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Math-2703-A2 · revision r1 · on Math-2703

OpenAI claim summary for Math-2703

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 result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 328; 2026-09-24).

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

Manuscript

Contents
  1. Nonexpansive fixed points in reflexive Banach spaces
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Nonexpansive fixed points in reflexive Banach spaces

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 328; 2026-09-24), the asserted result is that reflexivity of a real Banach space $X$ suffices to ensure a fixed point for every nonexpansive selfmap on a nonempty bounded closed convex part of $X$, using its original norm.

Mathematical statement of the claimed result

Manuscript 1.
For every real reflexive Banach space $X$, every nonempty norm-closed bounded convex $C\subseteq X$, and every nonexpansive self-map $F:C\to C$, the asserted implication is \[ (\forall a,b\in C:\|F(a)-F(b)\|\le\|a-b\|) \quad\Longrightarrow\quad\exists x\in C:\ F(x)=x. \] No compactness of $C$ in the norm topology is assumed.

Outline of the argument

For manuscript 1 (family 328; 2026-09-24), weak compactness provides a minimal invariant set and an adaptive anchor. Compact families of resolvent predictions are organized on an ordered tree; separating functionals and weighted leaf switching force the displacement to vanish.

Formal verification

For manuscript 1 (family 328; 2026-09-24), OpenAI lists Lean statements for this assertion: reflexivity of a real Banach space $X$ suffices to ensure a fixed point for every nonexpansive selfmap on a nonempty bounded closed convex part of $X$, using its original norm.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are ReflexiveFixedPoints.lean.

Their named declarations include OAI.ReflexiveFixedPoints.exists_fixedPoint_of_canonicallyReflexive.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources