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Math-208-A2 · revision r1 · on Math-208

OpenAI claim summary for Math-208

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.

Both selected constructions give a torsion-free group and nonzero finite sums over $\mathbb F_2$ whose product vanishes. In the second construction, $ab=1$ ensures $a\ne0$, while the source states $ac=0$ and $c\ne0$. Thus zero divisors can exist in the allowed class, providing a full disproof of the universal conjecture.

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

Manuscript

Contents
  1. Kaplansky's Zero-Divisor Conjecture for Torsion-Free Groups
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Manuscript 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Kaplansky's Zero-Divisor Conjecture for Torsion-Free Groups

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: A finitely presented torsion-free group is claimed to have nonzero elements $a,c\in\mathbb F_2[G]$ with $ac=0$, despite admitting a finite two-dimensional classifying complex; Another construction claims a torsion-free, finitely presented $G$ for which $\mathbb F_2[G]$ contains $a,b$ satisfying $ab=1$ and $ba\ne1$. The announced examples provide an affirmative answer to the existence question and refute the universal zero-divisor conjecture.

Mathematical statement of the claimed result

Manuscript 1.
For some torsion-free, finitely presented group $G$ admitting a finite two-dimensional classifying space, the source claims \[ \exists\alpha,\beta\in\mathbb F_2[G]\setminus\{0\},\qquad\alpha\beta=0. \] The coefficients are in the field with two elements. This is a zero-divisor example, without a claim here about integral units.
Manuscript 2.
The construction supplies a torsion-free, finitely presented group $G$, with a finite two-dimensional $K(G,1)$, and elements of its characteristic-two group algebra satisfying \[ a,b,c\in\mathbb F_2[G],\qquad ab=1,\quad ac=0,\quad c\ne0,\quad ba\ne1. \] In particular, $a\ne0$ and $e=ba$ obeys $e^2=e$, $e\notin\{0,1\}$. Thus this same example supplies failures of direct finiteness, absence of zero divisors, and triviality of scalar idempotents. These assertions concern $\mathbb F_2[G]$; the separate integral-unit assertion is not a consequence.

Outline of the argument

Finite labeled graphs encode supports whose product coefficients cancel modulo two. A random-label counting argument excludes planar path arrangements; cone surgeries turn that exclusion into asphericity and root separation, so the coefficients remain nonzero in a torsion-free group.

An unequal-side parity pattern supplies the one-sided inverse relation in labeled graphs. Weighted counts handle repeated edge traversals, and a separator reduces arbitrarily large planar arrangements to bounded path collections; cone attachment and root separation then preserve the inverse defect while proving torsion-freeness.

Formal verification

For family 196, the documented Lean theorem includes the torsion-free characteristic-two zero product, finite presentation, and a finite two-dimensional classifying model. The listed Comparator files are TorsionFreeZeroDivisors.lean. Named declarations include OAI.TorsionFreeZeroDivisors.main. For family 197, the documented Lean entries include the scalar inverse defects in the specified characteristics and an operator-invertible determinant counterexample. The characteristic-two search recipe is included, but the nonsofic consequence, the extra spectral-integral conclusion, and the later torsion-free construction are outside the selected statements. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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