# Surface cycle-class assumption reference

Family 040: Bloch’s conjecture for complex surfaces. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Track the claimed integral Albanese isomorphism for zero-cycles on projective surfaces with geometric genus zero.

## Applicability and commercial boundary

Zero-cycle equivalence and an Albanese map are not ordinary finite record deduplication; the new and classical cases have distinct hypotheses.

## 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

Inspect finite symbolic examples and any effective cycle-equivalence procedure.

## 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).

[Bloch’s conjecture for surfaces with p_g=0](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf).