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Math-281-A2 · revision r1 · on Math-281

OpenAI claim summary for Math-281

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.

Power-law failure of nodal upper bounds: One fixed smooth connected closed metric and eigenfunctions of unbounded positive eigenvalue have unbounded nodal-measure-to-square-root-eigenvalue ratio. This directly negates the universal metric-dependent-constant assertion in our definition.

Nodal upper length on surfaces: The sharp nodal upper bound holds for all smooth closed connected surfaces, while the recorded target ranges over all smooth closed dimensions n at least two.

Nodal upper-bound failures in dimensions three and four: One fixed smooth connected closed metric and eigenfunctions of unbounded positive eigenvalue have unbounded nodal-measure-to-square-root-eigenvalue ratio. This directly negates the universal metric-dependent-constant assertion in our definition.

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. Sharp Upper Bound in Yau's Conjecture on Nodal Sets of Laplace Eigenfunctions
  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

Sharp Upper Bound in Yau's Conjecture on Nodal Sets of Laplace Eigenfunctions

OpenAI's claim

Power-law failure of nodal upper bounds: The manuscript claims a fixed smooth metric on $S^4\times S^1$ with eigenfunctions whose nodal four-volume exceeds the square-root scale by a fixed positive power of the eigenvalue; this rules out even an arbitrary-small-power-loss version of the proposed general upper bound.

Nodal upper length on surfaces: The paper claims nodal length at most $C\sqrt\lambda$ for every real nonzero Laplace eigenfunction on a fixed smooth closed surface; combined with the established lower estimate, this settles the two-dimensional smooth setting.

Nodal upper-bound failures in dimensions three and four: The companion claims fixed smooth metrics on $S^3$ and $S^2\times\mathbb T^2$ with unbounded nodal-measure-to-$\sqrt\lambda$ ratios along exact eigenfunction sequences; the three-sphere metric may be chosen as close to round as prescribed in the smooth topology.

Mathematical statement of the claimed result

Mathematical claim 1.
On one fixed smooth metric $g$ on $S^4\times S^1$, the claim gives $\epsilon_0>0$ and nonzero real smooth eigenfunctions $u_j$ with \[-\Delta_gu_j=\lambda_ju_j,\qquad\lambda_j\to\infty,\qquad\frac{\mathcal H_g^4(\{u_j=0\})}{\lambda_j^{1/2+\epsilon_0}}\to\infty.\] Thus even a fixed positive power beyond the square-root nodal upper scale is exceeded along this sequence.
Mathematical claim 2.
Fix a smooth metric $g$ on a closed connected surface $M$. One finite $C(M,g)$ is asserted to work simultaneously for all nonzero real eigenfunctions solving $-\Delta_gu=\lambda u$, $\lambda>0$, satisfies \[\mathcal H_g^1(\{u=0\})\le C(M,g)\sqrt\lambda.\] The constant is shared by all eigenfunctions in all positive eigenspaces; the assertion is specific to surfaces.
Mathematical claim 3.
Every $C^\infty$ neighborhood of the round metric on $S^3$ is claimed to contain a smooth metric $g_\infty$ and real eigenfunctions $u_j$ with \[-\Delta_{g_\infty}u_j=\lambda_ju_j,\quad\lambda_j\to\infty,\quad\mathcal H_{g_\infty}^2(\{u_j=0\})/\sqrt{\lambda_j}\to\infty.\] The metric is fixed for the sequence. Taking its product with a fixed circle and extending $u_j$ constantly along that factor gives the stated corresponding four-dimensional failure.

Outline of the argument

Power-law failure of nodal upper bounds: A logarithmic envelope on the four-sphere is corrugated through a number of generations proportional to the logarithm of frequency, producing a power gain in its gradient integral. Complex phase and transport jets create highly accurate localized waves beneath that envelope. Exact metric and spectral corrections then realize genuine eigenfunctions while preserving the growing nodal score on one limiting metric.

Nodal upper length on surfaces: Subharmonic profile limits control local growth under affine subdivision. High-growth excursions along grid paths produce witness regions on which growth persists to terminal scale. A packing argument pays for witness regions outside their entry squares using summable collars, bounding the terminal average growth and hence nodal length.

Nodal upper-bound failures in dimensions three and four: Each construction starts with explicit high-frequency background modes and an untouched region that anchors spectral control. Corrugated logarithmic envelopes and finite complex-wave jets create quasimodes with a stable lower score from opposite-sign tests. Metric corrections make the waves exact, and a persistence argument passes to one fixed smooth metric without losing the score.

Formal verification

Family 350 describes Lean coverage for the surface upper bound and the three- and four-dimensional fixed-metric violations; the five-dimensional power-law refinement is separate.

Power-law failure of nodal upper bounds: The manuscript links NodalLength.lean, SmoothYau.lean (declaration OAI.YauCounterexamples.sphere_three), SmoothYau.lean (declaration OAI.YauCounterexamples.sphere_two_torus_two), YauCounterexample.lean (declaration OAI.Yau.Target.yau_nodal_set_upper_bound_counterexample).

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

Nodal upper length on surfaces: The manuscript links NodalLength.lean, SmoothYau.lean (declaration OAI.YauCounterexamples.sphere_three), SmoothYau.lean (declaration OAI.YauCounterexamples.sphere_two_torus_two), YauCounterexample.lean (declaration OAI.Yau.Target.yau_nodal_set_upper_bound_counterexample).

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

Nodal upper-bound failures in dimensions three and four: The manuscript links NodalLength.lean, SmoothYau.lean (declaration OAI.YauCounterexamples.sphere_three), SmoothYau.lean (declaration OAI.YauCounterexamples.sphere_two_torus_two), YauCounterexample.lean (declaration OAI.Yau.Target.yau_nodal_set_upper_bound_counterexample).

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