For every integer $d \ge 3$ there is a constant $D < \infty$ and a constant
$\varepsilon > 0$ with the following property. For every integer $m \ge 2$
there is a finite, connected, pure $d$-dimensional simplicial complex $X_m$
with $|X_m(0)| \to \infty$ such that each vertex lies in at most $D$ faces of
dimension $d$ and such that, for every integer $i$ with $0 \le i < d$ and
every cochain $f \in C^i(X_m; \mathbb{F}_2)$,
$$\|\delta_i f\|_{X_m,\,i+1} \ge \varepsilon\,\operatorname{dist}_{X_m,\,i}\bigl(f, B^i(X_m)\bigr).$$
Neither $D$ nor $\varepsilon$ depends on $m$, $i$, or $f$.

./pdf/oppenheim-valentinerbranth-2026-mathann-cosystolic-kms.pdf

An expander graph is a graph for which the edge boundary of any set of
vertices of size at most half the total is a fixed fraction of that set;
equivalently, the second eigenvalue of the graph Laplacian is bounded below
by a constant independent of the number of vertices. Coboundary expansion
lifts the condition to higher-dimensional complexes. Let $Y$ be a finite
pure $s$-dimensional simplicial complex, let $Y(i)$ denote its set of
$i$-faces, write $C^i(Y; \mathbb{F}_2)$ for the set of functions
$Y(i) \to \mathbb{F}_2$, and put
$$w_i^Y(\sigma) = \frac{|\{\tau \in Y(s) : \sigma \subseteq \tau\}|}{\binom{s+1}{i+1}\,|Y(s)|}, \qquad
\|f\|_{Y,i} = \sum_{\sigma : f(\sigma) \ne 0} w_i^Y(\sigma).$$
The interpretation is probabilistic: a random $s$-face is sampled uniformly
and then one of its $(i+1)$-vertex subsets is sampled uniformly, and
$\|f\|_{Y,i}$ is the probability that $f$ is nonzero on the sampled
$i$-face.

./pdf/lubotzky-1994-discrete-groups-expanding-graphs.pdf

For $f \in C^i(Y; \mathbb{F}_2)$ the coboundary is
$(\delta_i f)(\tau) = \sum_{\sigma \subset \tau,\, |\sigma| = i+1} f(\sigma)$,
summed in $\mathbb{F}_2$. The cocycles are $Z^i(Y) = \ker \delta_i$; the
coboundaries are $B^i(Y) = \operatorname{im}\delta_{i-1}$, with $B^0(Y)$
taken to consist of the constant functions; the cohomology is
$H^i(Y; \mathbb{F}_2) = Z^i(Y)/B^i(Y)$. For a subspace
$A \subseteq C^i(Y; \mathbb{F}_2)$ define
$\operatorname{dist}_{Y,i}(f,A) = \min_{a \in A} \|f - a\|_{Y,i}$.

The distinction between cocycles and coboundaries drives the whole
argument. A cocycle $f \in Z^i(Y)$ has $\|\delta_i f\| = 0$, so any
inequality of the form $\|\delta_i f\| \ge \varepsilon \operatorname{dist}_{Y,i}(f, Z^i(Y))$
is satisfied trivially by $f$; the inequality against $B^i(Y)$ fails
for $f$ precisely because $f$ is not a coboundary. An inequality with
$Z^i(Y)$ on the right is called a cosystolic inequality, and with
$B^i(Y)$ on the right a coboundary expansion inequality. Passing from
one to the other requires the vanishing of $H^i(Y; \mathbb{F}_2)$ in
every degree in which expansion is required.

Prior work splits into two lines, one random-combinatorial and one
algebraic. Linial and Meshulam in 2006, and Meshulam and Wallach in
2009, studied the random $k$-dimensional complex in which a complete
$(k-1)$-skeleton is fixed and $k$-faces are inserted independently at
a given rate, and located the threshold rate at which cohomology
vanishes. Those models possess complete lower skeleta and therefore
have unbounded vertex degree.

./pdf/linial-meshulam-2006-combinatorica-homological-connectivity-2-complexes.pdf
./pdf/meshulam-wallach-2009-rsa-homological-connectivity-k-complexes.pdf

Gromov in 2010 connected cohomological filling inequalities with
topological overlap, and asked for arbitrarily large complexes with a
uniform bound on vertex incidence and with uniform filling bounds.

./pdf/gromov-2010-gafa-singularities-expanders-topology-maps-2.pdf

The algebraic line begins with the Chevalley-group and coset-geometry
constructions of O'Donnell and Pratt in 2022, of Kaufman and Oppenheim
in 2023, and of Grave de Peralta and Valentiner-Branth in 2025. All
three construct coset complexes of algebraic or Kac-Moody-Steinberg
type and obtain expansion from representation-theoretic input.

./pdf/odonnell-pratt-2022-ccc-chevalley-groups.pdf
./pdf/kaufman-oppenheim-2023-ejc-coset-geometries.pdf
./pdf/grave-peralta-valentinerbranth-2025-ejc-kms-groups.pdf

Within the algebraic line, two distinct expansion conditions have been
attacked. The first is cosystolic expansion. Kaufman, Kazhdan, and
Lubotzky in 2016 established cosystolic and topological expansion for
two-dimensional skeleta of Ramanujan complexes, and Evra and Kaufman
in 2024 obtained bounded-degree cosystolic expanders in every ambient
dimension using a local-to-global theorem. Related contributions are
the building-like complexes of Lubotzky, Meshulam, and Mozes in 2016,
the strongly symmetric complexes of Kaufman and Oppenheim in 2021,
the coefficient-independent methods of Kaufman and Mass in 2021, and
the quantitative local-to-global results of Dikstein and Dinur in 2024.

./pdf/kaufman-kazhdan-lubotzky-2016-gafa-ramanujan-complexes.pdf
./pdf/evra-kaufman-2024-jams-bounded-cosystolic-expanders.pdf
./pdf/lubotzky-meshulam-mozes-2016-ggd-building-like-complexes.pdf
./pdf/kaufman-oppenheim-2021-icalp-strong-symmetry.pdf
./pdf/kaufman-mass-2021-isaac-coefficient-independent.pdf
./pdf/dikstein-dinur-2024-approxrand-coboundary-cosystolic.pdf

The second is coboundary expansion, which is stronger than cosystolic
expansion. Lubotzky and Meshulam in 2015 constructed random Latin-square
coboundary expanders in dimension two, and Lubotzky, Luria, and
Rosenthal in 2019 constructed Steiner-system coboundary expanders in
every dimension. Both families bound the number of top-dimensional
faces through a codimension-one face only. Latin-square complexes have
a complete tripartite 1-skeleton, and Steiner-system complexes have a
complete $(d-1)$-skeleton, so the number of top-dimensional faces
through a single vertex grows with the size. Those are not bounded
degree conditions at the vertex level.

./pdf/lubotzky-meshulam-2015-advmath-random-latin-squares.pdf
./pdf/lubotzky-luria-rosenthal-2019-dcg-random-steiner-systems.pdf

Combining bounded vertex degree with coboundary expansion was known
only in ambient dimensions one and two. Chapman and Lubotzky in 2025
constructed bounded-degree two-dimensional $\mathbb{F}_2$ coboundary
expanders. In higher ambient dimensions, Kaufman, Oppenheim, and
Weinberger in 2025, and Oppenheim and Valentiner-Branth in 2025, proved
degree-one coboundary expansion for coset complexes over suitable
non-Abelian coefficients. Neither family gives the vanishing in every
degree that the theorem requires.

./pdf/chapman-lubotzky-2025-advmath-stability-homomorphisms-2.pdf
./pdf/kaufman-oppenheim-weinberger-2025-stoc-coboundary-coset-complexes.pdf
./pdf/oppenheim-valentinerbranth-2025-arxiv-nonabelian-congruence-kms.pdf

The construction belongs to the Kac-Moody-Steinberg
cosystolic programme of Oppenheim and Valentiner-Branth, whose cosystolic
expansion theorem supplies the quantitative input. The specific local
groups used in the construction already appear in the affine-type-$A$ polynomial-matrix
example of that paper, where finite quotients by irreducible polynomials
are considered. The new ingredient is the passage to quotients
modulo powers $t^m$ of $t$ and the algebraic verification that the
resulting finite complexes have vanishing cohomology in every degree up
to $d - 1$, so that the cosystolic expansion inequality becomes ordinary
coboundary expansion.

./pdf/oppenheim-valentinerbranth-2026-mathann-cosystolic-kms.pdf

Fix an integer $d \ge 3$ and put $n = 2d$ and $N = n + 1 = 2d + 1$.
Write $S = \{0, 1, \ldots, N - 1\}$. Choose once and for all a prime $q$
satisfying the requirements collected later; each is a finite lower
bound depending only on $n$. Set $k = \mathbb{F}_q$ and $R = k[t]$.
For $a \ge 1$ put
$$\Gamma_a(m) = \ker\bigl(\operatorname{SL}_a(R) \to \operatorname{SL}_a(R/(t^m))\bigr).$$
Let $U \le \operatorname{SL}_N(R)$ be the inverse image, under evaluation
at $t = 0$, of the upper unitriangular subgroup of $\operatorname{SL}_N(k)$:
the set of matrices in $\operatorname{SL}_N(k[t])$ whose reduction at
$t = 0$ is upper triangular with diagonal entries $1$.

For $0 \le i < N$ define the simple generator
$x_i = e_{i,i+1}(1)$ when $1 \le i < N$, and $x_0 = e_{N,1}(t)$,
where $e_{a,b}(f) = I + f E_{a,b}$ and rows and columns are indexed
$1, \ldots, N$. The generator $x_0$ carries a factor of $t$ because its
associated edge is the wrap-around edge of the cycle
$1 \to 2 \to \cdots \to N \to 1$, and only with that factor does the
matrix belong to $\operatorname{SL}_N(k[t])$ while reducing to a
unitriangular matrix at $t = 0$. The generators $x_i$ satisfy the
relations of an affine root system of type $\widetilde A_n$ with node
set $S$; the corresponding Coxeter graph is the cycle with edges
$i - (i+1) \bmod N$.

For $J \subseteq S$ put $U_J = \langle x_j : j \in J \rangle$ and
$U_\varnothing = \{1\}$, and put
$H_i = U_{S \setminus \{i\}} = \langle x_j : j \in S, j \ne i \rangle$.

For a group $G$ and subgroups $(K_i)_{i \in S}$ of $G$, the coset
complex $\operatorname{CC}(G; (K_i)_{i \in S})$ has as vertices the
tagged left cosets $g K_i$ with $i \in S$; a finite set of such
vertices spans a face if the corresponding cosets have nonempty common
intersection. The chambers, that is the maximal faces, are the sets
$\{g K_i : i \in S\}$ with $g \in G$, so the complex is pure of
dimension $N - 1$ and its chambers are indexed by $G / \bigcap_i K_i$.

For $m \ge 2$ put
$$G_m = U / \Gamma_N(m), \quad T_m = \operatorname{CC}\bigl(G_m; (\overline H_i)_{i \in S}\bigr), \quad
X_m = \operatorname{skel}_d T_m,$$
where the bar denotes reduction modulo $t^m$. The group $G_m$ is the
finite image of $U$ in $\operatorname{SL}_N(R/(t^m))$, and $T_m$ is the
finite coset complex that will be analyzed.

The local structure of $U$ and its finite quotients is described by
the next lemma, which is the only place where the specific cycle
arrangement of the $x_j$ enters.

Lemma. For every proper $J \subsetneq S$, the group $U_J$ is the set of
matrices $g \in \operatorname{SL}_N(R)$ with $g_{aa} = 1$ for every
$a \in S$, and with
$$g_{ab} = 0 \text{ unless } g_{ab} \in V_{ab}(J),$$
where $V_{ab}(J) = k$ when the unique simple directed path in the cycle
$1 \to 2 \to \cdots \to N \to 1$ from $a$ to $b$ lies in $J$ and avoids
edge $0$, $V_{ab}(J) = kt$ when that path lies in $J$ and uses edge $0$,
and $V_{ab}(J) = 0$ otherwise. Moreover, the $U_J$ are finite, and
$$U_J \cap U_K = U_{J \cap K}, \qquad \bigcap_{i \in A} H_i = U_{S \setminus A} \quad (\varnothing \ne A \subseteq S),$$
and for every $m \ge 2$ the reduction $R \to R/(t^m)$ is injective on
each $U_J$ and preserves all these intersections. Finally
$|H_i| = M := q^{\binom{N}{2}}$ for every $i \in S$, and $|U_{\{i\}}| = q$.

Proof. Choose an edge $j_0 \notin J$ and delete it, turning the cycle
into a linear order $(j_0 + 1, j_0 + 2, \ldots, j_0 + N - 1)$ on $S$.
In this linear order, the set of matrices $g$ with $g_{aa} = 1$ and
$g_{ab} \in V_{ab}(J)$ for $a < b$ is a product of unitriangular groups,
one for each path component of $J$; unitriangular groups over a
commutative ring are groups, so this set is a subgroup of
$\operatorname{SL}_N(R)$. Every entry has degree at most one, so no
power $t^2$ or higher appears; the subgroup is therefore determined by
its image in $\operatorname{SL}_N(k \oplus kt) = \operatorname{SL}_N(k[t]/(t^2))$,
which is finite. This gives the inclusion
$\{g : g_{aa} = 1, g_{ab} \in V_{ab}(J)\} \subseteq \operatorname{SL}_N(R)$
as a finite subgroup.

Because $q$ is prime, raising $x_j$ to powers $e_{j, j+1}(a)$ with
$a$ ranging over $k$ produces every coefficient in the simple root
subgroup of $x_j$. The commutator identity
$[e_{a,b}(f), e_{b,c}(g)] = e_{a,c}(fg)$ for distinct $a,b,c$
composes path entries: at each step an additional edge of the cycle
is appended. Unitriangular elimination in the linear order that was
obtained by deleting $j_0$ then produces every matrix in the displayed
set, proving $U_J = \{g : g_{aa} = 1, g_{ab} \in V_{ab}(J)\}$.

For the intersection identities, note that the entry pattern of $U_J$
is described entirely by $J$: a matrix lies in $U_J \cap U_K$ exactly
when each $g_{ab}$ lies in $V_{ab}(J) \cap V_{ab}(K) = V_{ab}(J \cap K)$
together with $g_{aa} = 1$, which is the entry pattern defining
$U_{J \cap K}$. For the second identity, $H_i = U_{S \setminus \{i\}}$,
so $\bigcap_{i \in A} H_i = \bigcap_{i \in A} U_{S \setminus \{i\}} = U_{S \setminus A}$
by the first identity applied to the intersected subsets.

For the reduction step, every element of every $U_J$ has entries in
$k \cup kt$ and therefore degree at most one. Because $m \ge 2$, the
reduction $R \to R/(t^m)$ is injective on the degree-one piece, so it
is injective on each $U_J$ and preserves the entry-pattern description.

For the order computation, fix $i \in S$ and take $J = S \setminus \{i\}$,
that is $H_i = U_J$. Deleting edge $i$ turns the cycle into a linear
order on the remaining $N$ indices. In that order, the number of pairs
$(a, b)$ with $V_{ab}(J) \ne 0$ is exactly the number of pairs with $a$
strictly before $b$, which is $\binom{N}{2}$. Each such pair contributes
an entry in $k \cup kt$ with $q$ choices, giving $|U_J| = q^{\binom{N}{2}}$.
The generator $x_j$ has order $q$ because $q$ is prime and the entries
$e_{j,j+1}(a)$ with $a \in k$ and $e_{N,1}(bt)$ with $b \in k$ are
distinct as elements of $U$; in particular $|U_{\{i\}}| = q$.

The proof splits into two large pieces. The first is geometric: an
infinite complex $L$ is identified with the subcomplex of an affine
twin building opposite a chamber, and shown to be acyclic in a range.
The second is homological and finite: the local-group computations
just made let the cohomology of the finite quotient complexes $T_m$
be related to the group homology of $U$ and its congruence subgroups.

For the geometric piece, set $G = \operatorname{SL}_N(k[t, t^{-1}])$,
with $T$ the group of constant diagonal matrices of determinant one.
Define parabolic subgroups
$$B_+ = \{g \in G : g(t) \in \operatorname{SL}_N(k[t]),\ g(0) \text{ upper triangular}\},$$
$$B_- = \{g \in G : g(t^{-1}) \in \operatorname{SL}_N(k[t^{-1}]),\ g(\infty) \text{ lower triangular}\}.$$
Both are Borel subgroups of $G$; $B_+ = U \rtimes T$, and
$B_+ \cap B_- = T$.

./pdf/abramenko-vanmaldeghem-2001-mathz-1-twinnings-of-buildings.pdf
./pdf/abramenko-brown-2008-gtm248-buildings-theory-applications.pdf

Let $\Delta_+$ be the simplicial complex whose chambers are the left
$B_+$-cosets in $G$; the faces of a chamber are the parabolic cosets
it contains. Tits's theorem identifies $\Delta_+$ as a building of
type $\widetilde A_n$; similarly $\Delta_-$ is a building from the
$B_-$-cosets. The pair $(\Delta_+, \Delta_-)$ is an affine twin building
of type $\widetilde A_n$, which comes with a codistance map
$$\delta^* : \{ \text{chambers of } \Delta_- \} \times \{ \text{chambers of } \Delta_+ \} \to W \cup \{\infty\},$$
where $W$ is the affine Coxeter group of type $\widetilde A_n$. The
codistance is symmetric, satisfies $\delta^*(c, c_+) = 1$ exactly when
$c$ lies in $B_+ c_-$ for fixed positive base chamber $c_+$ and negative
base chamber $c_-$, and satisfies a panel axiom: if $c'$ is a chamber
adjacent to $c$ along an $s$-panel, then $\delta^*(c', c_+)$ lies in
$\{ \delta^*(c, c_+), \delta^*(c, c_+) s \}$, and when the second option
$ws$ has smaller Coxeter length than $w = \delta^*(c, c_+)$, every
chamber of that panel other than $c$ has codistance $ws$.

./pdf/abramenko-vanmaldeghem-2001-mathz-1-twinnings-of-buildings.pdf
./pdf/abramenko-1996-bullbelg-group-actions-twin-buildings.pdf

Let $L'$ be the union of all chambers of $\Delta_-$ of codistance $1$
from $c_+$, together with all their faces. Then $L'$ is a subcomplex
of $\Delta_-$.

The next lemma says that $U$ acts simply transitively on the chambers
of $L'$, and that the type-$i$ vertex stabilizer is exactly $H_i$, so
$L'$ is naturally isomorphic to $L = \operatorname{CC}(U; (H_i)_{i \in S})$.

./pdf/abramenko-vanmaldeghem-2001-mathz-1-twinnings-of-buildings.pdf
./pdf/abramenko-brown-2008-gtm248-buildings-theory-applications.pdf

Lemma. The group $U$ acts simply transitively on the chambers of $L'$,
and the map $u H_i \mapsto$ the type-$i$ vertex of $u c_-$ identifies
$L$ with $L'$.

Proof. By the codistance axioms, the chambers of $\Delta_-$ at codistance
$1$ from $c_+$ form a single $B_+$-orbit $B_+ c_-$. Because $B_+ = U T$
and $T$ stabilizes $c_-$, $U$ is transitive on these chambers. If
$u c_- = c_-$ with $u \in U$, then $u$ fixes the negative base chamber,
so $u \in B_+ \cap B_-^{\text{stab}(c_+)} = T$, and $U \cap T = \{1\}$.
Hence $U$ is simply transitive on the chambers of $L'$.

For the vertex stabilizers, in the lattice model of $\Delta_-$, vertices
are homothety classes of $k[[t^{-1}]]$-lattices $\Lambda \subseteq k[[t^{-1}]]^N$
of rank $N$, and an element $g \in G$ preserves a vertex of exponent
vector $b = (b_1, \ldots, b_N)$ precisely when $\deg(g_{ab}) \le b_a - b_b$
for every $a, b$. The base negative chamber has exponent vectors
$(0, \ldots, 0)$ at type $0$, and $(0, \ldots, 0, 1, \ldots, 1)$ with
$i$ zeros and $N - i$ ones at type $i$.

At type $0$, the bound gives $\deg(g_{ab}) \le 0$, so $g_{ab} \in k$.
Combined with $U$ being upper unitriangular at $t = 0$, the stabilizer
is the set of constant upper unitriangular matrices: this is $H_0$.

At type $i \ne 0$, split rows and columns after position $i$. The bound
$\deg(g_{ab}) \le b_a - b_b$ then gives $\deg(g_{ab}) \le 0$ in the two
diagonal blocks, $\deg(g_{ab}) \le -1$ in the upper-right block (so
$g_{ab} = 0$ there since $g_{ab} \in k[t]$), and $\deg(g_{ab}) \le 1$
in the lower-left block. Combined with $U$, the stabilizer is a pair
of constant upper unitriangular diagonal blocks, a zero upper-right
block, and arbitrary entries in $kt$ in the lower-left block. That is
exactly the entry pattern of $H_i$ from the local-group lemma with
$J = S \setminus \{i\}$, so the stabilizer is $H_i$.

The map $u H_i \mapsto$ the type-$i$ vertex of $u c_-$ is then well
defined by the second paragraph, is a bijection on vertices by the
first paragraph and the local-group computation, and preserves faces
because faces are intersections of vertex cosets. Consequently
$L \cong L'$ as simplicial complexes.

The filtration of $\Delta_-$ by codistance feeds the homology of $L$.
For a fixed element $w \in W$ of positive Coxeter length, the codistance
$w$ chambers are grouped into blocks by $J$-residues, where
$J = S \setminus D$ with $D = \{ s \in S : \ell(ws) < \ell(w) \}$. The
descent set $D$ is nonempty because $w \ne 1$, so $J$ is proper.
The parabolic $W_J$ is finite spherical: deleting any node from the
affine cycle $\widetilde A_n$ leaves a finite path. Let $w_J$ be the
longest element of $W_J$.

Within a block $R$ of type $J$, write $F$ for the common $D$-face
shared by all chambers of $R$, and write $c_{\max}$ for the unique
chamber of $R$ at codistance $w \cdot w_J$ from $c_+$. Then the
codistance-$w$ chambers of $R$ are exactly the chambers of $R$ opposite
$c_{\max}$: such chambers form a spherical opposition complex $O$
inside $R$, and the entire block, together with its faces, is the
simplicial join $F * O$.

./pdf/abramenko-vanmaldeghem-2001-mathz-1-twinnings-of-buildings.pdf

The reasoning is a walk along $J$-residues. Fix a codistance-$w$
chamber $c \in R$. Every $s \in J$ is an ascent of $w$, meaning
$\ell(ws) > \ell(w)$, and the panel axiom then forces $c$ to be the
shortest chamber of its $s$-panel. Iterating along a reduced expression
of $w_J$ reaches $c_{\max}$ at codistance $w \cdot w_J$. Any minimal
gallery in $R$ from $c_{\max}$ to a chamber $d$ has type equal to a
reduced word for the Weyl distance $v \in W_J$ from $c_{\max}$ to $d$;
applying the descending panel axiom at each step gives $\delta^*(d, c_+) = w \cdot w_J \cdot v$.
The equality $\delta^*(d, c_+) = w$ is equivalent to $v = w_J$ because
$w_J^2 = 1$. So the codistance-$w$ chambers of $R$ are precisely the
chambers at Weyl distance $w_J$ from $c_{\max}$, which is the definition
of the spherical opposition complex of $c_{\max}$ in $R$.

The intersection of $F * O$ with the previously added subcomplex is
exactly $\partial F * O$. A face of $F * O$ that
omits at least one vertex of $F$ lies in a $D$-descent panel of $R$,
hence in a codistance-$ws$ chamber for some $s \in D$ of strictly
smaller length, and therefore in a previously added block. A face
containing all of $F$ cannot lie in a chamber of a different $J$-residue,
because its $D$-face $F$ already determines the residue.

The relative chains of one block therefore collapse to the chains
of the opposition complex, shifted by $|D|$: a face of $F * O$ that
survives in $C_*(F * O, \partial F * O)$ is a face $F \cup \sigma$
with $F$ a complete face of $F$ and $\sigma \subseteq O$, and the
boundary map induced by deleting vertices of $F$ is zero while the
boundary map induced by deleting vertices of $\sigma$ is the augmented
boundary of $\sigma$ inside $O$. This gives
$$H_k(F * O, \partial F * O; \mathbb{F}_2) \cong \widetilde H_{k - |D|}(O; \mathbb{F}_2).$$

The opposition complex $O$ inside a spherical building of type $J$ is
acyclic in a certain range, provided $q$ is large enough. For a
spherical building of type $A_r$ over $\mathbb{F}_q$, and for a
chamber $c$ of that building, let $O$ be the complex of chambers
opposite $c$, together with all their faces. Then
$$\widetilde H_j(O; \mathbb{F}_2) = 0 \quad \text{for } 0 \le j < r - 1,$$
provided $q \ge 2^{r-1}$. This is a consequence of the integral cone
function of Oppenheim and Valentiner-Branth (Corollary 6.10 of the KMS
cosystolic paper): the cone function gives a chain contraction over
$\mathbb{Z}$ that descends to any coefficients, and the constant
$2^{r-1}$ accommodates the reduction.

./pdf/oppenheim-valentinerbranth-2026-mathann-cosystolic-kms.pdf

For a product of spherical buildings of type $A_{r_1} \times \cdots \times A_{r_k}$,
the opposition complex is the join $O_1 * \cdots * O_k$ of the factor
opposition complexes, and the augmented chain formula for joins carries
the vanishing to degree less than $(r_1 + \cdots + r_k) - 1$.

Within the codistance filtration, each $J$-residue is a product of
spherical buildings of type $A$ with total rank $|J|$, so its
opposition complex $O$ has dimension $|J| - 1$ and satisfies the
vanishing above for $q \ge 2^{|J|-1}$. Since $|J| \le |S| - 1 = n - 1$
and $|J| - 1 \le n - 2$, the requirement $q \ge 2^{n-1}$ covers every
proper residue.

Now pass to the pair $(T_{<w}, T_{\le w})$ where $T_{<w}$ is the union
of all blocks of codistance strictly less than $w$, and $T_{\le w}$ is
the union with the codistance-$w$ blocks. The relative homology of the
pair is the direct sum of the block contributions, and by the block
homology computation each block contributes vanishing in degrees below
$|D| + \dim O = |D| + |J| - 1 = |S| - 1 = n$, so
$$H_k(T_{\le w}, T_{<w}; \mathbb{F}_2) = 0 \quad \text{for } k < n.$$

The long exact sequence in homology of the pair $(T_{\le w}, T_{<w})$ then
gives, for $0 \le j \le n - 2$, that $H_j(T_{<w}) \to H_j(T_{\le w})$
is an isomorphism, because in that range both $H_j$ and $H_{j+1}$ of the
pair vanish.

The full negative building is the increasing union of the $T_{\le w}$
over all codistances $w$, and reduced homology of a simplicial complex
over a field commutes with such increasing unions. The affine building
$\Delta_-$ is a Euclidean building, hence a $\mathrm{CAT}(0)$ space,
hence contractible.

./pdf/bridson-haefliger-1999-gtm319-metric-spaces-nonpositive-curvature.pdf

Combining the vanishing of the relative homology of each step, the
isomorphism in each consecutive pair, and the contractibility of
$\Delta_-$, gives
$$\widetilde H_j(L; \mathbb{F}_2) = 0 \quad \text{for } 0 \le j < n - 1.$$
The complex $L$ is connected, because $\widetilde H_0(L; \mathbb{F}_2) = 0$
by that vanishing in degree zero.

./pdf/abramenko-brown-2008-gtm248-buildings-theory-applications.pdf

A corollary of the connectivity of $L$ is that the simple generators
$x_i$ generate $U$. In a connected pure complex any two chambers can
be joined by a chain of chambers in which consecutive chambers share
at least one vertex. Label the chambers of $L$ by elements of $U$ via
the simply transitive action, and walk such a chain from the base
chamber to reach $u$. Consecutive chambers sharing a vertex of type $i$
have labels $u, v \in U$ with $u H_i = v H_i$, hence $u^{-1} v \in H_i$.
So $u \in \langle H_i : i \in S \rangle$, and each $H_i$ is generated by
all $x_j$ with $j \ne i$. Hence $U = \langle x_j : j \in S \rangle$.

For a group $K$ and an abelian group $A$ with trivial $K$-action,
group homology $H_j(K; A)$ is the $j$-th left derived functor of the
coinvariant functor $M \mapsto M_K = M \otimes_{\mathbb{Z}[K]} \mathbb{Z}$.
Concretely, choose a projective resolution $P_* \to \mathbb{Z}$ of
the trivial module $\mathbb{Z}$ over the group ring $\mathbb{Z}[K]$,
and take the homology of $P_* \otimes_{\mathbb{Z}[K]} A$. Equivalently,
take a contractible free $K$-complex and quotient.

The complex $L$ is only $(n - 2)$-acyclic, not contractible, but the
odd-order stabilizers compensate: modulo bounded finite stabilizer
corrections, the simplicial chains of $L$ behave like a partial
projective resolution of the trivial $\mathbb{F}_2[U]$-module.

Lemma. Let $K$ be a group acting on a nonempty simplicial complex $Z$
without inversions, and suppose every stabilizer of a nonempty face
of $Z$ is a finite group of odd order. If $\widetilde H_j(Z; \mathbb{F}_2) = 0$
for $0 \le j \le h$, then
$$H_j(K; \mathbb{F}_2) \cong H_j(C_*(Z; \mathbb{F}_2)_K) \quad (0 \le j \le h),$$
where $C_*(Z; \mathbb{F}_2)_K$ is the chain complex of $K$-orbits of
faces with the induced boundary.

Proof. For each $r \ge 0$, the permutation module $C_r(Z; \mathbb{F}_2)$
is a direct sum of transitive permutation modules $\mathbb{F}_2[K/H]$
indexed by face stabilizers $H$. Because $|H|$ is odd, averaging over
$H$ splits the augmentation $\mathbb{F}_2[H] \to \mathbb{F}_2$, so
$\mathbb{F}_2$ is a projective $\mathbb{F}_2[H]$-module; by induction
the induced module $\mathbb{F}_2[K/H] = \mathbb{F}_2 \otimes_{\mathbb{F}_2[H]} \mathbb{F}_2[K]$
is a projective $\mathbb{F}_2[K]$-module. Hence every chain module
$C_r(Z; \mathbb{F}_2)$ is projective.

The augmented chain complex $C_*(Z; \mathbb{F}_2) \to \mathbb{F}_2$ is
exact through degree $h$ by hypothesis. Retain $C_0, \ldots, C_{h+1}$
together with their differentials, and complete to a full projective
resolution of $\mathbb{F}_2$ by choosing a projective module mapping
onto $\ker \partial_{h+1}$ and continuing upward. The resulting
$P_* \to \mathbb{F}_2$ agrees with $C_*(Z; \mathbb{F}_2)$ in degrees
$0, \ldots, h+1$ with identical differentials. Consequently
$H_j(K; \mathbb{F}_2) = H_j(P_* \otimes_K \mathbb{F}_2) = H_j(C_*(Z; \mathbb{F}_2)_K)$
for $0 \le j \le h$.

The two applications of this lemma are the main homology comparisons.
For the first, take $K = U$, $Z = L$, $h = n - 2$. Each nonempty face
of $L$ of type $A \subseteq S$ has stabilizer a conjugate of
$\bigcap_{i \in A} H_i = U_{S \setminus A}$ (from the local-group lemma),
which is finite of odd order because $q$ is odd. The complex $L$ is
$(n-2)$-acyclic by the acyclicity of $L$. Hence
$H_j(U; \mathbb{F}_2) \cong H_j(C_*(L; \mathbb{F}_2)_U)$ for $0 \le j \le n - 2$.

Because $U$ acts transitively on faces of each fixed type $A \subseteq S$,
the orbit complex $U \backslash L$ has exactly one face of each type.
In particular $U \backslash L$ is a single $n$-simplex. The orbit
boundary is the ordinary boundary with coefficient one at every face,
so the orbit chain complex is acyclic in positive degrees. Combining
with the previous paragraph,
$$H_j(U; \mathbb{F}_2) = 0 \quad (1 \le j \le n - 2).$$

For the second homology comparison, relate $L$ and $T_m$. Reduction
modulo $t^m$ induces a map $\pi : U \to G_m$, which descends to a map
of face sets $\widetilde \pi : L \to T_m$ by
$\widetilde \pi(u H_i) = \pi(u) \overline H_i$. This map is well-defined
because $H_i$ maps into $\overline H_i$. It is bijective on faces of
each type: injective because reduction mod $t^m$ is injective on every
$U_J$ for $m \ge 2$ (from the local-group lemma), and surjective because
$\pi : U \to G_m$ is surjective in the first place. The fiber over any
face is a $\Gamma_N(m)$-orbit. Consequently $\Gamma_N(m) \backslash L \cong T_m$
as simplicial complexes.

The subgroup $\Gamma_N(m)$ acts freely on $L$: an element of $\Gamma_N(m)$
fixing a face of type $A$ would lie in a conjugate of $U_{S \setminus A}$,
but $\Gamma_N(m) \cap U_J$ is trivial for every proper $J$ because the
reduction mod $t^m$ is injective on $U_J$.

Applying the projective chains lemma with $K = \Gamma_N(m)$, $Z = L$,
$h = n - 2$ then gives
$$H_j(\Gamma_N(m); \mathbb{F}_2) \cong H_j(C_*(L; \mathbb{F}_2)_{\Gamma_N(m)}) = H_j(T_m; \mathbb{F}_2)$$
for $0 \le j \le n - 2$. In particular $T_m$ is connected, since $L$
is connected by the acyclicity of $\widetilde H_0$.

Next, the homology of the congruence subgroups $\Gamma_a(m)$ stabilizes
in the matrix size $a$. The qualitative statement, due to Iwasa building
on Morrow, is that under suitable Tor-unitality and stable-range
hypotheses, the inclusion on the first $a$ coordinates induces a
pro-isomorphism of the inverse systems
$\bigl(H_j(\Gamma_a(m); \mathbb{Z})\bigr)_m \to \bigl(H_j(\Gamma_{a+1}(m); \mathbb{Z})\bigr)_m$
once $a$ is at least $2j + 1$.

./pdf/morrow-1404.4179v2-pro-unitality-pro-excision.pdf
./pdf/iwasa-1610.04998v2-homology-pro-stability-tor-unital-pro-rings.pdf

The following definitions make the statement precise. For a commutative
nonunital ring $A$, its integer unitization is $\mathbb{Z} \ltimes A = \mathbb{Z} \oplus A$
with multiplication $(u, a)(v, b) = (uv, ub + va + ab)$ and identity
$(1, 0)$. The projection to $\mathbb{Z}$ has kernel $A$, and
$\operatorname{GL}_a(A) = \ker (\operatorname{GL}_a(\mathbb{Z} \ltimes A) \to \operatorname{GL}_a(\mathbb{Z}))$.
A row $b \in (\mathbb{Z} \ltimes A)^\ell$ is $A$-unimodular if
$b \equiv e_1 \pmod{A}$ and there exists $c \equiv e_1 \pmod{A}$ with
$b \cdot c = 1$. The stable range satisfies $\operatorname{sr}(A) \le s$
if every $A$-unimodular row $b$ of length $\ell \ge s + 1$ can be
shortened: there exist $v_1, \ldots, v_{\ell-1} \in A$ with
$(b_1 + v_1 b_\ell, \ldots, b_{\ell-1} + v_{\ell-1} b_\ell)$
also $A$-unimodular. These are the stable-range conventions of
Vaserstein (1969), Section 2, pages 384--385 of Matematicheskii
Sbornik 8.

An inverse system $(A_m)$ of abelian groups is pro-zero if for every
index $m$ there is an $m' \ge m$ whose transition map $A_{m'} \to A_m$
is zero. A level map of inverse systems is a pro-isomorphism exactly
when its kernel and cokernel systems are pro-zero. The system $(A_m)$
of commutative nonunital rings is Tor-unital if for every $i \ge 1$
the inverse system $\bigl(\operatorname{Tor}_i^{\mathbb{Z} \ltimes A_m}(\mathbb{Z}, \mathbb{Z})\bigr)_m$
is pro-zero.

Morrow's theorem states that the powers of any ideal in a commutative
Noetherian ring form a Tor-unital inverse system. For the powers
$A_m = t^m k[t]$ of the ideal $(t) \subseteq k[t]$, this Tor-unitality
holds. Iwasa's theorem then states: if $(A_m)$ is a Tor-unital inverse
system of commutative nonunital rings, and if
$r = \max\{2, \sup_m \operatorname{sr}(A_m)\} < \infty$, then
$$\bigl(H_j(\operatorname{GL}_a(A_m); \mathbb{Z})\bigr)_m \to \bigl(H_j(\operatorname{GL}_{a+1}(A_m); \mathbb{Z})\bigr)_m$$
is a pro-isomorphism for $a \ge 2j + r - 1$.

./pdf/morrow-1404.4179v2-pro-unitality-pro-excision.pdf
./pdf/iwasa-1610.04998v2-homology-pro-stability-tor-unital-pro-rings.pdf

The stable range is small for the rings $A_m$. For an $A_m$-unimodular
row $b = (b_1, \ldots, b_{r'}) \in (\mathbb{Z} \ltimes A_m)^{r'}$ with
$r' \ge 3$ and $b \equiv e_1 \pmod{t^m}$, the polynomial $b_1$ is coprime
to $t$. For each irreducible divisor $p$ of $b_1$ with $b_{r'} \not\equiv 0 \pmod p$,
choose $v_p \in k$ avoiding the value $-b_2/b_{r'}$ modulo $p$ (only
one residue in $k$ is forbidden, and $q > 1$). By the Chinese remainder
theorem in $k[t]$ (a principal ideal domain) lift $(v_p)_p$ to a single
$v \in t^m k[t] = A_m$. Then
$b' = (b_1, b_2 + v b_{r'}, b_3, \ldots, b_{r'-1})$ is $A_m$-unimodular
and has the same reduction mod $t^m$: at a divisor where $b_{r'} \equiv 0$,
original unimodularity already makes some component of $(b_2, \ldots, b_{r'-1})$
nonzero, and at a divisor where $b_{r'} \not\equiv 0$ the choice of $v_p$
forces the modified second component nonzero.

For the unitization lift, if a row $b$ satisfies $b \cdot c = 1$ over
$R = k[t]$ with $b \equiv e_1 \pmod{t^m}$, then
$c' = e_1 + (1 - b_1)c$ also satisfies $b \cdot c' = 1$ and
$c' \equiv e_1 \pmod{t^m}$. The difference between $b \cdot c'$ and $1$
lies in $A_m$, so the equality holds in $\mathbb{Z} \ltimes A_m$. Hence
$\operatorname{sr}(A_m) \le 2$ for every $m$.

Combining the stable range bound with Iwasa's theorem (with $r = 2$),
the map $\bigl(H_j(\operatorname{GL}_a(A_m); \mathbb{Z})\bigr)_m \to \bigl(H_j(\operatorname{GL}_{a+1}(A_m); \mathbb{Z})\bigr)_m$
is a pro-isomorphism for $a \ge 2j + 1$.

The group $\operatorname{GL}_a(A_m)$ equals $\Gamma_a(m)$. A matrix over
$\mathbb{Z} \ltimes A_m$ that reduces to one congruent to $I$ modulo $t^m$
lifts back to a matrix in $\operatorname{GL}_a(\mathbb{Z} \ltimes A_m)$; its
determinant lies in $R^* = k^*$, and the only element of $k^*$ congruent
to $1$ modulo $t^m$ is $1$. Hence the determinant is $1$, and the
matrix lies in $\operatorname{SL}_a(k[t])$.

Lemma (stabilization). Let $h \ge 0$ and suppose $a \ge 2h + 1$. Then
the inclusion on the first $a$ coordinates induces surjections
$$H_j(\Gamma_a(m); \mathbb{F}_2) \to H_j(\Gamma_{a+1}(m); \mathbb{F}_2)$$
for every $0 \le j \le h$ and every $m \ge 1$.

Proof. The integral pro-isomorphism just obtained gives
pro-isomorphisms
$\bigl(H_j(\Gamma_a(m); \mathbb{Z})\bigr)_m \to \bigl(H_j(\Gamma_{a+1}(m); \mathbb{Z})\bigr)_m$
for $a \ge 2j + 1$. The universal coefficient theorem
$$0 \to H_j(G; \mathbb{Z}) \otimes \mathbb{F}_2 \to H_j(G; \mathbb{F}_2) \to \operatorname{Tor}_1^{\mathbb{Z}}\bigl(H_{j-1}(G; \mathbb{Z}), \mathbb{F}_2\bigr) \to 0$$
is natural in $G$, and its tensor and Tor functors extend to inverse
systems and preserve pro-isomorphisms. For $j \ge 1$, both integral
degrees $j$ and $j-1$ are in the stable range at $a \ge 2j + 1$, so
the levelwise universal coefficient sequences assemble into a
pro-isomorphism of $\mathbb{F}_2$-homology systems. Consequently the
cokernels of the levelwise maps $H_j(\Gamma_a(m); \mathbb{F}_2) \to H_j(\Gamma_{a+1}(m); \mathbb{F}_2)$
form a pro-zero inverse system.

For $m' \ge m$, the index $[\Gamma_a(m) : \Gamma_a(m')]$ is finite and
odd. Filtering matrices congruent to $I$ modulo $t^m$ by successive
powers of $t$ identifies each successive quotient with an additive
subgroup of matrices over $k$, whose order is a power of $q$ and
therefore odd. The transfer map
$\operatorname{tr} : H_j(\Gamma_a(m); \mathbb{F}_2) \to H_j(\Gamma_a(m'); \mathbb{F}_2)$,
summing over representatives of $\Gamma_a(m)/\Gamma_a(m')$, satisfies
$\operatorname{inc}_* \circ \operatorname{tr} = [\Gamma_a(m) : \Gamma_a(m')] = 1$
over $\mathbb{F}_2$. Hence $\operatorname{inc}_* : H_j(\Gamma_a(m'); \mathbb{F}_2) \to H_j(\Gamma_a(m); \mathbb{F}_2)$
is surjective, and every transition map between the cokernels is also
surjective.

A pro-zero inverse system with surjective transition maps is zero at
every level. For each $m$ choose $m' \ge m$ with zero transition
$A_{m'} \to A_m$; the same transition is surjective by the previous
paragraph; so its image is simultaneously $\{0\}$ and all of $A_m$.
Applying this to the cokernel system gives levelwise surjectivity of
$H_j(\Gamma_a(m); \mathbb{F}_2) \to H_j(\Gamma_{a+1}(m); \mathbb{F}_2)$
for $0 \le j \le h$.

./pdf/morrow-1404.4179v2-pro-unitality-pro-excision.pdf
./pdf/iwasa-1610.04998v2-homology-pro-stability-tor-unital-pro-rings.pdf

The next step uses the stabilization lemma to represent every homology
class on a chosen set of $N-2$ coordinates, so that the two remaining
coordinates are decoupled from the conjugation action.

Lemma (trivial conjugation). For $1 \le j \le d - 1$ and every $m \ge 1$,
the conjugation action of $\operatorname{SL}_N(R)$ on
$H_j(\Gamma_N(m); \mathbb{F}_2)$ is trivial.

Proof. The choice $N = 2d + 1$ gives $N - 2 = 2d - 1 = 2(d - 1) + 1$.
Apply the stabilization lemma twice, at $h = d - 1$: first with
$a = N - 2$, then with $a = N - 1$. In both cases the hypothesis
$a \ge 2h + 1$ is met because $N - 2 = 2(d-1) + 1$. Composing, the
natural map $H_j(\Gamma_{N-2}(m); \mathbb{F}_2) \to H_j(\Gamma_N(m); \mathbb{F}_2)$
is surjective for every $1 \le j \le d - 1$.

Signed permutation matrices of determinant one lie in $\operatorname{SL}_N(R)$,
normalize $\Gamma_N(m)$, and act transitively on the $(N-2)$-subsets of coordinates
within $\operatorname{SL}_N(R)$. Consequently the surjectivity extends
to every $(N-2)$-coordinate subgroup $\Gamma_T(m) \subseteq \Gamma_N(m)$.

Fix distinct $a, b \in S$ and take $T = S \setminus \{a, b\}$. The
elementary matrix $e_{ab}(f) = I + f E_{ab}$ acts on the source
$H_j(\Gamma_T(m); \mathbb{F}_2)$ trivially because it commutes with
every element of $\Gamma_T(m)$. Combined with the surjection from the
previous paragraph, the same matrix acts trivially on
$H_j(\Gamma_N(m); \mathbb{F}_2)$. Elementary matrices generate
$\operatorname{SL}_N(R)$ because $R$ is a Euclidean domain, so the
whole group $\operatorname{SL}_N(R)$ acts trivially.

The finite quotient $G_m$ also acts by conjugation, and the next
proposition assembles the triviality of that action with the homology
of $U$ and the homology of $T_m$ to show that the latter vanishes.

Proposition (vanishing cohomology). For every $m \ge 2$,
$$H_j(\Gamma_N(m); \mathbb{F}_2) = H_j(T_m; \mathbb{F}_2) = H^j(T_m; \mathbb{F}_2) = 0$$
for $1 \le j \le d - 1$.

Proof. The order of $G_m$ is odd. The quotient $U / \Gamma_N(1)$
is the upper unitriangular group over $k$, of odd order; and each deeper
quotient $\Gamma_N(i) / \Gamma_N(i+1)$ is an additive group of $N \times N$
matrices over $k$, again of odd order.

Choose a projective $\mathbb{F}_2[U]$-resolution $P_* \to \mathbb{F}_2$ of
the trivial module. Restriction to $\Gamma_N(m)$ leaves $P_*$ projective
because $\mathbb{F}_2[U]$ is free over $\mathbb{F}_2[\Gamma_N(m)]$, so
$(P_*)_{\Gamma_N(m)} = P_* \otimes_{\mathbb{F}_2[\Gamma_N(m)]} \mathbb{F}_2$
computes $H_*(\Gamma_N(m); \mathbb{F}_2)$.

The group $G_m$ acts by left multiplication on $\mathbb{F}_2[U]$, hence
on the chain complex and on its homology. Averaging over the odd-order
group $G_m$ shows that the coinvariant functor $M \mapsto M_{G_m}$
is exact on $\mathbb{F}_2[G_m]$-modules; combined with the tensor
identity $M_U = (M_{\Gamma_N(m)})_{G_m}$ for any $\mathbb{F}_2[U]$-module $M$,
taking homology gives
$$H_j(U; \mathbb{F}_2) \cong H_j(\Gamma_N(m); \mathbb{F}_2)_{G_m}.$$
By the trivial-conjugation lemma the right side equals
$H_j(\Gamma_N(m); \mathbb{F}_2)$ for $1 \le j \le d - 1$.

Combining with the vanishing $H_j(U; \mathbb{F}_2) = 0$ for $1 \le j \le n - 2$
(which contains $1 \le j \le d - 1$ because $d - 1 \le n - 2$ when $n \ge d + 1$),
$$H_j(\Gamma_N(m); \mathbb{F}_2) = 0 \quad (1 \le j \le d - 1).$$
By $H_j(T_m; \mathbb{F}_2) \cong H_j(\Gamma_N(m); \mathbb{F}_2)$ for $0 < j < n - 1$,
the homology of $T_m$ vanishes in the same range.

The cohomology follows by duality. Over the field $\mathbb{F}_2$ the
universal coefficient theorem reads
$$H^j(T_m; \mathbb{F}_2) = \operatorname{Hom}_{\mathbb{F}_2}\bigl(H_j(T_m; \mathbb{F}_2), \mathbb{F}_2\bigr),$$
and this vanishes because $H_j(T_m; \mathbb{F}_2) = 0$.

./pdf/iwasa-1610.04998v2-homology-pro-stability-tor-unital-pro-rings.pdf

The last ingredient is the quantitative expansion theorem, due to
Oppenheim and Valentiner-Branth, that supplies the cosystolic inequality
on the $(n-1)$-skeleton of $T_m$ with a uniform constant.

./pdf/oppenheim-valentinerbranth-2026-mathann-cosystolic-kms.pdf

A Kac-Moody-Steinberg group of type $\widetilde A_n$ is built from the
proper spherical subdiagrams of the affine cycle. For each proper
$J \subsetneq S$, the associated spherical local group is $U_J$. The
universal completion is the group obtained from the free product of
the $U_J$ by identifying $U_J$ with its image in $U_K$ whenever
$J \subseteq K$. A finite quotient $\varphi : \widehat U \to Q$ defines
a KMS complex when $\varphi$ is injective on every $U_J$ and preserves
all pairwise intersections $U_J \cap U_K = U_{J \cap K}$; the resulting
complex is $\operatorname{CC}(Q; (\varphi(U_{S \setminus \{i\}}))_{i \in S})$.

./pdf/oppenheim-valentinerbranth-2026-mathann-cosystolic-kms.pdf

The Oppenheim-Valentiner-Branth cosystolic expansion theorem (Theorem
1.2 of the cosystolic-KMS paper) states that for every integer $n \ge 3$ there are
$\varepsilon_0 > 0$, $\mu_0 > 0$, and $q_0 \in \mathbb{N}$ such that if
$q \ge q_0$ and $Y_i$ are the $(n-1)$-skeleta of a family of $n$-dimensional
$n$-classical KMS complexes over $\mathbb{F}_q$, then every $Y_i$ is an
$(\varepsilon_0, \mu_0)$-cosystolic expander with respect to any finitely
generated abelian group of coefficients. In $\mathbb{F}_2$ coefficients
this reads
$$\|\delta_i f\|_{Y, i+1} \ge \varepsilon_0 \operatorname{dist}_{Y,i}(f, Z^i(Y)) \quad (0 \le i < n - 1)$$
for every $f \in C^i(Y; \mathbb{F}_2)$, where the norm is the one
defined above with the weight functions $w_i^Y$.

The verification that $T_m$ satisfies the KMS hypotheses proceeds in
three steps. Dimension and classicalness: $T_m$ has dimension $N - 1 = n$,
and every proper subdiagram of the affine cycle $\widetilde A_n$ is a
disjoint union of type-$A$ paths, so every proper subdiagram has
irreducible components of type $A_r$, and $T_m$ is $n$-classical.

./pdf/oppenheim-valentinerbranth-2026-mathann-cosystolic-kms.pdf

Realization of the universal completion: for each proper $J \subseteq S$,
the map $U_J \to U$ extends to a homomorphism $\widehat U \to U$, which
is surjective by the generation corollary. On each $U_J$ this map is
given by the entry-pattern description of the local-group lemma.

Local injectivity and intersections: by the local-group lemma, the
composite $\widehat U \to U \to G_m = U / \Gamma_N(m)$ is injective on
each $U_J$ for $m \ge 2$ and preserves every pairwise intersection
$U_J \cap U_K = U_{J \cap K}$. Hence $T_m = \operatorname{CC}(G_m; (\overline H_i)_{i \in S})$
is a finite $n$-dimensional $n$-classical KMS complex.

./pdf/oppenheim-valentinerbranth-2026-mathann-cosystolic-kms.pdf

Well-definedness of $q$: the requirements are that
$q$ is an odd prime, that $q \ge 2^{n-1}$ (needed for the spherical
opposition vanishing), and that $q \ge q_0$ (needed for the cosystolic
expansion theorem). All are finite lower bounds depending only on $n$,
and arbitrarily large odd primes exist. Fix such a prime $q$ and keep
it fixed from now on.

The chosen $q$ gives a cosystolic inequality with a uniform constant
$\varepsilon_0 > 0$ on the $(n-1)$-skeleton $Y_m = \operatorname{skel}_{n-1}(T_m)$:
$$\|\delta_i f\|_{Y_m, i+1} \ge \varepsilon_0 \operatorname{dist}_{Y_m,i}(f, Z^i(Y_m)) \quad (0 \le i < n - 1).$$
This is the main input for the final assembly.

It remains to pass from the $(n-1)$-skeleton $Y_m$ to the $d$-skeleton
$X_m = \operatorname{skel}_d(T_m)$, and to replace distances to cocycles
$Z^i$ by distances to coboundaries $B^i$.

First, degree bounds. Every chamber of $T_m$ contains exactly one vertex
of each type $i \in S$. The common stabilizer of the base chamber's
vertices is $U_\varnothing = \{1\}$, so the chambers of $T_m$ are in
bijection with $G_m$. A vertex of type $i$ is a coset $g \overline H_i$;
the number of chambers through it equals $|\overline H_i| = |H_i| = M$,
independent of $i$ and $m$. Each chamber has $N$ vertices, of types
$0, \ldots, N-1$ one each.

A $d$-face containing a fixed vertex has $d+1$ vertices, so
$\binom{N-1}{d}$ choices of its other vertices. Each choice lies inside
at most $M$ chambers through the fixed vertex. Hence every vertex lies
in at most $M \binom{N-1}{d}$ faces of dimension $d$. Therefore
$D = M \binom{N-1}{d}$ works.

Second, growth. The subgroup $\{e_{12}(f) : f \in k[t]\}$ of $U$ is
isomorphic to the additive group of $k[t]$, and its image in
$G_m = U / \Gamma_N(m)$ has order $q^m$: the polynomials of degree
less than $m$ give distinct residue classes modulo $t^m$. Because
$|G_m| \ge q^m$, and the number of vertices of $T_m$ is $N |G_m| / M$,
$$|X_m(0)| = N |G_m| / M \ge N q^m / M \to \infty$$
as $m \to \infty$. The divergence rate depends only on $n$ and $M$.

Third, weight comparison. Let $T$ be a finite pure $n$-complex and let
$X = \operatorname{skel}_s(T)$ for some $0 \le s \le n$. Write
$\mu_i = w_i^T$ and $\nu_i = w_i^X$ for the intrinsic weights of $T$
and of $X$ on $i$-faces. For every $s$-face $\tau \in T(s)$, let
$c(\tau)$ be the number of $n$-faces of $T$ containing $\tau$. Suppose
$0 < a \le c(\tau) \le b$ for every $\tau \in T(s)$, and put
$$\Lambda = \frac{\binom{n+1}{s+1}\, |T(n)|}{|T(s)|}.$$
Then for every $i$-face $\sigma \in T(i)$ with $0 \le i \le s$,
$$\frac{\Lambda}{b}\, \mu_i(\sigma) \le \nu_i(\sigma) \le \frac{\Lambda}{a}\, \mu_i(\sigma).$$

Proof. The case $i = s$ follows from the definitions: on one hand
$\mu_s(\sigma) = c(\sigma) / \bigl(\binom{n+1}{s+1} |T(n)|\bigr)$, on
the other $\nu_s(\sigma) = 1 / |T(s)|$, so the ratio is $\Lambda / c(\sigma)$.

For $i < s$, use the two averaging identities
$$\mu_i(\sigma) = \binom{s+1}{i+1}^{-1} \sum_{\tau \in T(s),\, \sigma \subseteq \tau} \mu_s(\tau), \quad
\nu_i(\sigma) = \binom{s+1}{i+1}^{-1} \sum_{\tau \in T(s),\, \sigma \subseteq \tau} \nu_s(\tau).$$
The first is a double count of chamber--face incidences in $T$; the
second is the definition of the skeleton weights on $X$. Summing the
$i = s$ bounds over all $\tau \in T(s)$ containing $\sigma$ and dividing
by $\binom{s+1}{i+1}$ gives the asserted inequalities. The same
inequalities hold for norms and for distances to any subspace.

Apply this twice, with $T = T_m$.

For the first application, $X = Y_m$ and $s = n - 1$. Each $(n-1)$-face
of $T_m$ (a panel) lies in $|\overline H_i| = q$ chambers, so
$a = b = q$. The double count of (chamber, panel) incidences gives
$(n+1) |T_m(n)| = q |T_m(n-1)|$, so $\Lambda = q$ and both
$\Lambda / a$ and $\Lambda / b$ equal $1$. Hence the weights of $Y_m$
and of $T_m$ coincide in their common degrees:
$$\|f\|_{Y_m, i} = \|f\|_{T_m, i} \quad (0 \le i \le n - 1),$$
and the same equality holds for distances to any subspace of cochains.

./pdf/oppenheim-valentinerbranth-2026-mathann-cosystolic-kms.pdf

For the second application, $X = X_m$ and $s = d$. A $d$-face of $T_m$
is contained in at least $1$ chamber and in at most $M$ chambers through
any of its vertices; the exact count is $|U_{S \setminus A}|$ where $A$
is the type set of the face, which is bounded by $M$. So $a = 1$ and
$b = M$. Set $\Lambda_m = \binom{N}{d+1} |T_m(n)| / |T_m(d)|$. Then
$$\frac{\Lambda_m}{M}\, \|f\|_{Y_m, i} \le \|f\|_{X_m, i} \le \Lambda_m \|f\|_{Y_m, i} \quad (0 \le i \le d),$$
and the same inequalities hold for distances to any subspace of cochains.

Now assemble the main theorem. Put
$$\varepsilon = \varepsilon_0 / M, \qquad D = M \binom{N-1}{d}.$$
Both $\varepsilon$ and $D$ depend only on $n$ and $q$, hence only on
$d$. Fix $m \ge 2$ and $0 \le i < d$, and let $f \in C^i(X_m; \mathbb{F}_2)$.

Because $X_m = \operatorname{skel}_d(T_m)$, the cochain spaces
$C^i(X_m)$, $C^i(Y_m)$, $C^i(T_m)$ coincide for $0 \le i \le d$, and
the coboundary $\delta_i$ is the same. Hence $Z^i(Y_m) = Z^i(T_m)$ and
$B^i(X_m) = B^i(T_m)$ for $0 \le i < d$. By the vanishing cohomology
proposition, $H^i(T_m; \mathbb{F}_2) = 0$ for $1 \le i \le d - 1$, so
$Z^i(T_m) = B^i(T_m)$ in that range. In degree $i = 0$, connectedness
of $T_m$ gives $Z^0(T_m) = B^0(T_m)$ (both spaces consist of constant
cochains). Consequently
$$Z^i(Y_m) = B^i(X_m) \quad (0 \le i < d).$$

The weight comparison in degree $i+1$ gives
$$\|\delta_i f\|_{X_m, i+1} \ge \frac{\Lambda_m}{M} \|\delta_i f\|_{Y_m, i+1}.$$
The cosystolic expansion inequality on $Y_m$ gives
$$\|\delta_i f\|_{Y_m, i+1} \ge \varepsilon_0 \operatorname{dist}_{Y_m, i}(f, Z^i(Y_m)).$$
Substituting $Z^i(Y_m) = B^i(X_m)$ and using the weight comparison in degree $i$ gives
$$\operatorname{dist}_{Y_m, i}(f, Z^i(Y_m)) = \operatorname{dist}_{Y_m, i}(f, B^i(X_m)) \ge \frac{1}{\Lambda_m} \operatorname{dist}_{X_m, i}(f, B^i(X_m)).$$
Chaining the three inequalities and cancelling $\Lambda_m$ yields
$$\|\delta_i f\|_{X_m, i+1} \ge \varepsilon \operatorname{dist}_{X_m, i}(f, B^i(X_m))$$
with $\varepsilon = \varepsilon_0 / M > 0$ independent of $m$, $i$, and $f$.

The structural hypotheses for the $X_m$ have been verified: finite by the
local-group lemma; connected, pure, and bounded in vertex incidence by
$D$; with $|X_m(0)| \to \infty$ from the growth computation; and with the
uniform expansion inequality. Neither $D$ nor $\varepsilon$
depends on $m$, so the theorem holds for every $d \ge 3$.