The Casimir Effect as Dimensional Mode Counting

The Casimir effect is the only experimentally verified manifestation of vacuum energy. We present a pedagogical analysis showing that its mathematical core — the regularised sum ζ(−1) = −1/12 — coincides with the dimensional structure of three-dimensional space as −1/D(D+1) for D = 3. This interpretation is supported by the sign reversal of Casimir forces in cubic geometries (Ambjørn and Wolfram 1983), where constraining all three dimensions rather than one reverses the force from attractive to repulsive. We observe that the Casimir effect operates entirely at electron scales (100 nm – 100 μm), not at the Planck scale, implying that the natural cutoff for vacuum energy calculations is electromagnetic, not…