Full explainer:
The permanent and the determinant add up the same products, one entry from each row and column; only the signs differ. Determinants are easy, while the permanent is ♯P-complete (Valiant, 1979). How large must an n × n determinant of affine-linear forms, or a limit of them (the border complexity bdc), be to equal the m × m permanent? For two decades the best lower bound was m²/2 (Mignon–Ressayre, 2004). A preprint released by OpenAI on September 24, 2026 (not yet peer reviewed) proves bdc(perm_m) ≥ m³/(5,529,600 e) for every m ≥ 1408, and says its constants are not optimized. Keeping its framework and the exponent 3, our follow-on reaches m³/(32,000 e) for every m ≥ 13,000: smaller gadgets in its selector (an exclusive-choice gadget found by Codex, our referee model, and one merged gadget per index found by our rung improver, a Claude agent), so the selector has 6k + 2q rows instead of 11k − 2q; retuned parameters; and Codex’s stronger length bound (q + 1)². The cubic bound now beats m²/2 for every m ≥ 43,200, where the paper’s constant needs m ≥ 7,515,506.
What is what:
• OpenAI’s preprint theorem (1/(5,529,600 e)), formalized by OpenAI in Lean, not peer reviewed; we use its propositions as stated.
• Classical and cited: Valiant; Grenet’s 2^m − 1; Mignon–Ressayre’s m²/2 and its border version by Landsberg–Manivel–Ressayre.
• Kernel-checked in our proof checker Satori, from named axioms: cubic_merged (m³/(32,000 e) for every m ≥ 13,000), from axioms that state the paper’s Thm 2.3, Prop 6.1 + Lemma 7.1 and Prop 7.2, and our hand lemmas for the length bound and the 6k + 2q selector.
• By hand (in the family README, not formalized): that the new selectors compute the paper’s coefficient from their gadgets’ tables, and the length bound (q + 1)². A second referee model (Gemini, through agy) also found these steps valid.
• Exact: the exclusive gadget’s table is kernel-checked; the merged gadget’s 20 boundary states were checked exactly in ℚ(z) with Tungsten’s number fields. The whole selectors passed a randomized identity test modulo two primes for (k, q) up to (3, 1).
• The crossing with m²/2 is computed exactly for every m from 10,000 to 60,000 and follows from cubic_merged above that.
• Pictures: the 3 × 3 matrix is an arbitrary example; the selectors are the actual matrices at (k, q) = (3, 1).
Code, proofs and an interactive page:
The preprint, in OpenAI’s openai/math repository:
Tungsten, the language every computation and frame is written in:
Satori, our proof checker:
Animation in Tungsten · proofs in Satori · narration ElevenLabs.
Independent channel, not affiliated with @OpenAI.
#Shorts #math #openai #complexity
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