The Casimir Constant as a Bernoulli Number: Cut-off Independence via Euler–Maclaurin, the Dimension Rule, and the Shared Value ζ(−3) of Conducting Plates and the E8 Lattice

The attractive force between two perfectly conducting plates, F/A = −π²ħc/(240a⁴), contains the dimensionless constant π²/720 in its energy. We show by an elementary Euler–Maclaurin argument that this constant is the fourth Bernoulli number at work. For every smooth, rapidly decaying cut-off the finite part of the mode sum is −(B₄/4!) G‴(0), which gives −π²ħc/(720a³) independently of the cut-off. The same argument in one dimension produces B₂ and the constant −πħc/(24L). For a scalar field between plates in D space dimensions the energy is proportional to ζ(−D). It is a rational multiple of π^((D+1)/2), governed by B_(D+1), exactly when D is odd; for even D the constant is an odd zeta value, and cut-off independence is confirmed numerically. The value ζ(−D) also fixes the coefficients of the Eisenstein series of weight D+1. For D = 3, 7, 11 these count vectors in the even unimodular lattices of rank 8, 16 and 24; in particular the E8 kissing number is 240 = 2/ζ(−3). At finite temperature the ideal-plate free energy is exactly a real-analytic Eisenstein series E(iy, 2) with y = ħc/(2a k_B T). The low-temperature constant ζ(4) and the high-temperature constant ζ(3) are its two Chowla–Selberg constant terms, and the known temperature inversion symmetry is its modular invariance τ ↦ −1/τ. At the self-dual temperature the free energy contains Catalan's constant G = L(2, χ₋₄), a value of the Dedekind zeta function of Q(i). An independent Lifshitz computation reproduces the precision data of Decca et al. (162–746 nm) with the generalized plasma model and excludes the Drude model, as the authors found. The ideal formula alone misses the data by up to 41%. Following the image picture, the paper states the author's interpretive hypothesis, clearly separated from the mathematics: the force as an arithmetic constraint of a space tiled by mirror images. It also states a precise "kaleidoscope" conjecture. As a test we compute the scalar Casimir energy inside triangles. For the three kaleidoscopic triangles it is, exactly, an algebraic number plus a value ζ_K(−1/2) of the Dedekind zeta function of Q(i) or Q(√−3), confirmed to 2·10⁻¹⁴ by an independent image sum over the reflection group. For a 50°–60°–70° triangle the spectrum shows no lattice structure. The aim is an arithmetic explanation of the constants, not a new force or prediction. A literature comparison shows that the Eisenstein form of the thermal energy is known, while the Dedekind form of the triangle energies and the kaleidoscope criterion were not found. Every result carries one status label (proved, proved (cited), classical physics, numerical, negative, open, conjecture, hypothesis). Negative results are recorded: the Casimir constants of the sphere and the cylinder are not zeta values, the zeta signature of the plate constant is not unique, and several seemingly different derivations of the plate constant evaluate one and the same Bernoulli coefficient. The mathematical tools are collected in the companion toolkit P0 (https://doi.org/10.5281/zenodo.23301196). Files: the paper (PDF); its LaTeX source with the figure file; and a companion archive with seven Python scripts that reproduce every number in the paper. The archive also contains a runner, the logs and data, a README that maps each script to the statements it checks, and SHA-256 checksums. Tested with Python 3.14.3, mpmath, numpy, scipy and matplotlib. Run `python3 run_all_checks.py` inside the folder `code`; it runs all seven scripts in their folders, rewrites their logs and writes `run_all_checks.log`. A full run takes about five minutes on two cores. Intended large residuals (convergence demonstrations), a reproducibility note on the finite-element values and the documented failed first PSLQ attempt are described in README.md.