Hard Upper Bound on Spatial Dimensionality in Wave Confinement Theory

This work establishes an exact continuum scaling law for the Laplacian norm of a Gaussian test field in n spatial dimensions,

||Δψ_λ||²_{L²} = C_n × λ^(4−n),
where C_n = n(n+2) × (π/2)^(n/2) × σ^(n−4)

and connects it to a general instability mechanism in wave-like systems with curvature–feedback coupling.
The result implies that stable, self-localized waves in infinite flat space are only possible for n ≤ 3, providing a natural mathematical bound on the number of large, continuous spatial dimensions.

The analysis combines:

• Exact analytic integration in the continuum to confirm the scaling exponent 4−n.

• Numerical spectral simulations in 2D, 3D, and 4D to validate the scaling law and identify finite-domain artifacts.

• A discussion of physical implications for higher-dimensional field theories, extra-dimensional models, and confinement stability.

Keywords: spatial dimension bound, Laplacian scaling, Gaussian field, curvature–feedback instability, continuum PDE, quantum field theory constraints, extra dimensions.

Notes: The scaling law matches heuristic predictions from the six-mechanism collapse proof in the main text. A caution on numerical artifacts for n=4 periodic domains is included, along with the closed-form constants C_n for direct verification.