Distinct d-degree set relation in DSC is conjectural

Determine whether, for the deterministic simplicial complex (DSC) model, the set K^{(d)}(n) of distinct upper degrees of d-simplices at generation n equals {0} ∪ K^{(0)}(n − d) for all integers n ≥ d + 1 and d ≥ 1, where K^{(0)}(·) denotes the set of distinct vertex degrees of the corresponding generation.

Background

In the DSC model, the authors paper upper degrees of simplices and collect the distinct values into ordered lists K{(d)}(n). Empirically, they observe patterns linking lists across dimensions and generations.

They propose a concise relation connecting the distinct d-degree values at generation n to the distinct 0-degree (vertex-degree) values at generation n − d, augmented by 0, but they have not proved its validity for all n and d.

References

Another useful general relation K{(d)}(n) = 0 \cup K{(0)}(n-d) is conjecturally valid for all $n \geq d + 1$ and $d \geq 1$.

— Deterministic simplicial complexes (2507.07402 - Dorogovtsev et al., 10 Jul 2025) in Section “Upper degrees” (Unconstrained growth), around equation (K-stat)