Sharp robust spectral gaps for finite chiral chains

What is the smallest nonzero spectral gap of a finite open chiral chain under uncertainty in its bonds? Strong bonds have positive lower bounds βᵢ, boundary parameter s≥0, and weak-to-strong ratios qᵢ lie in fixed intervals [ℓᵢ,ρᵢ]; the target minimizes λmin(DβQ(q)ᵀQ(q)Dβ).

Why it matters and what counts as progress

Why it matters: A finite endpoint characterization replaces a continuous uncertainty optimization and can remain sharp for noncommuting matrix channels.

What counts as progress: Reduce the continuous minimum to ordered one-switch endpoint configurations and extend the sharp lower bound to matrix-valued channels with the same singular-value constraints.

How the problem developed

8 Oct 2026 · Sharp gap [1][2][3]
JD Jones, with GPT-6 Pro, GPT-6 Astra, and Codex assistance Proved that the worst scalar gap for a finite open chiral chain occurs at an ordered one-switch endpoint configuration and extended the sharp bound to finite-dimensional complex matrix channels.

AI contribution

GPT-assisted proof finds exact robust gaps for finite chiral chains

For a finite open chain with positive strong-bond lower bounds βᵢ, boundary parameter s≥0 and weak-to-strong ratios qᵢ∈[ℓᵢ,ρᵢ], the minimum scalar squared gap minq λmin(DβQ(q)ᵀQ(q)Dβ) is positive and equals the minimum over the ordered one-switch endpoint vectors q⁽ᵏ⁾, with lower endpoints before k and upper endpoints after k. The same value sharply bounds the squared nonzero gap for arbitrary finite-dimensional complex matrix channels satisfying the stated singular-value constraints.

AI involvementAI-led

ImportanceNotable

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

Human understandingFew