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Math-2665-A1 · revision r1 · on Math-2665

OpenAI claim summary for Math-2665

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: 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.

Harmonic growth comparison in dimension three: The comparison is restricted to dimension three but still assumes only nonnegative Ricci curvature. A fixed-factor excess at a chosen large integer degree violates the claimed Euclidean count, even though the constructed metric may depend on that degree.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

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

Manuscript

Contents
  1. Harmonic dimension comparison in dimension three
  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

Harmonic dimension comparison in dimension three

OpenAI's claim

Harmonic growth comparison in dimension three: The companion claims nonnegatively Ricci-curved metrics on $\mathbb R^3$ with more degree-$k$ harmonic growth modes than the Euclidean count by a fixed factor. The metric may depend on the large integer $k$, and prescribed asymptotic volume ratios and small global bi-Lipschitz distortion are available.

Mathematical statement of the claimed result

Mathematical claim 1.
For every $4/9<v<1$ and $1<c<9v/4$, all sufficiently large integers $k$ are claimed to admit a complete smooth $g_k$ on $\mathbb R^3$, Euclidean near the origin, with $\operatorname{Ric}_{g_k}\ge0$, asymptotic volume ratio $v$, and \[h_k(\mathbb R^3,g_k)\ge c(k+1)^2>(k+1)^2=h_k(\mathbb R^3,g_E).\] Here $h_k$ is the dimension of harmonic functions with growth $O((1+d(0,x))^k)$, and asymptotic volume ratio is $\lim_{R\to\infty}\operatorname{vol}B(0,R)/(\omega_3R^3)$. The metric is chosen after $k$, is Ricci-positive away from some compact region, and its pointed tangent cones at infinity include uncountably many isometry types.

Outline of the argument

Harmonic growth comparison in dimension three: A finite angular program is chosen before the radial tail and starting scale. Spectral rates are averaged by successively crossing cone geometries, then matched through the radial harmonic system. The transfer is controlled for that fixed program, producing enough independent global modes with the required pointwise integer growth; no uniform estimate in the growing degree is assumed.

Formal verification

Family 361 describes Lean coverage for an integer-growth dimension obstruction in dimension sixteen at a fixed degree; the three-dimensional construction is separate.

Harmonic growth comparison in dimension three: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

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