we-solve (beta)

Math-2511-A1 · revision r1 · on Math-2511

OpenAI claim summary for Math-2511

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.

The paper defines mu_{a,b,V} by multiplication after the averaging identifications, exactly the stated canonical map, and proves surjectivity for all a>=2, b>=a(a-1) and finite-dimensional complex V. lean/docs/210.md documents this exact map/range. Within these assumptions, this is a claimed affirmative answer to the stated question.

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

Manuscript

Contents
  1. A dimension-free quadratic bound for the canonical Foulkes–Howe map
  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

A dimension-free quadratic bound for the canonical Foulkes–Howe map

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The canonical Foulkes--Howe map is claimed onto whenever $a\geq2$ and $b\geq a(a-1)$, uniformly in the underlying dimension.

Mathematical statement of the claimed result

Result 1.
Let $V$ be any finite-dimensional complex vector space. Using averaging over permutations to identify symmetric powers with invariant tensors, multiplication in $(\operatorname{Sym}V)^{\otimes a}$ defines the canonical Foulkes--Howe map \[ \mu_{a,b,V}:\operatorname{Sym}^b(\operatorname{Sym}^aV) \longrightarrow\operatorname{Sym}^a(\operatorname{Sym}^bV). \] The asserted dimension-independent sufficient bound is \[ a\ge2,\quad b\ge a(a-1)\quad\Longrightarrow\quad\mu_{a,b,V}\text{ surjective}. \]

Outline of the argument

An annihilator formulation reduces surjectivity to vanishing of symmetric multilinear forms. Two successive differential procedures first free a slot from common factors and then eliminate the remaining common power; injectivity of directional differentiation supplies the quadratic degree condition.

Formal verification

For family 210, the formal scope is canonical surjectivity in the quadratic range, its $a=1$ case, and reverse embeddings. For sixth powers this only covers $b\geq30$, rather than every $b\geq6$. The listed Comparator files are FoulkesHowe.lean. Named declarations include OAI.Problem346.canonical_foulkes_howe_surjective. 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