OpenAI claim summary for Math-37

Baum-Connes Conjecture

OpenAI's claim

Manuscript 1 (family 285, 2026-09-23): For one finitely generated group without torsion, the coefficient-free reduced assembly map annihilates an infinite-order degree-zero K-homology class; this is claimed to refute rational injectivity. Manuscript 2 (family 285, 2026-09-23): OpenAI claims the existence of a nontrivial scalar projection inside the reduced C*-algebra of a finitely generated torsion-free group, contradicting the Kadison--Kaplansky assertion. Manuscript 3 (family 285, 2026-09-23): An irrational-trace scalar projection is constructed for a finitely generated discrete group, yielding a K_0 class that reduced Baum--Connes assembly with trivial coefficients cannot reach.

Mathematical statement of the claimed result

Mathematical claim 1.

OpenAI claims a discrete group, torsion-free and finitely generated, $G$, an injection $i:\mathbb Z^2\hookrightarrow G$, and $c\in H^2(BG;\mathbb Z)$ such that \[ \langle(Bi)^*c,[T^2]\rangle=1,\qquad i_*\beta=0\text{ in }K_0(C_r^*(G)), \] here $\beta\in K_0(C_r^*(\mathbb Z^2))$ is the Bott generator chosen to have rank zero. The class $h=(Bi)_*[D_{T^2}]\in K_0(BG)$ has infinite order but satisfies $\mu_G^r(h)=0$. Here $\mu_G^r:K_*(BG)\to K_*(C_r^*(G))$ is reduced coefficient-free analytic assembly and $[D_{T^2}]$ is the oriented torus spin-Dirac class. This asserts failure of reduced assembly injectivity without coefficients, even rationally.

Mathematical claim 2.

OpenAI claims a discrete group, torsion-free and finitely generated, $G$ whose reduced group $C^*$-algebra contains a scalar projection $e=e^*=e^2$ with \[ 0<\tau_G(e)<\tfrac12,\qquad\tau_G(x)=\langle x\delta_1,\delta_1\rangle. \] Thus $e$ is neither $0$ nor $1$. This concerns a scalar projection in the operator-norm completion of the left regular group algebra, and supplies a counterexample to the torsion-free projection assertion. The manuscript also uses the integral trace restriction on reduced assembly for torsion-free groups to place this projection class outside the assembly image.

Mathematical claim 3.

For some finitely generated discrete group $G$, OpenAI claims a projection $b_0$ in its reduced group $C^*$-algebra satisfying \[ \tau_G(b_0)\notin\mathbb Q,\qquad [b_0]\notin\mu_0^G\left(K_0^G(\underline EG)\right)\subset K_0(C_r^*(G)). \] Here $\tau_G$ is the canonical trace, $\underline EG$ the universal proper $G$-space and $\mu_0^G$ reduced coefficient-free analytic assembly. The obstruction uses the rational trace of every assembly class. No torsion-free hypothesis is asserted for this group.

Outline of the argument

Manuscript 1 (family 285, 2026-09-23): The construction embeds a torus subgroup with an integral cohomology detector while arranging disappearance of its Bott class analytically. Operators assembled on almost disjoint Cayley-graph sheets retain controlled spectral supports; estimates for the leftover forest permit the required homotopies in the reduced completion. A finite witness then restricts the construction to a finitely generated subgroup.

Manuscript 2 (family 285, 2026-09-23): A magnetic tower model produces a separated spectral band carrying a matrix projection of positive trace below one half. Graph sampling and small-cancellation recovery preserve this band after the model is represented on the group, with intersection losses bounded by trace estimates. A free-product compression converts the matrix projection into a scalar one while retaining torsion-freeness.

Manuscript 3 (family 285, 2026-09-23): Finite-field rules generate graph components on which a group-ring operator has kernels of explicitly counted pattern probabilities. Expander estimates separate those kernels from the remaining spectrum, putting their projection in the reduced C*-algebra rather than merely the von Neumann completion. Its irrational trace conflicts with the rational trace values available to assembly classes.

Formal verification

Family 285 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

• OpenAI, A torsion-free counterexample to reduced Baum–Connes injectivity, 2026-09-23; citation key OAI:A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, A torsion-free counterexample to the Kadison–Kaplansky projection conjecture, 2026-09-23; citation key OAI:A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, An irrational-trace counterexample to reduced Baum–Connes, 2026-09-23; citation key OAI:An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main; Statement source: thm:trace-image.
• OpenAI catalogue entry.