Emergent Spacetime from Geometric-Symmetry Matching

The dimensionality of macroscopic spacetime remains a fundamental open question in theoretical physics. In this work, we propose a geometric–symmetry constraint relating macroscopic spacetime dimensions to the isometry structure of compact internal manifolds. Motivated by Kaluza–Klein compactifications of higher‑dimensional gravity, we consider the matching between the geometric cross‑components of the metric linking the two spaces and the gauge generators arising from internal manifold symmetries. For a spacetime of dimension D and a compact internal R‑dimensional sphere S^R, there are exactly D·R independent metric cross‑components gμa. Motivated by a minimal saturation principle, equating this geometric linkage to the dimension of the internal isometry group Isom(S^R) = SO(R+1) yields the dimensional lock condition D·R = R(R+1)/2, which simplifies to 2D = R + 1. Combined with the maximal supergravity dimensional bound D + R ≤ 11, this condition uniquely selects (D, R) = (4, 7). This naturally reproduces the compactification structure of eleven‑dimensional supergravity and the 28 gauge fields of four‑dimensional N = 8 supergravity, suggesting a possible connection between spacetime dimensionality, internal manifold symmetries, and exceptional geometric structures.