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Math-241-A2 · revision r1 · on Math-241

OpenAI claim summary for Math-241

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: partially solved. 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 065, 2026-09-24): The main theorem has all genera, curve classes and cohomology insertions, but only smooth projective complete intersections; our Virasoro target is unrestricted in the smooth projective variety.

Manuscript 2 (family 065, 2026-10-05): Virasoro constraints are propagated only to projective bundles over a base already satisfying them; the catalogue target includes every smooth projective variety.

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

Manuscript

Contents
  1. Virasoro conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Mathematical claim 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Virasoro conjecture

OpenAI's claim

Manuscript 1 (family 065, 2026-09-24): Ordinary descendant Virasoro constraints are claimed for smooth projective complete intersections in every genus and curve class, retaining primitive and odd cohomology without semisimplicity. Manuscript 2 (family 065, 2026-10-05): Full descendant Virasoro constraints are claimed to pass from any smooth projective base to projectivizations of arbitrary nonsplit algebraic bundles, including all genera, integral curve classes, and odd insertions.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $X\subset\mathbb P^N_{\mathbb C}$ be any smooth complete intersection. Let $F_g^X$ be its connected ordinary, unreduced genus-$g$ descendant potential, with cotangent classes at stable-map markings, arbitrary cohomology insertions and separate Novikov symbols for actual integral curve classes. Set $Z_X=\exp(\sum_{g\ge0}\hbar^{g-1}F_g^X)$. OpenAI claims the formal coefficientwise identities \[ L_k^X Z_X=0\qquad(k\ge-1). \] Here $L_k^X$ are the manuscript's normally ordered Virasoro operators built from the Poincaré pairing, first Hodge grading $\mu|_{H^{p,q}}=p-\dim(X)/2$ and $R=c_1(TX)\cup$, with its stated scalar normalization. The quantifiers include all genera, individual curve classes and primitive or odd insertions; these are descendant rather than ancestor constraints.
Mathematical claim 2.
For a smooth connected projective complex variety $B$, an algebraic vector bundle $E$ of rank $r\ge2$, and $X=\mathbb P_B(E)$ parametrizing lines in the fibers, OpenAI claims \[ \left[L_k^B Z_B=0\text{ for all }k\ge-1\right]\ \Longrightarrow\ \left[L_k^X Z_X=0\text{ for all }k\ge-1\right]. \] For each target $Y$, $Z_Y$ is its ordinary descendant partition function $\exp(\sum_g\hbar^{g-1}F_g^Y)$ with all cohomology classes, stable-map cotangent classes and actual effective integral curve labels. The normally ordered $L_k^Y$ use $\mu_Y|_{H^{p,q}}=p-\dim(Y)/2$, $R_Y=c_1(TY)\cup$, and zero-mode scalar $\chi(Y)/16-\operatorname{str}(\mu_Y^2)/4$; other modes have no scalar correction. The conclusion is coefficientwise in every genus, curve class and descendant list.

Outline of the argument

Manuscript 1 (family 065, 2026-09-24): Relative caps create and detect bounded seam-contact states with coefficientwise finite inversions. Pairing each primitive error tensor with its conjugate realizes a squared norm through these tests; Hodge--Riemann positivity then kills the full tensor without limiting the number of primitive insertions.

Manuscript 2 (family 065, 2026-10-05): Localization on the completion by a trivial summand separates the base and projective-bundle blocks. The logarithm of an equivariant shift cancels coefficient differentiation in the grading, giving quantizable modes; ancestor bounds continue their identities coefficientwise, and quantum Riemann--Roch identifies the ordinary constraints on both components.

Formal verification

Family 065 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