Abstract. Every cubical -polytope with vertices and edges satisfies for , proving
Jockusch’s 1993 lower bound conjecture. The resulting lower bounds for all face
numbers are attained by stacked cubical polytopes. When , the right-hand side
increases by times the imbalance between the graph’s two color classes. The proof
uses a generic linear system on the graph for which infinitesimal rigidity of the
simplicial vertex figures makes one facet determine every solution throughout the
boundary. The same argument gives the strengthened bound for cubical PL spheres.
AI disclosure. AI assistance was used in this research, with models used including GPT-6 Astra, GPT-5.6 Sol, and Kimi K3. All of this paper’s primary results have been formalised in Lean4 here.
1. Introduction
A convex -polytope is cubical if every facet is combinatorially a -cube. Starting with a
cube and successively attaching cubes along facets gives the stacked cubical polytopes,
whose face numbers satisfy linear relations in the number of vertices. Jockusch [4]
conjectured that these relations give lower bounds for every cubical polytope. Klee [6]
proved the edge bound for cubical -spheres in 2011, including the boundaries of cubical
-polytopes. We establish the bounds in every dimension and strengthen them by a term
measuring the imbalance in the graph’s bipartition.
Write for the number of -faces. When , the graph of is bipartite; its boundary sphere
has zero first homology over , so every edge cycle is a sum of square boundaries and has
even length. Let be its two color classes.
holds. The first bound is attained by cubes and stacked cubical polytopes.
The edge inequality implies all the conjectured face-number bounds by the cubical
McMullen–Perles–Walkup reduction; these are stated in Corollary 5. The proof of the
edge inequality uses the normalized system of bipartite rigidity [5], with parameters
independent of the polytope’s coordinates. Its solution space injects into the solution space
on a single facet, whose dimension is , giving the edge bound by counting unknowns and
equations. The imbalance term comes from the same count; exchanging the two colors
gives a second inequality, and their maximum retains the difference between the color-class
sizes.
The injection is a local-to-global consequence of rigidity. At each vertex, infinitesimal
rigidity of the simplicial vertex figure forces vanishing on one incident facet to propagate
to every neighbor. Connectedness of the facet adjacency graph then propagates vanishing
through the whole boundary. Convexity enters only through vertex-figure rigidity; generic
rigidity of simplicial PL spheres gives the same strengthened bound for cubical PL
spheres.
2. Local propagation
Assume . Choose vectors , , and consider the linear system
with unknowns and . Denote its solution space by .
The unequal numbers of unknowns at the two colors are deliberate. If one facet
determines a solution, the number of equations must be at least minus the dimension of
the solution space on that facet. To prove such determination, we must transfer
information between black vertices. Subtracting the two equations of a square at the white
vertex opposite eliminates the scalar unknown and gives an equation for the difference of
two black vectors. With a suitable choice of the parameters, these are the rigidity
equations of the vertex figure at .
By the Dehn–Whiteley rigidity theorem, the vertices and edges of every convex
simplicial -polytope, , form an infinitesimally rigid framework in [7, Theorems 3.1
and 8.6]. Thus any motion of its vertices has the form , with . If the motion vanishes at
the vertices of one facet, then annihilates their -dimensional space of differences. Its
rank is at most one and, being even, is zero. The whole motion consequently
vanishes.
Lemma 2.For generic , the following holds for every white vertex and every facet containing it. If a solution of (1) has at the neighbors of in , then at every neighbor
of .
Proof.Fix and . The vertex figure at is a simplicial -polytope. Its vertices are
indexed by the black neighbors of , and the neighbors in form a facet of . An edge
of corresponds to a square of with cyclic order
Different edges have different opposite vertices . Otherwise two squares would
intersect in a face containing their common opposite vertices , although no proper
face of a square contains such a pair. The squares, and hence the edges of , would
coincide.
After pinning on , an edge equation at gives . Together with the differences of
the two equations at each opposite square vertex , the equations at therefore imply
To show that this pinned system generically has zero kernel, it suffices to exhibit
one specialization of full column rank. Choose the vertex coordinates of and
specialize
These assignments are consistent because the labels are distinct and different
from . The first equations put all in ; the second become the infinitesimal rigidity
equations of . The pinned-facet consequence of the rigidity theorem forces every to
vanish.
A full-column minor of this local matrix is therefore a nonzero polynomial in the
coordinates of the . Taking the product of one such minor for each of the finitely
many pairs gives a nonempty polynomial open set on which all the assertions
hold simultaneously. The individual specializations need not agree, since it is their
nonvanishing conditions that are being intersected. □
We also choose so that the vectors indexed by the white neighbors of any black vertex
within any facet are independent. These are further nonempty polynomial open conditions,
compatible with the lemma.
3. Determination by one facet
For a facet , let be the solution space of the equations restricted to its vertices and
edges.
Proposition 3.For generic and every facet , restriction gives an injection
Proof.Suppose a solution vanishes on . Let be an adjacent facet and their common
ridge. At every white vertex of , all black neighbors in have zero vectors. Lemma 2
makes all its black neighbors in zero. Within the cube , every vertex outside the
facet is adjacent to its matched vertex in . Thus all black vectors on vanish, and
its edge equations then make all white scalars vanish. The facet adjacency graph is
connected, so this propagates through the boundary and proves injectivity.
Put . The facet has vertices of each color and edges, giving scalar unknowns. Its
edge equations are independent. Indeed, a linear relation among their rows, restricted
to the vector columns of a black vertex , reads
The vectors are independent by our choice of , so every coefficient at is zero.
Repeating this at all black vertices proves row independence. Hence
□
Proof of Theorem 1. Write . System (1) has scalar unknowns and equations.
Proposition 3 gives
Interchanging and gives the second asymmetric bound. Their maximum is
This proves the stated inequality for . For , quadrilateral incidence and Euler’s
formula give and , hence . □
Corollary 4.The strengthened edge bound in Theorem 1holds for every finite
cubical PL -sphere, , whose cells intersect in common faces.
Proof.The link of each vertex is a simplicial -sphere. By Fogelsanger’s rigidity
theorem [3, Theorem 3.14], its graph has an infinitesimally rigid generic placement
in . Use this placement for the coordinates in Lemma 2; its facet vertices are affinely
independent, so the same pinning argument applies. The common-face condition
again makes the opposite square vertices distinct. The graph is bipartite by the
homology argument above, and the facet adjacency graph is connected. The rest of
the proof is unchanged. □
4. The remaining face numbers
The edge bound controls the higher face numbers by summing the simplicial lower bounds
over the vertex figures. This is the cubical McMullen–Perles–Walkup reduction observed
by Jockusch and recalled in [1].
Corollary 5.Let be a cubical -polytope, , with vertices. Then
All these bounds are attained by stacked cubical polytopes.
Proof.For , apply the simplicial lower bound theorem [2] to the -dimensional vertex
figures and sum over vertices. Since an -face has vertices, this gives, for ,
Substitute Theorem 1 and use Pascal’s identity to obtain the first bound. For
facets the corresponding simplicial bound is
Its sum gives . The dimension-three case follows from and .
A -cube attains every bound. Attaching another cube along a facet adds vertices,
adds -faces for , and replaces one facet by others. These are exactly the increments
in the displayed lower bounds. □
For , the same substitution adds to the -face bound and to the facet bound, where
.
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