A Mechanical Derivation of a Vacuum Density Scale

Quantum Field Theory (QFT) predicts an enormous vacuum‑energy density when summing zero‑point modes up to a high‑energy cutoff. This value, typically around 1096 kg/m³, depends strongly on the chosen cutoff scheme and is known to be circular because the cutoff itself is defined using ħ, G and c. In this note, I present a purely mechanical derivation of a vacuum‑density scale Λlocal . obtained from the relation G= c²/(Λlocal . ẟ²) where ẟ is a geometric protrusion length derived directly from the α‑tail described in earlier derivations of the fine‑structure constant. The key result is that , derived without using , Planck units, or QFT assumptions, naturally falls near the QFT Planck‑cutoff estimate. This coincidence is not presented as proof of any model but as a potentially meaningful indication that mechanical interpretations of vacuum structure may encode physically relevant scales. New version: correction of typos in section 2.2.