Reduced assembly, scalar projections, and irrational traces

Source and verification

This article explains family 285 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

This family studies three ways in which topology and reduced operator algebras can diverge: a nonzero geometric index class can disappear, a torsion-free group can acquire a nontrivial scalar projection, and a projection can have an irrational trace that no assembly class can attain. The three manuscripts use different groups. Their common analytic issue is realizing spectral information inside a norm-closed algebra, with estimates uniform over infinitely many graph pieces.

🏷️ Reduced operators, projections, and traces

Let $G$ be a countable discrete group. On the Hilbert space ${\ell }^2(G)$, with orthonormal basis ${\delta }_h$, left translation is the unitary $\lambda (g){\delta }_h={\delta }_{gh}$. The reduced group $C^{\ast }$-algebra $C_r^{\ast }(G)$ is the operator-norm closure of finite sums ${\sum }_g{a}_g\lambda (g)$. Thus membership requires approximation in the norm controlling every unit vector, rather than convergence on individual vectors.

Its canonical trace is ${\tau }_G(A)=⟨A{\delta }_1,{\delta }_1⟩$. It extracts $a_1$ from a finite sum and satisfies ${\tau }_G(AB)={\tau }_G(BA)$ and ${\tau }_G(1)=1$. It is faithful: a positive operator of trace zero is zero. Indeed, its positive square root kills ${\delta }_1$; commutation with right translations then makes it kill every basis vector.

A projection satisfies $p=p^{\ast }=p^2$. Its trace acts as a dimension, even when it is not an integer. For a matrix projection, the convention here is

$$\begin{array}{c}{\tau }_{G,N}(p)=\sum _{j=1}^N{\tau }_G(p_{jj}),\\ {\tau }_{G,N}(I_N)=N.\end{array}$$

This is the unnormalized matrix trace. Dividing by $N$ would change the crucial smallness hypothesis below.

Continuous functional calculus supplies a useful membership test. If a self-adjoint $T\in C_r^{\ast }(G)$ has spectrum contained in $\{0\}\cup [\epsilon ,M]$ with $\epsilon >0$, the function equal to one at zero and zero on the other spectral part is continuous on that spectrum. Consequently its kernel projection belongs to $C_r^{\ast }(G)$. Without the gap, a kernel projection is guaranteed only in the larger group von Neumann algebra, the weak operator closure.

🏷️ Stable classes and the assembly map

For a unital $C^{\ast }$-algebra $A$, the group $K_0(A)$ records formal differences $[p]-[q]$ of finite-matrix projections, identifying projections after stabilization and continuous deformation. A partial isometry $v$, satisfying $v^{\ast }v=p$ and $v{v}^{\ast }=q$, also identifies their classes. A specified trace on $A$, extended by the unnormalized matrix sum, therefore induces a homomorphism ${\tau }_{\ast }:K_0(A)\to \mathbb{R}$.

The Baum–Connes assembly map sends geometric elliptic-index data to such operator-theoretic classes. For a torsion-free group its degree-zero form is ${\mu }_G^r:K_0(BG)\to K_0(C_r^{\ast }(G))$. Here $BG$ is a classifying space, and $K$-homology is formed as a direct limit over finite subcomplexes: a class has compactly supported geometric data. When finite subgroups occur, the domain is instead $K_0^G(\underline{E}G)$, equivariant $K$-homology with $G$-compact support on the universal space for proper actions.

All assembly maps in the stated counterexamples have coefficient algebra $\mathbb{C}$ with trivial action. Allowing a general coefficient algebra changes the question. Likewise the maximal group completion, defined using all unitary representations, differs from the reduced completion defined by the regular representation.

Three manuscript conclusions

The family asserts the following existence results.

• A finitely generated torsion-free group $G_{\mathrm{inj}}$ has an infinite-order class $h\in K_0(B{G}_{\mathrm{inj}})$ with ${\mu }^r(h)=0$. This is failure of rational injectivity. (OpenAI, 2026)
• A finitely generated torsion-free group $G_{\mathrm{proj}}$ has a scalar projection $e\in C_r^{\ast }(G_{\mathrm{proj}})$ with $0<\tau (e)<1/2$, contradicting the Kadison–Kaplansky projection conjecture. (OpenAI, 2026b)
• A finitely generated group $G_{\mathrm{irr}}$ has a scalar projection of irrational canonical trace, outside the image of reduced assembly, even after rationalization. No torsion-free conclusion is asserted for this group. (OpenAI, 2026c)

These are manuscript claims whose selected analytic mechanisms are explained below. The construction of the groups and the complete proofs are not independently verified here.

🏷️ An index class that survives geometrically

For $G_{\mathrm{inj}}$, the first manuscript constructs an embedding $i:{\mathbb{Z}}^2\to G_{\mathrm{inj}}$ and an integral degree-two cohomology class $c$ whose restriction pairs to one with the oriented torus. Let $L$ be the complex line bundle with first Chern class $c$. The pushed-forward torus spin Dirac class $h=(Bi{)}_{\ast }[D_{T^2}]$ is detected by

$$⟨[L]-[1],h⟩={\int }_{T^2}(Bi{)}^{\ast }c=1.$$

This is the index pairing between $K$-theory and $K$-homology. On the torus, the untwisted Dirac index is zero and the twisted index is the degree of the line bundle. Since a homomorphism to $\mathbb{Z}$ takes $h$ to one, $h$ has infinite order.

Fourier transform identifies $C_r^{\ast }({\mathbb{Z}}^2)$ with continuous functions on the dual torus. Its rank-zero Bott class $\beta$ is a difference of matrix projections of equal rank representing the nontrivial two-dimensional bundle class. Torus assembly sends the Dirac class to this generator, up to the fixed orientation convention. Naturality reduces the analytic conclusion to $i_{\ast }(\beta )=0$ in the reduced algebra. (OpenAI, 2026a)

Bounded families and finite cancellation

The source arranges localized Bott representatives at heights $0,1,2,\dots$. It forms bounded operator families on the even heights, odd heights, positive even heights, and the root height alone. Rotations followed by adjacent profile rescalings on disjoint pairs move the even family to the odd one; the same operations on the shifted pairs move the odd family to the positive even one. Fixed adjacent scale ratios give uniform continuity bounds. The resulting class relations are

$$\begin{array}{rl}{x}_{\mathrm{ev}}& =x_{\mathrm{od}}=x_{\mathrm{ev}+},\\ x_{\mathrm{ev}}& =x_0+x_{\mathrm{ev}+}.\end{array}$$

Subtracting gives $x_0=0$. This uses finite additivity among four actual classes. It does not define an infinite sum of Bott classes in $K_0$.

The difficult analytic task is putting those bounded operators and both homotopies into the reduced algebra. Uniform expansion isolates constant vectors on graph pieces. If $P$ projects onto constants and a normalized incidence operator has centered norm at most $\rho <1$, powers of its adjoint-square suppress nonconstant modes by ${\rho }^{2k}$. The manuscript’s extraction estimate adds deletion loss of order $k\sqrt{{\epsilon }_{\mathrm{del}}}$.

Finite sampled group sums approximate the prescribed profiles in full regular operator norm. After assigning disjoint domains to graph sheets, the unused edges form a forest. A weighted forest adjacency of degree at most $D$ and edge-weight norm at most $w$ has norm at most $2w\sqrt{D}$: root each tree, bound the parent-to-child operator by $w\sqrt{D}$, and add its adjoint. This controls the operator contribution outside the chosen domains.

Height and approximation accuracy are different indices. For accuracy $\nu$, write $A_{\nu }=C_r^{\ast }(G_{\nu })$. The construction works in the norm sequence quotient

$$Q_r=\prod _{\nu }{A}_{\nu }/\underset{\nu }{⨁}{A}_{\nu },$$

where the product consists of bounded sequences and the denominator consists of sequences whose norms tend to zero. Errors vanishing with $\nu$ become exact identities there. Only the source algebra generated by the four profiles and two uniformly continuous paths is shown to map into $Q_r$; an arbitrary bounded model operator need not do so. Matrix sizes and path continuity bounds remain uniform along this accuracy sequence.

Returning to a single group

An equality in the quotient must yield an actual finite witness. A stabilized $K_0$ equality uses finitely many projection and partial-isometry entries. Lift them to bounded sequences, correct approximate projections by continuous functional calculus, and compress an approximate partial isometry into the corrected corners. If $z^{\ast }z$ and $z{z}^{\ast }$ are sufficiently close to the respective corner identities, polar normalization

$$z(z^{\ast }z{)}^{-1/2}$$

gives an exact partial isometry in every sufficiently late coordinate. This supplies Bott vanishing for one group, not merely a limiting assertion.

Finally approximate the finite witness by finitely supported group sums. Their supports, together with the required embedded subgroup generators, generate a finitely generated subgroup. Reduced subgroup inclusions are isometric because the ambient regular representation splits into copies of the subgroup’s regular representation. The same spectral corrections therefore work within that subgroup.

The source also distinguishes its reduced conclusion from maximal assembly: the degree-two detector gives nonvanishing of the maximal image through an external low-degree index theorem. Thus the asserted kernel arises on passing to the reduced completion; it does not establish a counterexample to maximal rational injectivity or to the classical Novikov conjecture. Finite presentation and exactness are not conclusions. (OpenAI, 2026a)

🏷️ From a small matrix projection to a scalar projection

The second manuscript first constructs a finitely generated torsion-free group $\Lambda$ and a finite self-adjoint matrix over $\mathbb{C}[\Lambda ]$ with a separated upper spectral band. Its projection $p$ belongs to $M_N(C_r^{\ast }(\Lambda ))$ and has unnormalized trace $a\in (0,1/2)$. (OpenAI, 2026b)

Spectral separation and trace control require different comparisons. Disjoint assigned graph domains give a bounded direct-sum operator with a uniform upper-band gap. Full norm approximation and a small forest remainder transfer the gap to the regular operator. But the domain assignment may fail to commute with group translations, so that comparison alone does not preserve canonical trace.

For the trace bound, the source instead uses every indexed local upper projection $E_Q$. The restriction map $A\xi =(E_Q(\xi {|}_Q){)}_Q$ is bounded and equivariant. Its target has small equivariant dimension, estimated using weighted local traces, including every matrix index. On $\ker A$, the quadratic form of the global operator lies strictly below its upper band. Thus projection to $(\ker A{)}^{\perp }$ injects the upper spectral space into a space of small trace. Equivariant polar decomposition gives the desired trace bound. Nonvanishing and faithfulness give strict positivity.

The scalar reduction is a separate useful lemma. Enlarge the group to $K=\Lambda \ast {\mathbb{F}}_N$, with free generators $t_1,\dots ,t_N$, and set $U=(\lambda (t_1),\dots ,\lambda (t_N))$. The source proves

$$\begin{array}{c}\Vert U\xi \Vert \ge c_a\Vert \xi \Vert (\xi \in \mathrm{ran}p),\\ c_a=\sqrt{1-a}-\sqrt{a}>0.\end{array}$$

Here is why small trace enters. Set $a_j={\tau }_{\Lambda }(p_{jj})$. On each left $\Lambda$-orbit, the projection identity and Cauchy–Schwarz bound the squared absolute value of a coordinate of a vector in $\mathrm{ran}p$ by $a_j$ times its orbit mass. Split coordinate $j$ according to whether the reduced free-product word begins with $t_j^{-1}$. These are the inputs where multiplication by $t_j$ cancels a letter.

Their total input mass is at most $a\Vert \xi {\Vert }^2$. The other inputs produce disjoint outputs beginning with the respective $t_j$, giving output norm at least $\sqrt{1-a}\Vert \xi \Vert$. Weighted Cauchy–Schwarz and uniqueness of the cancellation preimages bound the cancelling output norm by $\sqrt{a}\Vert \xi \Vert$. The triangle inequality proves the displayed lower bound. Ordinary boundedness of $U$ would not give this sharper cancellation estimate.

Consequently $B=p{U}^{\ast }Up$ is invertible in the corner with identity $p$. Define

$$\begin{array}{rl}V& =Up{B}^{-1/2},\\ V^{\ast }V& =p,e=V{V}^{\ast }.\end{array}$$

Then $e$ is a scalar projection in $C_r^{\ast }(K)$, and traciality gives ${\tau }_K(e)={\tau }_{K,N}(p)=a$. It is neither zero nor one. The free product remains finitely generated and torsion-free. The change of group and the unnormalized trace bound are essential; an arbitrary matrix projection does not already contradict a conjecture about scalar projections. (OpenAI, 2026b)

🏷️ A uniform gap with irrational kernel trace

The third construction has $G_{\mathrm{irr}}=V⋊\Gamma$, where $V$ is a quotient of a finite-field lamp module. Fourier transform on $V$ describes operators as local rules on its compact dual $\Omega$, equipped with Haar probability.

For each $\omega \in \Omega$, the operator becomes

$$T_{\omega }=dI-{\mathrm{Adj}}_{\omega }.$$

The enabled graph has degree at most $d$. Designated finite expander copies of sizes $n_i$ become isolated components when their coordinate pattern is valid. Their positive Laplacian spectra have a common gap $\rho$, and their kernels consist of constant vectors.

Finite linear codes of dimension $k_i=\lfloor n_i/2\rfloor$ provide the pattern laws. Both a code and its dual have large minimum support: the primal condition forces many discrepancies on an invalid copy; the dual condition prevents relations on other copies from creating extra constraints on this copy. The source proves its marginal is exactly uniform on the code, so a prescribed valid pattern has probability $p^{-k_i}$. This is a marginal statement, with no independence assertion across copies. (OpenAI, 2026c)

The estimate on the remaining graph

Gaps on valid components do not control the complement. The key global estimate is that every finite vertex set $F$ in that complement satisfies

$$d|F|-2|E(F)|\ge \frac{c}{2}|F|$$

for a uniform $c>0$. The left side counts boundary edges plus missing degree.

The proof requires simultaneous overlap control. Relative to one root, all relevant earlier intersections of a copy lie in one small intrinsic tree ball. Finite-support cycles decompose uniquely into copy-cycle coordinates. Eliminating these coordinates in root-distance order bounds their dimensions by exterior vertex sets that can be charged disjointly. Girth bounds the extra cycles created by attachments to the tree ball. Euler’s formula then gives the displayed deficit. A collection of pairwise small-overlap estimates would not supply this global count.

To turn it into an operator estimate, put $q(x)=d-\mathrm{deg}(x)\ge 0$. The quadratic form includes both edge differences and missing-degree potential:

$$\begin{array}{rl}⟨T_{\omega }\xi ,\xi ⟩& =\sum _{\{x,y\}\in E}|\xi (x)-\xi (y){|}^2\\ & +\sum _{x}q(x)|\xi (x){|}^2.\end{array}$$

Apply the finite-set deficit to the superlevel sets of $|\xi {|}^2$ and integrate. The resulting edge terms are differences of squares. Cauchy–Schwarz, using $(s+t{)}^2\le 2(s^2+t^2)$ together with the $q$ terms, gives

$$\frac{c}{2}\Vert \xi {\Vert }^2\le \sqrt{⟨T_{\omega }\xi ,\xi ⟩}\sqrt{2d}\Vert \xi \Vert .$$

Thus the complement has gap $c^2/(8d)$. The estimate extends from finitely supported vectors by density. Together with the expander gap, it gives a uniform $\epsilon =\min \{\rho ,c^2/(8d)\}>0$ for every $\omega$. The family of evaluation representations is faithful, so this controls the spectrum in the reduced algebra itself. Hence $b_0=1_{\{0\}}(T)$ belongs to $C_r^{\ast }(G_{\mathrm{irr}})$.

Exact trace and the assembly obstruction

On a valid component of size $n_i$, the constant-vector projection has diagonal entry $1/n_i$. Exactly $n_i$ indexed copies of that type contain the identity vertex, each with validity probability $p^{-k_i}$. Valid copies containing that vertex are mutually exclusive. Integrating the diagonal therefore gives

$$\tau (b_0)=\sum _i{n}_i\frac{1}{n_i}{p}^{-k_i}=\sum _i{p}^{-k_i}.$$

The construction chooses $p>2$ and strictly increasing $k_i$ with $k_{i+1}-k_i\to \infty$. The base-$p$ expansion has ones at these positions and zeros elsewhere. It has infinitely many ones and arbitrarily long intervening zero blocks, so it cannot be eventually periodic. It is irrational.

The manuscript invokes Lück’s trace theorem: traces of degree-zero assembly classes lie in ${\Lambda }_G=\mathbb{Z}[1/|F|:F\le G\text{ finite}]\subseteq \mathbb{Q}$, without assuming assembly is surjective. Therefore $[b_0]$ is outside the assembly image. Finite rational linear combinations of rational traces are still rational, giving the rationalized obstruction as well. The crossed-product description used to construct $T$ does not change the final statement: this is coefficient-free assembly for the whole group $V⋊\Gamma$. (OpenAI, 2026c)

🏷️ Scope of the exposition

The selected review covers the bounded-family cancellation and finite-witness interfaces, the separated-band and scalar-compression arguments, and the finite-pattern, global-gap, and trace calculations. The labeling, pruning, sampling, geometric realization, and uniform parameter constructions behind these interfaces are substantial inputs. Their complete proofs and the cited external index theorems have not been independently checked here.

🏷️ Key steps and proof outlines

Make the infinite construction obey a finite error budget

The injectivity construction has two different limits. Height $i$ runs through infinitely many graph pieces inside one construction; accuracy $\nu$ runs through increasingly accurate constructions. Fix the height-to-height scale ratio first. This supplies a common spectral gap and a common modulus of continuity for the Bott homotopies. For target error ${ϵ}_{\nu }$, choose the extraction power $k_{\nu }$ so the centered error ${\rho }^{2k_{\nu }}$ is small, then choose the deletion fraction so $k_{\nu }\sqrt{{\epsilon }_{\mathrm{del}}}$ is small. Only afterward choose the graph parameters large enough to realize those bounds. (OpenAI, 2026a)

The completion section makes the last step quantitative: after all accuracy-dependent constants are fixed, the remaining geometric errors are bounded by expressions of the form

$$C_{\nu }{2}^{-345H_{0,\nu }},C_{\nu }\frac{2^{-344.5H_{0,\nu }}}{1-2^{-344.5b}}.$$

Here $b$ is the fixed height increment and $H_{0,\nu }$ the initial height. Increasing that initial height controls the entire infinite tower by a geometric series. A separate approximation for each height, without this summability and uniform path continuity, would not produce the asserted quotient homotopy.

At each accuracy only finitely many mesh points of each path are realized. Mesh refinement together with the common continuity modulus supplies the complete path in the norm quotient. Once finite additivity cancels the root Bott class there, a finite stabilized witness is extracted at a sufficiently late coordinate. Thus the quotient serves to organize vanishing errors; the eventual counterexample is one actual group.

Transfer a spectral band without losing its projection

The projection companion uses a useful general compression lemma. Suppose $A=A^{\ast }$, $\Vert A\Vert \le 2$, and a reducing projection $E$ satisfies

$$\Vert (A-I)E\Vert \le a,A{|}_{(1-E)\mathcal{H}}\le (1-b)I.$$

Let $J$ retain the assigned graph domains and assume $\epsilon =\Vert (1-J)E\Vert \le 1/2$. Then $EJE\ge (1-{\epsilon }^2)E$, so

$$V=JE(EJE{)}^{-1/2}$$

is a partial isometry from $E\mathcal{H}$ onto its compressed image. Moreover $\Vert V-E\Vert \le 2\epsilon$. If $E^J=V{V}^{\ast }$, every vector in $(J-E^J)\mathcal{H}$ is orthogonal to $E\mathcal{H}$. Consequently the comparison operator

$$B_J=E^J+(J-E^J)A(J-E^J)$$

has an exact upper eigenvalue one and the same complementary upper bound $1-b$. Estimating its diagonal and off-diagonal blocks gives $\Vert JAJ-B_J\Vert \le a+20\epsilon$. (OpenAI, 2026b)

For a further perturbation of size $u$, a contour of radius $r_0\le b/2$ around one yields a spectral-projection error at most $u/(r_0-u)$. This follows directly by integrating the resolvent identity and its Neumann-series bounds. If $u<r_0/2$, the projections are less than one apart and polar normalization makes them equivalent. Equivariance preserves trace when present; the separate equivariant trace comparison above remains necessary when the domain assignment itself is not equivariant.

An invalid coded component still has a positive deficit

For the irrational-trace construction, consider an invalid copy $P$ of size $n$, with prescribed degree $d$. The code’s minimum distance says that the mismatch set $M$ has $|M|>\eta n$. At each mismatch at least one prescribed incidence is missing. For $B\subseteq P$, the deficit therefore dominates both $|M\cap B|$ and the original edge boundary $|{\partial }_{P}B|$.

If $n-|B|\ge \eta n/2$, expansion gives $|{\partial }_{P}B|\ge (\eta \kappa /2)|B|$, where $\kappa$ is the expansion constant. Otherwise $|M\cap B|>\eta n/2\ge (\eta /2)|B|$. Both cases yield

$$d|B|-2e_{\omega ,P}(B)\ge c|B|,c=\frac{\eta }{2}\min (\kappa ,1).$$

This local estimate is then combined through the source’s cycle-coordinate charging argument, rather than added over overlapping copies. Euler’s identity converts the resulting cycle-rank bound into the global deficit needed for the spectral gap. The three papers therefore have distinct final proof obligations: a finite homotopy witness, an actual small-trace spectral projection followed by scalar compression, and a global complementary gap followed by an exact irrational trace. (OpenAI, 2026c)

See Also

• Matrix functional calculus discusses norm control for functions of operators; here spectral separation makes a discontinuous ambient indicator continuous on the actual spectrum.
• Markov type and superreflexivity provides another setting where graph geometry and uniform analytic estimates interact, with different invariants and conclusions.

🏷️ Manuscripts and verification scope

• A torsion-free counterexample to reduced Baum–Connes injectivity — pinned manuscript and companion source.
• A torsion-free counterexample to the Kadison–Kaplansky projection conjecture — pinned manuscript and companion source.
• An irrational-trace counterexample to reduced Baum–Connes — pinned manuscript and companion source.

No family scope page was present at lean/docs/285.md in the pinned snapshot. This does not establish that all related formalization code is absent. 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 torsion-free counterexample to reduced Baum–Connes injectivity.

🐻  OpenAI 2026b. A torsion-free counterexample to the Kadison–Kaplansky projection conjecture.

🐻  OpenAI 2026c. An irrational-trace counterexample to reduced Baum–Connes.