A mathematically natural SIRSN

Definitions and mathematical statement

In the usual planar spatial-route setting, a random route $\mathcal R(z,z')$ joins every pair of points of $\mathbb R^2$, with joint consistency/compatibility. For distinct $z_1,\ldots,z_k$, its finite-point span is the embedded network \[ \operatorname{span}(z_1,\ldots,z_k)=\bigcup_{1\leq i<j\leq k}\mathcal R(z_i,z_j). \] Similarity covariance means that for translations, rotations, and dilations $S(z)=aRz+b$, $a>0$, the joint law of transformed routes equals the law of routes between the transformed endpoints. A scale-invariant random spatial network (SIRSN) also has route-compatibility and quantitative finiteness/integrability axioms; the supplied item does not reproduce those axioms.

Construct a SIRSN whose law is mathematically natural. The source offers an explicit formula for the law of $\operatorname{span}(z_1,\ldots,z_k)$ as an example of such naturality, rather than the sole stated criterion. Such a formula should work for arbitrary finite point configurations and specify the embedded-network distribution, not merely a mean length or one marginal statistic.

The supplied summary identifies an Aldous--Kendall improper-Poisson-line construction as a natural candidate and says explicit span distributions remain unavailable. That report does not turn the optional example into a mandatory source requirement. The meaning of mathematically natural and the representation expected of an explicit law are not fixed, so the task remains a construction/tractability program with the full SIRSN conditions to be checked.

Short English statement

Construct a mathematically natural scale-invariant random spatial network, ideally with explicit finite-point span laws. The exact SIRSN axioms and criterion of naturality are not reproduced, and the explicit-law example is optional in the original wording.

Sources

• David Aldous, Open problems, page identified as sirsn.html, linked paper Open Problem 27, maintained index https://www.stat.berkeley.edu/~aldous/Research/OP/index.html; accessed 2026-07-29. The supplied extraction includes HTML and linked primary PDF text but no separate PDF identifier.
• Aldous--Kendall improper Poisson line construction, as the candidate-model attribution in the supplied research summary; no complete publication citation is given.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $9700027$, source code catalogue entry, A mathematically natural SIRSN. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.