Kingy AI
Subscribe, 2.1M subscribersWork with Kingy

Breakthrough Tracker record

Pierce–Birkhoff now has a hand-checkable R30 example and a claimed formal R4 counterexample

Lai, Lim and Ren’s paper gives a continuous piecewise-quadratic function on R30 that cannot be expressed by finite polynomial min/max combinations. Their repository now also claims a four-variable construction with degree-232 partition tests, degree-40 pieces and a Lean endpoint excluding every finite min/max representation.

← Back to the filtered Breakthrough Tracker

Stable ID
math-pierce-birkhoff-counterexample-2026
Revision
math-pierce-birkhoff-counterexample-2026.v2
Field
Mathematics · Real algebraic geometry, semialgebraic splines and formal verification
Evidence
Tier 1 · Peer reviewed: No
Record state
Provisional · Provisional hand-checkable counterexample
Last checked

AI role

The authors used a custom multi-agent harness spanning GPT-5.6, GPT-6, Claude Opus 5 and Claude Fable 5.1 for construction and review. They report that ChatGPT helped reduce the example to dimension 30.

Record details

Problem or result
Pierce–Birkhoff representation of continuous piecewise-polynomial functions
Authors
Zehua Lai, Lek-Heng Lim and Junyu Ren
Institutions
Chinese Academy of Sciences; University of Chicago
Result date
Paper submitted September 9; four-variable artifact added October 4, 2026

Why it matters

The paper’s short R30 construction refutes the 1956 conjecture, while the new R4 artifact would greatly reduce the dimension and supplies machine-checkable evidence for its encoded endpoint.

Limits

The paper is unrefereed and still presents the hand-checkable R30 construction. The repository reports a clean build and kernel replay for R4, but this maintenance pass did not reproduce the large build; human expert review of the four-dimensional argument remains pending.

Sources

  1. Primary: Lai, Lim and Ren, Pierce–Birkhoff counterexample
  2. Artifacts: Lean formalizations of Pierce–Birkhoff counterexamples
  3. Formal artifact: Four-variable Pierce–Birkhoff formalization checkpoint

Correction and revision history

  1. 2026-09-10 — Added after primary-source, scope, status, AI-role and limitation review.
  2. 2026-10-05 — Added the repository’s four-variable formal counterexample claim while preserving the R30 paper claim and provisional status.

Machine-readable: JSON v1 · CSV v1 · Schema v1

This individual record remains noindex until a story-specific featured image passes Kingy’s rendered-pixel visual review. The source-linked tracker hub remains the public index.