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Math-2950-A1 · revision r1 · on Math-2950

OpenAI claim summary for Math-2950

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.

The uniform estimate on all normalized univalent maps implies $B_b(-1)<1/4$ for bounded maps. Affine normalization changes an integral mean only by a constant factor, leaving its limiting growth exponent unchanged, so the disk-to-disk equality $B_b(-1)=1/4$ in this question is claimed false.

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

Manuscript

Contents
  1. The inverse-first-power universal integral-means spectrum
  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

The inverse-first-power universal integral-means spectrum

OpenAI's claim

Inverse first-power means: OpenAI announces a universal growth exponent below $1/4$ for circular means of the inverse derivative of normalized injective holomorphic disk maps. Applied to bounded maps, this gives a strict inequality at the spectrum parameter $-1$, contrary to Kraetzer’s proposed value.

Mathematical statement of the claimed result

Mathematical claim 1.
Write $\mathcal S=\{f:\mathbb D\to\mathbb C:\ f\text{ holomorphic and injective},\ f(0)=0,\ f^\prime(0)=1\}$ and $M_p(f,r)=(2\pi)^{-1}\int_{-\pi}^{\pi}|f^\prime(re^{i\theta})|^p\,d\theta$. The claim is that some $0<\epsilon<1/4$ and finite $C$, independent of $f$ and $r$, satisfy \[M_{-1}(f,r)\le C(1-r)^{-1/4+\epsilon}\quad(f\in\mathcal S,\ 1/2\le r<1).\] In particular the bounded-image subclass has spectrum $B_b(-1)=\sup_f\limsup_{r\uparrow1}\log M_{-1}(f,r)/\log(1/(1-r))<1/4$. The theorem bounds this exponent; it does not determine its exact value.

Outline of the argument

Inverse first-power means: Affine rebasings of upper-half-plane maps define a positive averaging operator. Hypothetically excessive growth produces a scale-compatible weighted probability law and identities for logarithmic-derivative moments. Critical points of a target potential are paired geometrically; their signed Jacobians have an incompatible expected sign, and the equality case is separately eliminated.

Formal verification

Family 072 describes Lean coverage for the Brennan exponent interval, inverse-square means, and the strict inverse-first-power spectrum comparison.

Inverse first-power means: The manuscript links Brennan.lean (declaration OAI.Brennan.main_theorem), BrennanSharp.lean (declaration OAI.Brennan.Sharp.sharp_endpoints), StrictMeans.lean (declaration OAI.StrictInverseFirstPower.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