Coarse assembly: persistent topology and vanishing indices

Source and verification

This article explains family 307 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Coarse assembly turns a large-scale topological class into an operator index. Family 307 claims that an infinite-order class can have zero index even for a separated union of finite graphs with uniformly bounded degrees. The September manuscript treats the reduced Roe algebra; the October manuscript gives a different construction for the stronger maximal-algebra statement. Both separate a topological persistence argument from an averaging argument that destroys the index. (OpenAI, 2026) (OpenAI, 2026a)

🏷️ Preliminaries: scale, topology, and controlled operators

A metric space is uniformly discrete if distinct points are uniformly separated. It has bounded geometry if every fixed-radius ball contains a uniformly bounded number of points. A coarse disjoint union of finite graphs places the components increasingly far apart: at any fixed distance, only finitely many pairs of different components interact.

The Rips complex $P_r(X)$ fills a simplex on each finite vertex set of diameter at most $r$. Increasing $r$ permits more simplices and can make formerly nontrivial cycles bound. The coarse topological domain is

$$K{X}_q(X)=\underset{r}{\underrightarrow{\lim }}{K}_q^{\mathrm{lf}}(P_r(X)),$$

with locally finite analytic $K$-homology $K_q^{\mathrm{lf}}(Z)=K{K}_q(C_0(Z),\mathbb{C})$. These are complex $K$-groups, with degree modulo two. Informally, $K$-homology records generalized elliptic cycles, while $K$-theory records stable vector bundles or projections in even degree and stable unitary classes in odd degree. An index pairing between compatible $K$-theory and $K$-homology classes is an integer.

For example, a map of an oriented circle to $S^1$ pairs with its fundamental $K$-homology class by its winding number. Locally finite classes on a countable disjoint union form a product of component groups, so an entire infinite family of circle or torus classes can define one class. The argument must then show that this class survives every later Rips scale.

Use the module ${\ell }^2(X)\otimes {\ell }^2(\mathbb{N})$. An operator has propagation at most $R$ if its matrix entry from $y$ to $x$ is zero when $d(x,y)>R$. It is locally compact if multiplication on either side by any function in $C_0(X)$ gives a compact operator. For a finite-propagation matrix here, this means every vertex entry is compact; it imposes no common rank bound.

The reduced Roe algebra $C^{\ast }(X)$ completes these operators in their concrete operator norm. The maximal Roe algebra $C_{\max }^{\ast }(X)$ uses the supremum of norms over all representations. Coarse assembly sends a class in $K{X}_q(X)$ to its index in the corresponding algebra’s $K$-theory.

Stronger manuscript claim

There is a coarse disjoint union $X$ of finite connected graphs with uniformly bounded degrees and an infinite-order $\alpha \in K{X}_1(X)$ such that

$${\mu }_X^{\max }(\alpha )=0.$$

Hence maximal coarse assembly fails rational injectivity. Its image under the maximal-to-reduced quotient also has zero reduced index. (OpenAI, 2026b)

This is a failure of injectivity, not merely a missing analytic class in the image. Vanishing of a reduced index alone would not establish vanishing of its maximal lift. The earlier manuscript proves reduced failure using its own disk-based construction. (OpenAI, 2026a)

🏷️ The reduced example: winding survives every scale

In the September construction, the base $D_j$ is a triangular-lattice disk of radius $j$, with boundary cycle $C_j$. A vertex of depth $k$ carries a finite congruence quotient $Q_k$ of $S{L}_3(\mathbb{Z})$; $Q_0$ is trivial. Cayley edges connect a fiber internally. Neighboring fibers are joined by twisted matchings or reduction maps, whose kernels have at most $2^9$ elements. Thus the total graph degree stays bounded even as the fibers grow.

Choose a transcendental $2$-adic unit $t$ and compatible twists

$$g_k=\mathrm{diag}(t,t^{-1},1)(\mathrm{mod}{2}^k).$$

Crossing a fixed ray from the disk center applies right multiplication by $g_k$ or its inverse. A path’s signed crossing count is its charge. Away from the origin this agrees with winding; fiber steps have charge zero. The Cayley action is on the left, so it commutes with these right twists.

For each fixed length bound $A$, sufficiently late graphs have no closed path of length at most $A$ with nonzero charge. Indeed, such a path must project near the center: otherwise the entire projected loop lies in an open half-plane and has winding zero. All its fiber levels are consequently large. Reducing to their common minimum level gives

$$wz{g}_k^m=z,$$

where $|m|\le A$, $m\ne 0$, and $w$ is a word of length at most $A$. Taking traces would identify $t^m+t^{-m}+1$ modulo a high power of two with an integer trace of one of finitely many words. Transcendence rules out each limiting equality, so all are excluded beyond one finite threshold.

At Rips scale $r$, assign to an edge the charge of a graph path of length at most $r$ joining its endpoints. The short-loop statement at length $3r$ makes this independent of the path and makes its sum around every Rips triangle zero. It is therefore an integral cocycle. Its restriction to $C_j$ has winding one, so it defines a $K$-theory test pairing to one with that boundary class.

For every fixed $r$ and every nonzero integer $n$, a sufficiently late finite component detects $n{\alpha }_r$ as the nonzero integer $n$. The component may change with $r$. If $n\alpha$ vanished in the filtered colimit, it would vanish at some finite scale, contradicting this test. The class has infinite order before rationalization.

🏷️ Reduced averaging and the filled-disk factorization

Let ${\xi }_u$ be the normalized constant vector in the fiber over $u$. Define an isometry $V$ sending the basis vector at $u$ to ${\xi }_u$, with the infinite multiplicity factor unchanged.

The fibers have unbounded diameters, so $V$ need not cover a coarse map from the disks into the graph union. Instead, property (T) supplies a uniform spectral gap for averaging on every finite fiber. Powers of the lazy averaging operator approximate the projection $P$ onto fiber constants in operator norm, with finite propagation at every finite power.

An operator $T$ of bounded propagation on the base can be lifted block by block using controlled contractions that send normalized constants to normalized constants. Across a reduction with kernel size $h$, the lift of one basis vector is the sum of its $h$ children divided by $\sqrt{h}$. Decomposing the matrix support into finitely many matchings bounds the norm of the lifted operator $A_T$.

Sandwiching gives

$$(P\otimes 1)A_T(P\otimes 1)=VT{V}^{\ast }.$$

Finite-propagation approximations to $P$ show that this lies in the reduced Roe algebra. The amplified projection $P\otimes 1$ itself is not locally compact: its diagonal entries are nonzero multiples of the identity on an infinite-dimensional space. Local compactness is proved for the sandwiches.

On the boundary the fibers are singletons, so $V$ agrees exactly with the boundary inclusion. The entire family of boundary classes bounds in the union of filled disks. Assembly naturality is applied only to the actual boundary inclusions; the transfer supplies the remaining operator-algebra map. This yields zero reduced index for the same persistent class. (OpenAI, 2026a)

🏷️ The maximal example: two torus ends

The October construction uses three-dimensional discrete tori

$$\begin{array}{c}{B}_j=(\mathbb{Z}/l_j\mathbb{Z}{)}^3,\\ l_j=100(j+1).\end{array}$$

Over each lies a fiber $Q_{2j}$. Set $m=2^j$ and choose

$$\begin{array}{c}U=1+m{E}_{12},\\ V=1+m{E}_{23}.\end{array}$$

Their commutator is $1+m^2{E}_{13}$. They commute modulo $m^2$, so they give consistent twists around the first two base circles, but their commutator remains visible modulo $m^3$.

The full graph has heights from $-3j$ to $3j$. In the central interval $[-j,j]$ the quotient is constantly $Q_{2j}$; outside it, the quotient level decreases by one at every step until it is trivial at each end. The end slices are copies of $B_j$. Gradual reduction keeps degree bounded. The long central region is needed to prevent bounded-length paths near its middle from escaping through an easier route near an end.

Write $Y$ for the union of middle slices, $B$ for the torus union, $f:Y\to B$ for projection, and $i_{\pm }:B\to X$ for the two end inclusions. Choose a torus class $\beta$ and define

$$\alpha =(i_{+}{)}_{\ast }\beta -(i_{-}{)}_{\ast }\beta .$$

Persistence now uses line-bundle divisibility instead of boundary winding.

🏷️ Roots of a line bundle detect the endpoint difference

On the continuous base torus $T_j$, let $u_1,u_2,u_3$ be its integral circle cohomology classes. Choose a line bundle ${\mathcal{L}}_j$ with first Chern class $u_1{u}_2$, and let $e_3$ be the odd $K$-class of the third circle. Then

$${\theta }_j=([{\mathcal{L}}_j]-1)e_3$$

pairs to one with a suitable ${\tau }_j\in K_1(T_j)$. Their sequence defines $\beta$ in the locally finite product at an initial scale.

Fix a later scale $r$. For sufficiently large $j$, interpolation retracts the base Rips complex to $T_j$. On middle Rips edges, let $a,b$ be the first two seam cocycles. Lifting transport from $Q_{2j}$ to $Q_{3j}$ produces a cochain $c$ satisfying

$$\delta c=-a⌣b(\mathrm{mod}{2}^j).$$

The key multiplication defect is the visible commutator, whose $(1,3)$ coefficient is one after division by $m^2$. Short Cayley words lift consistently because reduction is injective on any fixed finite word ball at sufficiently high levels. The calculation only commutes kernel elements past the right twists, not past arbitrary Cayley words.

The integral coefficient exact sequence now makes the pulled-back Chern class divisible by $2^j$, even if integral cohomology has torsion. Line-bundle classification gives a $2^j$th root ${\mathcal{L}}^{\prime }$ of the pullback.

Bounded geometry gives a dimension bound $D_r\ge 1$ for the middle Rips complexes at this fixed scale. With $z=[{\mathcal{L}}^{\prime }]-1$, one has $z^{D_r+1}=0$, so

$$(1+z{)}^{2^j}-1=\sum _{h=1}^{D_r}(\genfrac{}{}{0ex}{}{2^j}{h})z^h.$$

From $h\left(\genfrac{}{}{0ex}{}{2^j}{h}\right)=2^j\left(\genfrac{}{}{0ex}{}{2^j-1}{h-1}\right)$, each coefficient is divisible by $2^{j-\lfloor {\log }_2{D}_r\rfloor }$. Thus the pullback of ${\theta }_j$ has that growing divisor in integral $K$-theory.

If a fixed nonzero multiple $n$ of the endpoint difference vanished, Mayer–Vietoris for the upper and lower halves would provide a class in their overlap carrying both endpoint classes. When $j>2r$, short paths in that overlap stay in the central product region, so forgetting height maps it to the middle Rips complex. Pairing would make $n$ divisible by the growing power of two above, impossible for sufficiently large $j$.

This proves nonvanishing for every fixed $n\ne 0$ at every scale. Rationalization is taken only after obtaining an infinite-order class in the actual Rips colimit.

🏷️ Universal-norm control for the analytic transfer

The maximal argument cannot use concrete operator equalities without additional control. Its basic bound is

$$\Vert T{\Vert }_{\mathrm{op}}\le \Vert T{\Vert }_{\max }\le c_X(R)\Vert T{\Vert }_{\mathrm{op}}$$

for propagation at most $R$. To prove it, partition the matrix support into at most $2N_R-1$ matchings, where $N_R$ bounds radius-$R$ ball cardinalities. A matching operator has diagonal $T^{\ast }T$, and representations are contractive on the diagonal algebra of bounded compact entries. Summing gives the bound. On a fixed finite vertex support, the two norms agree exactly.

Property (T) supplies the projection $P$ onto invariant vectors in the maximal group algebra, hence as a multiplier of the maximal Roe algebra of $Y$. Transport a base entry along a shortest horizontal path to define $L(T)$. This lift preserves adjoints but need not preserve products.

For fixed propagations, its multiplicative defect is supported on only finitely many components: in every sufficiently large torus the relevant short paths have identical seam totals. The defect annihilates fiber constants concretely. Finite-support norm agreement promotes this to the maximal equality

$$P(L(T)L(T^{\prime })-L(T{T}^{\prime }))=0.$$

Consequently $\Phi (T)=PL(T)$ extends to a maximal-algebra homomorphism. After projection to $B$, the averaging approximants have one fixed base propagation bound, so the preceding norm estimate identifies the composite with conjugation by an isometry over the identity. It induces the identity on $K$-theory. In particular, $f_{\ast }$ is surjective on analytic $K_1$.

🏷️ Moving copies one height at a time

The two end maps $i_{-}f$ and $i_{+}f$ have distances growing with $j$, so bounded-distance invariance does not compare them. Instead place orthogonal copies of an operator at every height except the top, and compare them with the copies shifted upward by one step.

Although the number of copies is unbounded, orthogonality controls their norm and consecutive heights are only one edge apart. A simultaneous sine-cosine rotation in a doubled module gives a continuous homotopy between the two collections. Its propagation increases by at most one, so fixed-propagation norm control makes the homotopy continuous in the maximal norm.

Cross-component compression introduces compact defects, so first perform this argument modulo compact operators. Cancel the common interior copies on $K_1$ to identify the two endpoints in that quotient. Since $K_1(\mathcal{K})=0$, the six-term exact sequence makes the map from the Roe algebra’s $K_1$ to the quotient’s $K_1$ injective. The endpoint maps therefore agree before quotienting as well.

Choose an analytic class $b$ over $Y$ projecting to ${\mu }_B^{\max }(\beta )$. Naturality and endpoint equality give

$$\begin{array}{rl}{\mu }_X^{\max }(\alpha )& =(i_{+}f{)}_{\ast }b-(i_{-}f{)}_{\ast }b\\ & =0.\end{array}$$

The persistence and vanishing arguments concern the same endpoint-difference class, but use different properties of its fibers.

Links

• Family 304 also uses divisibility to contradict a fixed nonzero integer. Here the growing divisor comes from line-bundle roots at each fixed Rips scale.
• Family 306 uses finite coefficient quotients to preserve nonvanishing; this family instead proves integral persistence before rationalizing.

Both manuscripts’ main section bodies were read for this exposition. The review checks the displayed geometric, pairing, transfer, and sliding mechanisms against the sources; the external property-(T), $K$-theory, duality, and canonical-index foundations are imported, and no independent formal verification of the complete research claims is asserted. The family scope page describes the earlier reduced coarse-assembly formalization, despite its maximal-assembly heading; it does not document a formalization of the later maximal theorem.

🏷️ Key steps and proof outlines

The two counterexamples separate a topological persistence test from an analytic cancellation. A useful way to follow the proofs is to keep track of which estimates are uniform in the component index $j$, and which are allowed to depend on a fixed Rips scale $r$.

Why a growing divisor detects a rational class

In the maximal construction, bounded geometry gives a dimension bound $D_r$ for all the complexes $P_r(Y_j)$ at one fixed scale. A $2^j$-th root ${\mathcal{L}}_{j,r}^{\prime }$ of the relevant pulled-back line bundle yields, with $z=[{\mathcal{L}}_{j,r}^{\prime }]-1$,

$$z^{D_r+1}=0,[p_{Y,j}^{\ast }{\mathcal{L}}_j]-1=\sum _{h=1}^{D_r}\left(\genfrac{}{}{0ex}{}{2^j}{h}\right)z^h.$$

Here nilpotence follows by factoring the classifying map through a finite skeleton of complex projective space. The elementary identity

$$h\left(\genfrac{}{}{0ex}{}{2^j}{h}\right)=2^j\left(\genfrac{}{}{0ex}{}{2^j-1}{h-1}\right)$$

shows that every displayed coefficient is divisible by $2^{j-\lfloor {\log }_2{D}_r\rfloor }$. Thus a geometric root becomes an integral divisibility statement in $K$-theory, before any rationalization.

The next step explains why that divisibility constrains the endpoint difference. Cover $P_r(X_j)$ by the subcomplexes at heights at least $-r$ and at most $r$. Every simplex lies in one of them because height is $1$-Lipschitz. Their overlap $I$ projects to $P_r(Y_j)$: a path of length at most $r$ starting in the overlap remains in the product collar when $j>2r$.

If $n$ times the endpoint difference vanished, Mayer–Vietoris exactness would provide a class $\eta \in K_1(I)$ whose two images are the endpoint classes multiplied by $n$. Pairing with the pulled-back dual class gives

$$n=⟨h^{\ast }{p}_{Y,j}^{\ast }{\theta }_j,\eta ⟩.$$

The right side has the growing divisor above. For any fixed nonzero integer $n$, choose $j$ so that $j-\lfloor {\log }_2{D}_r\rfloor >v_2(|n|)$. This is impossible. The order of quantifiers matters: fix $r$ and $n$, then choose $j$. It proves persistence at every finite scale, hence infinite order in the coarse direct limit. It does not require one component to work at every scale.

The reduced construction has a different detector: a charge cocycle survives on sufficiently large components at each fixed scale and pairs nontrivially with a winding class. Its transfer and disk-filling argument must therefore be checked separately; the line-bundle calculation is specific to the maximal companion.

Turning concrete estimates into maximal-norm estimates

An estimate in one Hilbert-space representation alone does not bound the maximal Roe norm. The missing uniform estimate comes from a simple decomposition. If $T$ has propagation at most $R$, let

$$N_R=\underset{x}{\sup }|B(x,R)|.$$

View its matrix support as a bipartite graph of rows and columns. Greedy edge coloring splits it into at most $2N_R-1$ matchings. Write $T={\sum }_a{T}^{(a)}$ accordingly. Each summand has at most one nonzero entry per row and column, so $(T^{(a)}{)}^{\ast }{T}^{(a)}$ is diagonal. Every representation is contractive on the diagonal $C^{\ast }$-algebra of bounded families of compact entries. Consequently,

$$\Vert T{\Vert }_{\mathrm{op}}\le \Vert T{\Vert }_{\max }\le (2N_R-1)\Vert T{\Vert }_{\mathrm{op}}.$$

This proves convergence in maximal norm whenever concrete convergence has a common propagation bound. If instead the vertex support is contained in one finite set, the operator lies in a compact-operator $C^{\ast }$-algebra and the two norms agree exactly. These two conclusions have different uses: the first controls continuous rotations; the second makes finite-component multiplicative defects vanish in the maximal completion.

An explicit rotation, followed by exact cancellation

Let $D_{j,t}$ embed a copy of the component module at height $t$. To move all copies up one step, use the doubled target module and the isometries

$$W_{\lambda ,j,t}=\left(\begin{array}{c}\cos \lambda D_{j,t}\\ \sin \lambda D_{j,t+1}\end{array}\right),0\le \lambda \le \pi /2.$$

Different height copies remain orthogonal in each target summand. Therefore their direct sum $W_{\lambda }$ is an isometry even though the number of copies grows without bound. Conjugating the repeated block operator by $W_{\lambda }$ gives $F_{\lambda }(T)$, with

$$\begin{array}{rl}\mathrm{prop}(F_{\lambda }(T))& \le R+1,\\ \Vert F_{\lambda }(T)-F_{\mu }(T){\Vert }_{\mathrm{op}}& \le 2\Vert T{\Vert }_{\mathrm{op}}|\lambda -\mu |.\end{array}$$

The preceding matching estimate promotes this to maximal-norm continuity. Compression to components has compact multiplicative defects, so the resulting maps are homomorphisms after quotienting by the compact ideal. At the endpoints the two stacks consist of the same interior copies plus, respectively, the lower and upper endpoint. Additivity on $K_1$ cancels the common interior term.

Finally, $K_1(\mathcal{K})=0$ makes the map from $K_1$ of the Roe algebra into $K_1$ of its compact quotient injective. Thus endpoint equality holds in the original algebra. Combined with the transfer’s surjectivity and assembly naturality, this kills the analytic image of the same class whose topological persistence was established above. The proof avoids estimating a direct movement over the full, unbounded cylinder length.

🏷️ Manuscripts and verification scope

• Failure of rational injectivity for maximal coarse assembly — pinned manuscript and companion source.
• A counterexample to the coarse Novikov conjecture — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. Despite its maximal-assembly heading, the scope page documents the earlier reduced, locally compact Roe-algebra result: an infinite-order degree-one coarse K-homology class with zero ordinary reduced assembly image. It does not document the later maximal theorem. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A counterexample to the coarse Novikov conjecture.

🐻  OpenAI 2026b. Failure of rational injectivity for maximal coarse assembly.