Symbolic proof of the realizable-ray single-crossing property

Establish a symbolic proof that, for every realizable unit-direction signed-distance rate triple associated with a trivalent planar junction, the derivative of the logarithmic ratio governing the normalized field has at most one sign change, specifically a positive-to-negative crossing, thereby proving the single-interval property of the normalized super-level sets without relying on numerical certification.

Background

The normalized partition-of-unity field near a trivalent junction is analyzed through a planar blow-up model. Along each realizable ray, the authors reduce the threshold condition to the behavior of a logarithmic function whose derivative is denoted by g′g'. Establishing that g′g' has at most one sign change of type positive-to-negative would imply that each radial super-level set is a single interval anchored at the junction, and hence that the planar super-level region is connected.

The paper reports numerical verification over more than 10610^6 realizable rays, with no exceptions, but does not provide the requested symbolic argument. This proof would close the remaining analytic gap in the normalized-field comparison used to transfer the sharp-limit result from the offset cover to the normalized cover.

References

The $g'$ single-crossing on the realizable curve is certified numerically by two independent sufficient criteria --- no $-\to+$ crossing of $g'$, and $g''<0$ at every zero --- verified over $>106$ realizable rays with zero exceptions (worst observed $g''=-5.4\times10{-6}$); a one-line symbolic proof remains open.

— A limit theorem linking continuous plate models to the topology of the plate boundary network  (2609.39187 - Kim, 30 Sep 2026) in Section 6, Lemma 5 (Planar Junction single-interval / connectedness), concluding discussion of the junction-ball comparison