Lukic’s multipoint weighted-entropy conjecture
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Does Lukic’s localized decomposition condition imply finite weighted entropy in the unrestricted multipoint setting? The counterexample target uses critical angles 0 and π/2 of multiplicity three and αₙ=c(n+1)^(-3/20)(1+(−i)ⁿ), 0<c<1/2.
Why it matters and what counts as progress
Why it matters: It tests the implication in the presence of mixed resonances; restricted cases and a corrected criterion remain separate questions.
What counts as progress: An exact sequence satisfying the localized conditions but failing the entropy conclusion, or a corrected criterion excluding the mixed resonance.
How the problem developed
- 29 Jul 2026 · Counterexample [1]
Jun Yan, with a counterexample found by GPT-5.6 Gave a two-point, multiplicity-three counterexample: the sequence has a Lukic decomposition, but the corresponding weighted entropy is −∞.
AI contribution
A Mixed-Resonance Counterexample to the Lukic Conjecture
For critical angles 0 and π/2 of multiplicity three, Yan gives αₙ=c(n+1)^(-3/20)(1+(-i)ⁿ), 0<c<1/2. It satisfies Lukic’s localized decomposition conditions but has weighted entropy −∞, refuting that direction of the unrestricted multipoint conjecture.
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