Robustness of the four-row permanent inequality

How robust is the four-row permanent inequality when the probability law on the 24 permutations departs from uniform? For balanced laws at total-variation distance r≤1/24 and nonnegative rows with uniform mean square one, the target deficit coefficient is (1−24r)/9 times the sum of row variances.

Why it matters and what counts as progress

Why it matters: Sharp robustness constants identify precisely how nonuniform permutation laws affect the four-row inequality and where it can fail.

What counts as progress: Prove the sharp variance coefficient at each radius, characterize equality and determine the exact admissible radius.

How the problem developed

4 Aug 2005 · Theorem [1]
Eric Carlen, Elliott Lieb and Michael Loss Established a permanent inequality for four nonnegative functions under the uniform law on permutations, providing a baseline for later robustness questions.
13 Sept 2021 · Theorem [2]
Laura Bristiel and Clément Caputo Proved uniform-law permanent inequalities for all exponents in a broad function-space setting.
8 Oct 2026 · Sharp inequality [3]
JD Jones For balanced laws on S₄ within total-variation distance r≤1/24 of uniform, proved the sharp stability coefficient (1−24r)/9 for normalized nonnegative rows.

AI contribution

GPT-assisted proof finds the sharp robustness radius for a four-row permanent inequality

For a balanced probability law ν on the 24 permutations of four labels at total-variation distance at most r≤1/24 from uniform, and nonnegative rows fᵢ normalized by the uniform mean of fᵢ² equal to one, 1−Eν∏ᵢfᵢ(σ(i))≥(1−24r)/9 ΣᵢVar(fᵢ). The coefficient is sharp at each radius and 1/24 is the exact admissible radius.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean + comparator + NanoDa + con-ron + exact computation)

Human understandingFew