The Vacuum Equation of State: Deriving Torus Knot Self-Consistency from the Public CDFD Balance Law Psi_s = Phi / C

Capstone of the formal 10-paper CDFD mass-spectrum sequence. Earlier papers used a successful but ungrounded torus-knot self-consistency condition to fix each family’s vacuum twist. Here that condition is recast as a model theorem inside the public flow–constraint notation. The CDFT vacuum medium is mapped onto three public CDFD quantities: Flow (Phi), Constraint/capacity (C), and Adaptive Operating Ratio (Psi_s). Continuous flow energy is identified with the squared-amplitude L2 norm, transport capacity with the squared absolute-amplitude sum, and equilibrium deformation with total vacuum twist. Applying the public balance law Psi_s = Phi / C recovers the same self-consistency condition used in the torus-knot spectrum, so the Koide-like ratio Q_n becomes the geometric face of a transport-limited vacuum equation of state. The lepton/meson spectra of Papers 01–09 are thereby tied internally to m_e and the proposed vacuum balance. Status: internal model theorem and consistency check — not a demonstrated vacuum-engineering technology. Series role: Vacuum-engineering control-law bridge — control of Phi, C, S, and M_s. Open science workspace & reproducibility:https://github.com/bampita-bico/CDFD-Part-I-Release