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Math-323-A2 · revision r1 · on Math-323

OpenAI claim summary for Math-323

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: 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 main theorem constructs an infinite ordinary finitely presented torsion group, with all quantifiers on the same group and no uniform exponent required. Within these assumptions, this is a claimed negative answer to the stated question.

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

Manuscript

Contents
  1. Finiteness of Finitely Presented Periodic Groups
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Result 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Finiteness of Finitely Presented Periodic Groups

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: An infinite ordinarily finitely presented periodic group and an infinite-dimensional finitely presented nil algebra over $\mathbb F_2$ are claimed to exist; The periodic group of the companion is claimed residually finite, with an infinite finite-index finitely presented subgroup whose element orders are finite powers of two and unbounded. Together they give a claimed negative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
For a constructed unital associative algebra $R$ over $\mathbb F_2$, the Steinberg group $G=\operatorname{St}_{12}(R)$ is claimed to satisfy \[ |G|=\infty,\qquad G\text{ finitely presented},\qquad \forall g\in G\ \exists n\ge1:\ g^n=1. \] The companion corollary also assigns $G$ Kazhdan property $(T)$, hence nonamenability. Periodicity here permits element orders to vary.
Result 2.
The source claims existence of a group $G$ with an ordinary finite presentation such that \[ |G|=\infty,\qquad G\text{ residually finite},\qquad \forall g\in G\ \exists k\ge0:\ g^{2^k}=1. \] Thus all elements have finite order, and each order is a power of two; no common bounded exponent is asserted.

Outline of the argument

Finite local rules encode a hierarchy and a faithful computation tableau in admissible words. The resulting graded algebra has nil behavior without global nilpotence; a Steinberg-type group construction then transports these finite relations into an infinite group while forcing each element to have finite order.

Phases of long graded words make finite-degree truncations detect group elements. Faithful Steinberg strips, cyclic-shift comparisons, and long-entry commutator calculations treat the central kernel as well, so residual finiteness survives passage to the periodic two-group subgroup.

Formal verification

For family 247, the selected entry gives an infinite ordinarily finitely presented periodic group. A uniform exponent, the nil-algebra conclusions, and the later residual-finiteness strengthening are not part of it. The listed Comparator files are PeriodicGroup.lean. The inventory does not supply a named declaration for these Comparator entries. 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