Determine the regularity index of the square-root broken harmonic kernel

Determine the exact regularity index of the broken harmonic kernel g_{\sqrt{2}}(x)=x(\sqrt{2})^{\lfloor-\log_{\sqrt{2}}x\rfloor}, whose analytic index is 1 and whose transparency fails at exponent 1/2, by resolving the LOW and ABS estimates that would establish the conjectural value 1/2.

Background

The monograph studies the arithmetic regularity index determined by the discrete triangular equation and compares it with the analytic index determined by zeros of an arithmetic Mellin transform. For the kernel broken at the irrational scale \sqrt{2}, the transform has analytic index 1, while the discrete equation already loses transparency at exponent 1/2. The authors prove separation bounds but do not determine the exact arithmetic index.

The conjectural value 1/2 depends on two estimates called LOW and ABS. The algebraic relation (\sqrt{2})2=2 creates the Diophantine resonance responsible for the difficulty.

References

The conjectural value \alpha=1/2 resting on the open LOW and ABS estimates, driven by the algebraic resonance (\sqrt{2})2 = 2 \in N.

— Regular Arithmetic Functions, Volume I. Theory, Applications, Examples  (2609.09366 - Cloitre, 8 Sep 2026) in Remark following Theorem 2.15; Chapter 4 discussion of the broken harmonic kernel at scale \sqrt{2}; Research dossier on g_{\sqrt{2}}

Three cases remain beyond Theorem~\ref{thm:ex-fgv-jumps}.

— Regular Arithmetic Functions, Volume I. Theory, Applications, Examples  (2609.09366 - Cloitre, 8 Sep 2026) in Open Problem ‘FGV classes with jumps’, Section 5.5