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Math-1608-A1 · revision r1 · on Math-1608

OpenAI claim summary for Math-1608

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 315, 2026-09-25): The Euler and signature corollaries prove chi>=|signature| and nonnegativity for aspherical four-manifolds, but do not determine every possible Euler characteristic or the least one in the hyperbolic class.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Kirby Problem 4.19
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Kirby Problem 4.19

OpenAI's claim

The Singer vanishing assertion is claimed in dimension four, including finite aspherical integral Poincare complexes and nonorientable cases: universal-cover $L^2$-Betti numbers vanish in every degree except two. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $Q$ be an integral Poincaré complex that is aspherical, connected and finite, with formal dimension four, allowing an arbitrary orientation character. OpenAI claims \[ b_j^{(2)}(\widetilde Q)=0\ (j\ne2),\qquad \chi(Q)=b_2^{(2)}(\widetilde Q)\ge0. \] Here $b_j^{(2)}$ is the von Neumann dimension of the homology of $\mathcal N\pi_1(Q)\otimes_{\mathbb Z\pi_1(Q)}C_*(\widetilde Q)$. The claims include closed connected aspherical topological four-manifolds; zero Euler characteristic is equivalent to vanishing of every $L^2$-Betti number. For oriented such manifolds $M$, the manuscript additionally claims $\chi(M)\ge|\sigma(M)|$, where $\sigma$ is the real intersection-form signature. These are dimension-four conclusions.

Outline of the argument

Manuscript 1 (family 315, 2026-09-25): Positive first $L^2$ cohomology would supply a bounded finite-energy function on a Cayley graph. Its translates define a boundary with controlled approach paths, and annular separation arguments organize cut points into dendritic blocks. Poincare duality imposes dimension and cohomology constraints excluding circles, so a nondegenerate block would be a dendrite with a forbidden cut point.

Formal verification

Family 315 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