IRIS face-count conjecture for simple three-polytopes
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The IRIS face-bound question proposes a lower bound on hexagonal faces of simple three-dimensional polytopes from their other face counts. The focused counterexample target has ten faces: four triangles, three pentagons and three heptagons, but no hexagons, below the proposed bound 1/5.
Why it matters and what counts as progress
Why it matters: It tests a specific combinatorial inequality in a finite, exactly describable polytope class.
What counts as progress: An exact simple-polytope counterexample to the proposed face inequality, or a corrected inequality with a proved domain of validity.
How the problem developed
- 2025 · Conjecture [1]
Davila, De Loera and collaborators The IRIS workshop paper proposed Conjecture 6.1, a face-count inequality for simple 3-polytopes with at least three faces of size seven or more.
- 11 Jun 2026 · Counterexample [2]
Demonstrandum research team A public write-up reported five exact counterexamples with ten faces and none with fewer; the examples refute the inequality as printed.
AI contribution
AI-found polytopes refute the printed IRIS face bound
Five simple 3-polytopes with 10 faces violate the printed IRIS Conjecture 6.1. One has four triangular, three pentagonal and three heptagonal faces, but no hexagonal faces, below the conjectured lower bound of 1/5.
Sources
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