Explain apparent nondifferentiability at the single-insertion degeneration

Determine why the solution to the Dyson–Schwinger equations with two insertion places appears to become nondifferentiable when one insertion place degenerates into the other by setting the insertion-power parameter s to −1, and develop a combinatorial understanding of this behavior.

Background

The conclusion reports numerical computations for Dyson–Schwinger equations with two insertion places. When the insertion places degenerate into one by setting the power parameter s to −1, the resulting function appears not to be differentiable. The paper does not explain this phenomenon and explicitly notes the absence of a combinatorial understanding, making the source and mathematical status of the apparent nondifferentiability unresolved.

References

Interestingly, when the two insertion places degenerate into one by setting the power of one of the insertions (the parameter $s$) to $-1$, then the function does not appear to be differentiable based on the result of the numerical computation. It is not clear why this should be, and we certainly have no combinatorial understanding in this direction at present.

— The algebraic structure of Dyson--Schwinger equations with multiple insertion places  (2501.12350 - Olson-Harris et al., 21 Jan 2025) in Section Conclusion