Existence and structure of boundary critical 2-relation parameters

Establish, for every integer n≥1, the existence and uniqueness of the asserted backwards critical 2-relation parameters on the upper and lower boundary arcs of the Klein combination locus, together with the five dynamical and canonical-map properties listed in Conjecture 2.??.

Background

The conjecture proposes a sequence of boundary parameters a_n and their complex conjugates a'_n. Each is expected to generate a parabolic critical cycle, admit pinched fundamental domains, define a discrete correspondence, and map under the appropriate branch of Ψ to the modular torsion point (σρ)n(i)(\sigma\rho)^n(i). These claims are not established in general; the case n=1 is treated in the paper and n=2 is cited as known from other work.

References

For each $n \ge 1$ there exists a (unique) backwards critical $2$-relation parameter $a_n\in \partial\cap \mathcal U$ with the following properties, which also, with obvious changes, apply to the complex conjugate of $a_n$, the parameter $a'_n\in \partial\cap\mathcal L$.

— Tessellating the discreteness locus for the modular mating family of correspondences  (2608.17243 - Bullett et al., 18 Aug 2026) in Conjecture 2.??, Section 3.1 “Critical relations on $\partial$”

What is the structure of $\partial$ between each $a_n$ and $a_{n+1}$, and what is the structure of the corresponding part of $\partial(\Psi())$?

— Tessellating the discreteness locus for the modular mating family of correspondences  (2608.17243 - Bullett et al., 18 Aug 2026) in Section 3.3 “A possible scenario for $\Psi(^{cut})\subset H$”, second key open question

The computer plot suggests the conjecture that the set of all green and purple points accumulates on $\partial$, and indeed everywhere outside $\partial$; it also provides numerical evidence supporting our conjecture that the Klein combination locus $$ is a topological punctured disc, with inward pointing cusps at $a=-1$ and $a=7$, and possibly at other critical relation parameters on $\partial$.

— Tessellating the discreteness locus for the modular mating family of correspondences  (2608.17243 - Bullett et al., 18 Aug 2026) in Section 2.??, subsection “Critical relations in $\widehat{C}\setminus$”