Audit — the 'Cohomological Stability Certificate'

Rewritten 2026-07-10 against onn_ws/ONN. This page previously stated a "Cohomological Stability Certificate" (ONN loss as an explicit Lyapunov function, cohomological invariants doubling as stability certificates). That object is not present in the source; it is corrected here.

The claim as originally stated#

That the ONN total loss $\mathcal{L}_{\mathrm{total}}$ is an explicit, computable Lyapunov function with class-$\mathcal{K}_\infty$ bounds, and that the learned ontology's cohomological invariants serve simultaneously as representational quality measures and closed-loop stability certificates — bridging Massera-type existence theorems to a constructive certificate.

What the audit found#

• There is no "cohomology $=$ Lyapunov" certificate in the source. The results that remain are control-theoretic and model-specific: a phase-margin gate $\Delta t_{\max} = \varphi_{\mathrm{PM}}/\omega_c$ for a constructed linear plant with an integrating controller, and a separate single-mode scalar recurrence certificate $\rho+|K_c|<1$. The latter does not cover coupled modes or map $K_c$ to the deployed PD controller. Neither result follows from a cohomology class. See the delay-margin certificate.
• The curvature $\leftrightarrow$ persistent-homology bound "is proved by no theorem" and was removed in the audit.
• The 99.75% figure does not exist in the source and is removed; the only "optimality/coupling" content is a conjecture that is itself largely refuted (the interface is $\lVert W \rVert$-bounded and modular, independent of H-preservation and $\lambda_2$).
• More broadly, the premise that cohomological structure is a representational resource is exactly what the Two Ceilings No-Go disproves on the information axis.

Takeaway#

The cohomological stability certificate, as claimed, is withdrawn. What remains is narrower: model-level control analysis that owes nothing to cohomology and does not yet establish coupled multimode or deployed-system stability.