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Math-2717 · mathematics · Group theory

Coefficient-preserving solutions of nonsingular group-equation systems

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every group G and finite list $w_{1},\dots ,w_m\in G*F_n,1\leq m\leq n$, whose exponent-sum matrix has row rank m over Q, is $G\to (G*F_n)/\langle \langle w_{1},\dots ,w_m\rangle \rangle $ injective; equivalently, are the equations simultaneously soluble in an overgroup containing G?

Short English statement

For every group G and finite list $w_{1},\dots ,w_m\in G*F_n,1\leq m\leq n$, whose exponent-sum matrix has row rank m over Q, is $G\to (G*F_n)/\langle \langle w_{1},\dots ,w_m\rangle \rangle $ injective; equivalently, are the equations simultaneously soluble in an overgroup containing G?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2717-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2717

The October theorem has arbitrary G, 1<=m<=n and rational row rank m, and explicitly proves coefficient injectivity and equivalent simultaneous overgroup solvability. lean/docs/256.md only links the earlier one-variable unimodular paper, not the systems theorem. Within these assumptions, this is a claimed affirmative answer to the stated question.

Coefficient-preserving solutions of nonsingular group-equation systems

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Any finite equation system over a group with full-rank exponent-sum matrix is claimed simultaneously solvable after embedding into a larger group.

Mathematical statement of the claimed result

Result 1.
Let $G$ be any group, $1\le m\le n$, and $w_i\in G*F_n$. For the exponent-sum matrix $E=(e_{ij})$, where $e_{ij}$ counts the total exponent of the $j$-th free generator in $w_i$, assume $\operatorname{rank}_{\mathbb Q}E=m$. The source claims injectivity of \[ G\longrightarrow (G*F_n)/\langle\!\langle w_1,\ldots,w_m\rangle\!\rangle. \] Equivalently, all equations $w_i=1$ can be solved simultaneously in an overgroup without identifying distinct coefficient elements of $G$.

Outline of the argument

Simultaneous unitary fixed-space constructions are measured by a traced spectral phase. Subadditivity and weighted-cycle calculations rule out the planar coefficient obstruction associated with a nonsingular system, allowing the coefficient group to embed in the solution overgroup.

Formal verification

For family 256, the selected statement is coefficient injectivity for a single relator of exponent sum $1$ or $-1$ in $A*\mathbb Z$. It underlies the Kervaire consequence but does not include the newer general nonsingular-system theorem. 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

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