bibkey: baumelertselentiswolf2025graphical
authors: Ämin Baumeler; Eleftherios-Ermis Tselentis; Stefan Wolf
year: 2025
title: Graphical Tests of Causality
doi: 10.1016/j.jlamp.2026.101133
url: https://arxiv.org/abs/2511.22552v1
claim: "After Definition 13, page 24: We know that Q̃_n^src = Q_n^src for n ≤ 4. In general, this identity is unsolved."
strata_touched:
  - D5/S0/FiniteGeometry/Polytopes/BaumelerTselentisWolfSourceDigraphPolytope
license: citation-only
triage: anchor

Graphical Tests of Causality

Verified locator

DOI: 10.1016/j.jlamp.2026.101133

Source: https://arxiv.org/abs/2511.22552v1

Journal of Logical and Algebraic Methods in Programming 151 (2026), 101133. The locators below refer to the arXiv v1 pagination.

Source statements

Section 2, page 6: “For a digraph D with V(D) = [n], we use α(D) ∈ {0, 1}^{n(n−1)} for the adjacency vector with α(D)_{i,j} = 1 ⇔ i j ∈ A(D).”

Theorem 3, page 17: “The set of digraphs Γ(P_n^causal) is the set of all digraphs on n vertices with at least one source vertex:” Γ(P_n^causal) = {D | V(D) = [n], ∃i ∈ [n] : deg_D^in(i) = 0}.

Definition 8, page 17: “The polytope of source digraphs is Q_n^src := conv(Γ(P_n^causal)).”

Definition 9, page 17: “For any function f : [n] → [n] without fixed-points, i.e, with ∀i : f(i) ≠ i, the kefalopoda κ_f is” V(κ_f) = [n], A(κ_f) = {f(i) i | i ∈ [n]}.

Definition 13, page 24: “The polytope Q̃_n^src ⊆ ℝ^{n(n−1)} is the set of vectors that satisfy the following inequalities:” (58)–(60). The three families are (−1_{i,j}, 0), (1_{i,j}, 1) for n ≥ 3, and (α(κ_f), n − 1) for every fixed-point-free f; an inequality (w, c) means w · x ≤ c.

After Definition 13, page 24: “We know that Q̃_n^src = Q_n^src for n ≤ 4. In general, this identity is unsolved.”

Mathematical scope

The Lean statement ranges over every natural n ≥ 2. Vertices are Fin n; arcs are ordered pairs (s, r) with s ≠ r; an arc set is a finite set of such pairs. The adjacency function is one on the arc set and zero elsewhere. A source is a vertex receiving no arc. The real convex hull of these vectors is compared with precisely the three inequality families. The coordinate upper bound is assumed only for n ≥ 3, as in (59).

The proof uses incoming-column maxima and a convex mixture of fixed-source unit boxes. It does not formalize facet dimension or the lifted correlation polytope of Definition 14. Literature-search completeness and the independent full journal-text comparison are ASSUMED-UNVERIFIED.