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Math-117-A2 · revision r1 · on Math-117

OpenAI claim summary for Math-117

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.

Integral negative scalar curvature: The integral estimate rules out nonnegative scalar curvature when real simplicial volume is nonzero. It does not obstruct positive scalar curvature on every aspherical manifold with zero simplicial volume.

Pointwise scalar curvature and width: The explicit the stated result excludes positive scalar curvature on all closed aspherical manifolds of dimension at least four; the standard dimension-two/three cases are included in the surrounding universal-cover discussion.

Positive scalar curvature and inessentiality: The explicit aspherical corollary rules out positive scalar curvature for every closed smooth aspherical manifold, including nonorientable ones; this is the entire recorded obstruction.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

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

Manuscript

Contents
  1. Positive Scalar Curvature on Aspherical Manifolds
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Mathematical claim 2.
  6. Mathematical claim 3.
  7. Outline of the argument
  8. Formal verification
  9. Status caveat
  10. Sources

Positive Scalar Curvature on Aspherical Manifolds

OpenAI's claim

Integral negative scalar curvature: For each $n\ge3$, the manuscript claims a dimension-only positive lower bound on the integral of $(\mathrm{Scal}^-)^{n/2}$ in terms of real simplicial volume; the bound is uniform over closed oriented manifolds and their metrics.

Pointwise scalar curvature and width: The paper claims dimension-only control of the full fibers of a continuous map from any complete scalar-positive $n$-manifold to an $(n-2)$-complex, for $n\ge4$; it also gives the stated macroscopic-dimension consequence on universal covering spaces when the closed base admits a positive-scalar-curvature metric.

Positive scalar curvature and inessentiality: OpenAI claims that positive scalar curvature forces the rational fundamental class to vanish under the classifying map, without spin or group restrictions; consequences include the aspherical obstruction, flatness rigidity at nonnegative scalar curvature, and zero real simplicial volume in the nonnegative case.

Mathematical statement of the claimed result

Mathematical claim 1.
For each integer $n\ge3$, some dimension-dependent $a_n>0$ is claimed to satisfy \[\int_M\left(\max\{0,-\operatorname{Scal}_g\}\right)^{n/2}\,dV_g\ge a_n\|M\|\] for each smooth metric $g$ on a closed smooth $n$-manifold $M$ which is oriented and connected. Here $\|M\|$ is the infimum of the sum of absolute coefficients over finite real singular cycles representing its fundamental class. Thus nonnegative scalar curvature forces $\|M\|=0$, but this implication alone does not treat all aspherical manifolds with vanishing simplicial volume.
Mathematical claim 2.
For a closed connected oriented smooth $n$-manifold, $n\ge2$, admitting a metric with $\operatorname{Scal}>0$, the classifying map $c_M:M\to B\pi_1(M)$ is claimed to satisfy \[(c_M)_*[M]=0\in H_n(B\pi_1(M);\mathbb Q).\] Its explicit aspherical corollary rules out positive scalar curvature on every closed smooth aspherical manifold, including nonorientable ones by passage to an orientable cover.
Mathematical claim 3.
For every $n\ge4$ there is $C_n<\infty$ such that a connected complete smooth boundaryless $n$-manifold with $\operatorname{Scal}_g\ge1$ has a continuous map $f:M\to K$, where $K$ is a simplicial complex with the usual polyhedral topology and $\dim K\le n-2$, with \[\sup_{y\in K}\operatorname{diam}_g f^{-1}(y)\le C_n.\] Entire fibers, rather than their individual components, are bounded. For smooth Riemannian $(M,g)$ of dimension $n\ge4$, assume $M$ is closed and connected and $\operatorname{Scal}\ge\sigma^2>0$, the corollary gives $\operatorname{FillRad}(M,g)\le C_n/(2\sigma)$. A separate stated corollary covers all closed positive-scalar-curvature dimensions $n\ge2$: the universal-cover geometry obtained by lifting that metric has continuous macroscopic dimension at most $n-2$. The filling radius uses the integral fundamental class when the manifold is orientable and its mod-two fundamental class otherwise.

Outline of the argument

Integral negative scalar curvature: The companion vanishing result forces a negative Yamabe minimizer, which is normalized to scalar curvature minus one; Hölder then reduces the desired integral estimate to a volume bound. A bounded cocycle is represented through an equivariant probability map from the cover. Weighted graph deformations increase map energy while retaining the volume budget, and a coordinate-by-coordinate degree estimate pays for the detecting cocycle, yielding the uniform comparison.

Pointwise scalar curvature and width: A separated net supplies labeled inner and outer balls in the original metric. Compatible walls and inserted cylinders are constructed while a nearby collapse map keeps track of those original positions. The final walls have no intersection of $n-1$ different labels, so their sign coordinates define an $(n-2)$-dimensional map; shared coordinates place the entire fiber in one uniformly bounded original ball.

Positive scalar curvature and inessentiality: An essential rational class is detected on an auxiliary cover at a finite word-metric scale. Compactly supported cohomology and relative duality produce a finite-dimensional comparison with a nonzero local degree. Transverse realization and scalar-curvature reduction contradict that degree; the nonnegative aspherical rigidity statement then uses splitting after the positive-curvature obstruction is established.

Formal verification

Integral negative scalar curvature: The pinned catalogue lists no Lean formalization for this result.

Pointwise scalar curvature and width: The pinned catalogue lists no Lean formalization for this result.

Positive scalar curvature and inessentiality: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources