Bound bifurcation-line slopes

Determine bounds on the slopes of the straight lines in the shifted Alladi–Erdős map’s bifurcation diagram.

Background

The paper proves that attractor elements depend linearly on the shift parameter along families determined by the prime/composite transition structure of cycles. It does not determine general bounds for the slopes of these line-like branches in the bifurcation diagram.

References

The work opens several questions. Can attractors corresponding to different values of $A$ be grouped into classes so that attractors in a given class are ``similar"? Is there a regularity in the sequence of primes and composites in these attractors? Is it possible to identify the complete basin of a given attractor analytically? For a fixed $A$, do basin sizes stabilize in the limit $N\to\infty$? Which arithmetic properties distinguish shifts with unusually large residual basin mass or high normalized entropy? What are the bounds on the slopes of the lines in the bifurcation diagram?

— Attractors and basins generated by repeated sums of prime factors of natural numbers  (2609.29232 - Shekatkar et al., 24 Sep 2026) in Section Discussion and conclusions