# Hyperkähler fibration existence ledger

Family 041: Hyperkähler SYZ and projective-space bases. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Record when a nonzero isotropic nef line bundle becomes semiample and induces a Lagrangian fibration.

## Applicability and commercial boundary

Hyperkähler and line-bundle hypotheses must be checked; existence does not provide an efficient numerical fibration finder.

## Initial business decision

Research reference or conditional symbolic/verification engineering. No recurring buyer need, practical implementation or profitability has been demonstrated. Defer a standalone commercial product until a constructive example and a measurable workflow improvement exist. These records also identify unsupported inferences that an evidence platform could flag.

## Next verification action

Read the section construction and input representation for an explicit example.

## Evidence scope

The repository catalog statement was individually read. Main-paper construction, effectiveness and selected formal-scope comparison remain queued for this record. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Projective-space bases of Lagrangian fibrations](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf).