# Theorem dependency and correction alerts Edition: 8 October 2026. Initial decision: **Prototype now**. The product interpretation and pricing are hypotheses; independent proof and buyer validation remain open. ## Research finding The 7 October history records a sign error that invalidated one manuscript and two dependent papers. Other corrections changed hypotheses and companion citations. The release preserves prior versions, enabling a concrete regression fixture for dependency tracking. ## Problem and buyer Research teams and engineering teams adopting new algorithms need to know whether a correction invalidates a result they rely on. A changed PDF filename or bibliographic citation does not explain the affected assumptions or downstream implementation. ## What the finding could enable A monitored dependency graph can show the path from a changed lemma to a manuscript claim and then to a software component or experiment. It enables scoped review tasks and correction alerts tied to actual use rather than a generic news feed. ## Technical and commercial limits A citation is not always a logical dependency. Automated extraction must label inferred edges and allow human confirmation. The actual three withdrawn manuscripts require explicit withdrawal fixtures; family membership alone must not imply invalidity of unrelated papers. ## Minimal architecture Revision watcher -> citation and theorem-link parser -> dependency graph -> change classification -> affected-component report. Store graph edges with evidence spans, revision hashes and reviewed status. Keep historical snapshots instead of replacing the source record. ## Existing alternatives and differentiation Crossmark already distributes correction and retraction metadata. The proposed differentiation is theorem-level impact across Git repositories and software usage; bibliographic status alerts by themselves are weak differentiation. ## Monetization hypothesis Hypothesis: AUD 200-1,000 monthly for a research team and AUD 5,000-20,000 annually for an institutional integration. Buyers and budgets remain unvalidated. An audit service around one dependency-heavy project is the narrow entry point. ## Validation experiment Use the repository's withdrawal history as a retrospective fixture. Recover the three stated withdrawals and their stated dependency relation without flagging every paper in the same field. Have a mathematician review ambiguous edges. ## Conditions to reject or defer Reject a general subscription if alerts are too noisy, logical edges cannot be reviewed economically, or free correction metadata meets the user's needs. ## Next concrete action Extract the withdrawn manuscript notices and their archived-version links, then build an explicit three-paper fixture and a dependency impact report. ## Source evidence Repository sources are pinned to revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. Selected scope notes and manuscript statements have been reviewed to the extent described above; these links do not represent successful kernel checks. ### Family 032 Subject: The rational Hodge conjecture for CM abelian varieties. - [The rational Hodge conjecture for CM abelian varieties](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf) - [Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026/paper.pdf) - [Weil classes and Hodge classes on abelian powers](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Weil-classes-and-Hodge-classes-on-abelian-powers-October-6-2026/paper.pdf) - [Abelian covers, Gale correspondences, and the Hodge conjecture for powers](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-October-6-2026/paper.pdf) - [A Conditional Reduction for Algebraic Kuga–Satake Correspondences](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Conditional-Reduction-for-Algebraic-Kuga-Satake-Correspondences-September-10-2026/paper.pdf) ### Family 149 Subject: Classwise permanence for weakly reversible mass-action systems. - [Uniform Permanence in Weakly Reversible Mass-Action Systems](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf) - [Boundedness and persistence of weakly reversible mass-action systems](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf) - [Formal scope notes](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/149.md) ### Family 376 Subject: Universal computation in forced Navier–Stokes flows. - [Finite Instructions and Solenoidal Shear Flows](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Finite-Instructions-and-Solenoidal-Shear-Flows-September-27-2026/manuscript.pdf) - [A Fixed Particle Test for Computation in a Forced Viscous Flow](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/manuscript.pdf) - [Universal Computation with Eventually Stationary Navier–Stokes Forcing](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/manuscript.pdf) - [Geometric Programs for Solenoidal Forcing](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Geometric-Programs-for-Solenoidal-Forcing-September-27-2026/manuscript.pdf) - [Prefix Instructions and Incompressible Flows](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/manuscript.pdf) - [Computation under Rapidly Vanishing Navier–Stokes Forcing](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Computation-under-Rapidly-Vanishing-Navier-Stokes-Forcing-September-27-2026/manuscript.pdf) - [Velocity-Field Detection of Computation in Forced Navier–Stokes Flows](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026/manuscript.pdf) - [Scalar Potentials and Slow Clocks for Forced Fluid Computation](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Scalar-Potentials-and-Slow-Clocks-for-Forced-Fluid-Computation-September-27-2026/manuscript.pdf) - [Incompressible Box Transport and Finite Computation](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/manuscript.pdf) - [Formal scope notes](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/376.md) - [Repository revision and withdrawal history](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/history.md) ## Current alternative sources Primary documentation reviewed on 8 October 2026. Product availability demonstrates alternatives, not demand or willingness to pay for this proposal. - [Crossref Crossmark correction metadata](https://www.crossref.org/services/crossmark/)