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Math-1832-A1 · revision r1 · on Math-1832

OpenAI claim summary for Math-1832

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 at unsolvedmath.com: partially solved. 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 witness is one group with finite generators carrying two essentially free actions preserving probability. Their costs lie in disjoint ranges, so the cost cannot be independent of the action as requested. This is presented as a claimed negative answer.

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

Manuscript

Contents
  1. Some Questions — Question 41
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Some Questions — Question 41

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. A finitely generated amalgam is claimed to have free probability-preserving actions of different costs, refuting fixed price.

Mathematical statement of the claimed result

Result 1.
Let $A=F(a,b_1,\ldots,b_{99})$, $w=ab_1ab_2\cdots ab_{99}a$, $J=\langle b_1,\ldots,b_{99},w\rangle$, and \[ \Gamma=A*_J(J\times\langle t\rangle),\qquad\chi(a)=1,\quad\chi(b_i)=\chi(t)=0. \] Let $X$ be the free conull part of the Bernoulli shift $[0,1]^\Gamma$, and $Y_M=X\times\mathbb Z/M\mathbb Z$ with action $(x,r)g=(xg,r+\chi(g))$. Both carry their natural probability measures. Set $K=e^{96}96^{99}/95^{95}$, choose $0<\alpha<1/200$ with $K\alpha^3<1/2$, and put $\eta=\alpha/100$. The asserted orbit-equivalence-relation costs satisfy \[ \operatorname{Cost}(\Gamma\curvearrowright X)\ge1+\eta,\qquad \operatorname{Cost}(\Gamma\curvearrowright Y_M)\le1+99/M\quad(M>100). \] An integer $M>\max\{100,99/\eta\}$ therefore separates the costs of two essentially free probability-preserving actions of the same group.

Outline of the argument

Finite-height extensions of the amalgam yield graphings with costs approaching one. A relative graphing-compression inequality and Bernoulli finite-permutation models give a separate cost lower bound, supported by a planar word lemma; the two action constructions therefore have unequal costs.

Formal verification

Family 259 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources