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Math-2664 · mathematics · Geometry

Euclidean comparison for integer-growth harmonic dimensions

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every complete smooth nonnegative-Ricci n-manifold and integer $k\geq 0$, must the real vector space of harmonic functions bounded in absolute value by some function-dependent constant times (1+distance from a fixed $point)^k$ have dimension at most that space on Euclidean $R^{n}$?

Short English statement

For every complete smooth nonnegative-Ricci n-manifold and integer $k\geq 0$, must the real vector space of harmonic functions bounded in absolute value by some function-dependent constant times (1+distance from a fixed $point)^k$ have dimension at most that space on Euclidean $R^{n}$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2664-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2664

Higher-dimensional harmonic-growth obstruction: The comparison concerns the exact integer growth degree under nonnegative Ricci curvature. A single metric with more independent modes than the Euclidean count suffices to refute it; unique tangent geometry and stronger sectional-curvature assumptions are not imposed here.

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

Euclidean comparison for integer-growth harmonic dimensions

OpenAI's claim

Higher-dimensional harmonic-growth obstruction: The first paper claims an even-dimensional complete metric on Euclidean space with nonnegative Ricci curvature and more integer-growth harmonic functions than Euclidean harmonic polynomials. A neighborhood of the origin keeps the Euclidean metric, whereas the volume ratio at infinity is strictly positive and smaller than one.

Mathematical statement of the claimed result

Mathematical claim 1.
The claim gives some even $n\ge8$, integer $k\ge2$ and complete smooth metric $g$ on $\mathbb R^n$ with $\operatorname{Ric}_g\ge0$ and \[h_k(\mathbb R^n,g)>h_k(\mathbb R^n,g_E).\] The space counted by $h_k$ consists of harmonic functions growing at most as a constant times $(1+d(0,x))^k$. The metric agrees with the Euclidean one in an origin neighborhood, while $0<\operatorname{AVR}(g)<1$. Thus the proposed Euclidean integer-growth dimension comparison fails at some $(n,k)$.

Outline of the argument

Higher-dimensional harmonic-growth obstruction: Berger-link spectra provide a finite surplus of candidate modes. A real control scheme realizes successive angular choices by a complete Ricci-nonnegative metric and tunes full-period transmission matrices exactly. Invertible regular fundamental solutions and degree-controlled radial growth yield independent smooth harmonic functions, while changing link limits gives nonunique tangent cones.

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.

Higher-dimensional harmonic-growth obstruction: The manuscript links HarmonicGrowth.lean (declaration OAI.HarmonicCounterexample.main).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

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

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