Gravity as a Projection into S4: Derivation of the Hierarchy from Geometric Volume Suppression

Gravity is uniquely weaker than the Standard Model forces by 34 orders of magnitude. In the Operator-Derived Dimension (ODD) framework, this hierarchy is a geometric consequence of projecting a 5D bulk onto a 4D hypersphere (S4). While gauge forces are confined to the intersection seam (N = 3), gravity propagates through the full encapsulating bulk volume (N = 4).
We derive Newton’s constant G explicitly from the bulk curvature scale k and the anchor mass Manchor ≈ 173 GeV. The hierarchy arises from an exponential volume suppression factor e−S4 , where S4 is the geometric action of the hypersphere. We calculate that a geometric action of S4 ≈ 78 naturally reproduces the observed Planck scale (G ∼ 10−39 GeV−2) from the electroweak scale, unifying gravity with the Standard Model geometry.