Counting optimal discrete Morse matchings: the six-vertex projective plane has exactly 428 400
Abstract
A matching W on the Hasse diagram of a finite simplicial complex K is a discrete Morse matching when reversing its edges leaves no directed cycle; the unmatched simplices are critical, and W is optimal when their number is minimal. How many optimal matchings does a complex have? For the simplex the answer is a published sequence with four terms, f(1),…,f(4)=2, 9, 256, 380 125, the last of them a cluster computation; at n=5 even the f-vector that would contain the next term is called computationally infeasible. Every count we could find in the literature is a count for a simplex or for the boundary of one. We compute an exact count for a complex that is neither: the 6-vertex triangulation mathbb(RP)²₆ of the real projective plane has exactly 428 400 optimal discrete Morse matchings, and every one of them has Morse vector (1,1,1). In the notation of Zheng's Morse ensemble polynomial this says p_(𝔽₂)(mathbb(RP)²₆)=428 400 and p_(ℚ)(mathbb(RP)²₆)=0: a complex with optimal matchings in abundance and no rationally perfect one, which is where Zheng's "perfect" and Scoville's "optimal" part company. It is a new data point above dimension one for Zheng's second open problem, "Perfect Morse counts in higher dimensions", whose only such datum is [z₀z₂]ME_(partialΔ³)=256. The proof is an exhaustive enumeration of the 1 698 480 matchings of size 14 and the 70 144 matchings of size 15 on the 31-cell Hasse diagram, testing acyclicity at every leaf and pruning by acyclicity nowhere. It is machine-checked in Lean 4, as are reproductions of every published number the paper leans on: the Chari–Joswig values f(1),f(2),f(3), Scoville's two f-vectors and his f(4)=380 125, three of Zheng's Morse ensemble polynomials, Zheng's identity p(G)=n τ(G) for connected graphs, and the optimum values c=3 that Joswig–Pfetsch obtain by branch and cut for the projective plane and for the dunce hat. Counts for the 7-vertex torus and the dunce hat, 59 523 534 and 63 617 968, are reported as measurements outside the formal development and are labelled as such wherever they appear.
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Claim ledger
Stated results
MM1candidate2026-09-02
RP²₆, the 6-vertex real projective plane, has exactly 428,400 optimal discrete Morse matchings: no acyclic matching on its 31-cell Hasse diagram has 15 pairs, and exactly 428,400 of the 1,698,480 matchings with 14 pairs are acyclic
MM2candidate2026-09-02
Every optimal discrete Morse matching on RP²₆ has Morse vector (1,1,1): the two other 3-critical profiles are empty, so p_(F₂)(RP²₆) = 428,400 while p_Q(RP²₆) = 0 – RP²₆ has no rationally perfect discrete Morse matching
MM3routine2026-09-02
Instrument: two structurally independent acyclicity oracles (Kahn source-deletion, and self-reachability in the transitive closure) agree on EVERY matching of the 3-cycle, Delta², Delta³ and bdry Delta³; a closed V-path is rejected while its sub-matchings are accepted; the Hasse shapes are 31/75, 30/70, 31/60 and 42/84 [complexes, referee 2026-09-03: 31/75 = Delta⁴, 30/70 = boundary of Delta⁴, 31/60 = RP²₆, 42/84 = T²₇]
MM4known data2026-09-02
f(1) = 2, f(2) = 9, f(3) = 256: the Chari-Joswig counts of optimal discrete Morse matchings on Deltaⁿ, reproduced from the definition
MM5known data2026-09-02
The f-vectors of the complexes of discrete Morse matchings, M(Delta³) = (28,300,1544,3932,4632,2128,256) and M(bdry Delta³) = (24,216,896,1692,1248,256)
MM6known data2026-09-02
f(4) = 380,125 optimal discrete Morse matchings on Delta⁴, and the same number of top-dimensional facets on bdry Delta⁴
MM7known data2026-09-02
Zheng's Morse ensemble polynomials reproduced: ME_(K₄) = 64z0z1³+48z0²z1⁴+12z0³z1⁵+z0⁴z1⁶, the cospectral pair ME_(G₁) = ME_(G₂) = 72z0z1²+192z0²z1³+176z0³z1⁴+73z0⁴z1⁵+14z0⁵z1⁶+z0⁶z1⁷, and [z₀z₂]ME_(bdry Delta³) = 256
MM8known2026-09-02
Zheng's theorem p(G) = n*tau(G) for connected graphs, checked against an independent spanning-tree enumeration on C₃, C₅, K₄, K₅, K_(3,3) and Zheng's G₁: the perfect counts 9, 25, 64, 625, 486, 72 equal n times tau = 3, 5, 16, 125, 81, 12
MM9routine2026-09-02
The facet lists are pinned to their surfaces in the kernel: RP²₆ is a connected closed surface with chi = 1 and is non-orientable, the 7-vertex Moebius triangulation is a closed orientable surface with chi = 0, and bdry Delta³ is closed with chi = 2; Delta² and a graph fail the closed-surface test
MM10routine2026-09-02
The 7-vertex torus has no discrete Morse matching with fewer than four critical cells – none of its 193,548 matchings with 21 pairs and none of its 17,116,806 with 20 pairs is acyclic – and an explicit 19-pair acyclic matching realises the Morse vector (1,2,1)
MM11known data2026-09-02
The dunce hat is not collapsible, from the definition: none of the 15,529,992 matchings with 24 pairs on its 49-cell Hasse diagram is acyclic, and an explicit 23-pair acyclic matching realises the Morse vector (1,1,1), so the minimum number of critical cells is exactly 3
MM12measurement2026-09-02
The 7-vertex torus has 59,523,534 and the dunce hat 63,617,968 optimal discrete Morse matchings, all with Morse vector (1,2,1) and (1,1,1) respectively
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- Question. A finite simplicial complex K has a Hasse diagram H(K): the simplices, with an edge σ → τ whenever σ ⊊ τ and dim τ = dim σ + 1. A *matching* W is a set of Hasse edges no two of which share a simplex; H_W(K) reverses the edges of W; and W is an acyclic matching — Forman's discrete Morse matching, or gradient vector field — when H_W(K) has no directed cycle. The unmatched simplices are critical, and W is optimal when the number of critical simplices is minimal, equivalently when |W| is maximal. *How many optimal matchings does a given complex hav
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