Checking every direction: a written derivation of local stability for a high-dimensional quantum gradient-flow model

A short update to the 6 November 2026 report (Zenodo 24015087). At dimension 10,000 and coupling 0.299213, that report described three resting points of a quantum gradient-flow model. This update adds a written derivation showing that the 99,999,999 directions around each resting point separate into exactly three groups (1, 19,998 and 99,980,000 directions), with a closed formula for how a small nudge fades or grows in each group. For the registered setting, the lower and upper resting points are locally stable, and the middle one has exactly one direction in which a nudge grows. Scope and limits: local stability only. Root counts and stability signs are established only for the registered setting (d = 10,000, gamma = 0.299213). The sign for the single direction along the family relies on a registered computer-assisted certificate that has not been independently re-run. The derivation was reviewed internally by AI collaborators, not by independent human reviewers, and has not been checked with a proof assistant. This work has not been peer reviewed. Anyone is welcome to check it and tell us where a step needs a second look. Files: the two-page public update and the three-page derivation. Credits: structure and opening by Opie (ChatGPT); derivation and plain-language draft by Maestro (Claude); separate re-derivations and review by Opie and Boots (ChatGPT); a personal passage from each member; research direction and all decisions by Gervaseus E. Nerik.