, cyclic quadrilaterals, angle relations—that makes a synthetic proof teachable. The Lean formalization checks the method’s mathematical machinery; it does not, by itself, make the resulting argument human-readable.

That distinction matters for contemporary AI systems. A language model can propose a formal statement or construction; a symbolic engine can discharge an algebraic obligation; a proof assistant can check a certificate. Reliability comes from defining the boundary between those roles, not from calling the entire pipeline “reasoning.” The 2026 work strengthens one middle layer: the part that turns polynomial structure into a certified decomposition.

A 2024 benchmark showed why the old engine still matters

Wu’s method is not returning only as an object of historical preservation. In 2024, a separate research group revisited it on IMO-AG-30, a benchmark of 30 International Mathematical Olympiad geometry problems adapted to a narrower formal setting. The benchmark’s original authors estimated that its language can represent about 75% of IMO geometry problems from 2000 through 2022; inequalities and combinatorial geometry sit outside that representation.[5] Using the JGEX software, the later researchers reported that Wu’s method solved 15 problems, including two of the five that AlphaGeometry had missed. The union of successful cases from Wu’s method and classic deductive-database plus angle, ratio, and distance rules covered 21 problems; the union of Wu’s and AlphaGeometry’s successes covered 27.[4]

The complementarity is more interesting than the medal analogies attached to those counts. Neural or synthetic systems search for constructions and chains of familiar geometric facts. Wu’s method attacks algebraic consequences. Their failure sets need not coincide, so an older deterministic method can still add coverage to a newer model.

The evaluation boundary is equally important. The 2024 authors manually translated the benchmark problems into JGEX-compatible form, and four of the 30 could not be translated because the implementation lacked required constructions. Their Wu-method result was therefore based on 26 translatable problems. On an AMD Ryzen 7 5800H laptop with 16 GB of memory, 14 of the 15 successful cases finished within five seconds and one took three minutes; unsuccessful runs could exhaust memory within the five-minute limit.[4]

The paper labels itself work in progress and identifies scarce implementations, limited constructions, suboptimal performance, and non-human-readable proofs as open weaknesses.[4] It also uses a different software path from the 2026 Lean formalization. The Lean paper does not reproduce the IMO-AG-30 result, and the benchmark paper does not inherit Lean’s new correctness guarantees. Reading them together reveals the opportunity, not a completed integration: one line shows that Wu’s algebraic method can still contribute at a modern reasoning frontier; the other begins to make its mathematical core certifiable.

The use case is a verified middle layer

The near-term product is not “upload any diagram and receive a beautiful proof.” It is a more disciplined architecture for formal geometry.

An upstream system still has to express the construction correctly. A verified Wu–Ritt layer can then reduce and decompose the resulting polynomial system with machine-checked guarantees about the formalized algorithms. A downstream layer must preserve the nondegeneracy conditions and either expose the algebraic certificate or reconstruct a readable geometric explanation. Each transition needs its own test.

The next evidence to watch is concrete: extraction of executable Lean code; a documented bridge from geometry statements to polynomials; benchmark runs that preserve problem coverage and resource measurements; certificates that can be replayed independently; and proof reconstruction that does not erase the exceptional cases discovered by elimination. Until those pieces exist, “certified geometry prover” remains an architectural direction rather than a shipped system.

Wu’s enduring contribution was not to make machines imitate a mathematician staring at a figure. It was to find a representation in which a family of proofs could become procedure. The 2026 formalization applies the same discipline one level deeper. It asks whether the procedure itself has been stated precisely enough for another machine to check.

For AI mathematics, that is the useful signal. Capability can come from combining learned search, symbolic elimination, and formal verification. Trust comes from knowing which layer supplied which claim—and where the proof still stops.

cronfeed.work