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Math-2586-A1 · revision r1 · on Math-2586

OpenAI claim summary for Math-2586

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.

Manuscript 1 (family 057, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

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

Manuscript

Contents
  1. Two-step linear images of special quasi-projective groups
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Two-step linear images of special quasi-projective groups

OpenAI's claim

Every complex linear representation of a smooth special quasi-projective fundamental group is claimed to have virtually two-step-nilpotent image, while the paper also constructs nonspecial general quasi-Albanese fibers. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For a smooth complex quasi-projective $V$, specialness requires the following after any proper birational modification: each dominant algebraic fibration onto a normal quasi-projective base $Z$ with $\dim Z>0$ and connected general fiber has $\kappa(Z,\Delta_f)<\dim Z$, where $\Delta_f$ denotes the induced orbifold boundary. The orbifold dimension is computed on smooth compactifications with infimum multiplicities, and removed divisors have infinite multiplicity. For every connected such $V$, every $d\ge1$ and every $\rho:\pi_1(V)\to\operatorname{GL}_d(\mathbb C)$, OpenAI claims \[ \exists H\le\rho(\pi_1V):\quad[\rho(\pi_1V):H]<\infty,\qquad[H,[H,H]]=1. \] Thus every complex linear image is virtually nilpotent of class at most two. Virtual abelianity of the entire open fundamental group is not the stated conclusion.

Outline of the argument

Manuscript 1 (family 057, 2026-09-24): At each canonical nilpotent stage, the fiber Lie algebra equals the commutator algebra. Its abelianization has only weight minus two, making the next conjugation bracket incompatible with the allowed weights; universality then transfers the resulting class-two bound to arbitrary unipotent representations.

Formal verification

Family 057 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources