The Casimir Opposition: Deriving α = |ζ(−1)| from Spectral Mode Regularisation

Paper 186 established that three golden rectangles with one opposing at strength α=1/12 predict the median strain eigenvalue ratio r=0.080 in turbulent flow, confirmed by DNS to 0.0003. This companion paper identifies the origin of the value α=1/12. We hypothesise that each spectral mode of the three-sphere S³ contributes a back-reaction against symmetry-breaking strain proportional to its frequency. Under this hypothesis, the total opposition is the regularised sum ∑l = ζ(−1) = −1/12, giving α=|ζ(−1)|=1/12. The hypothesis is falsifiable: alternative scaling laws (energy ∝ l², eigenvalue ∝ l(l+2)) produce different predictions that the DNS rejects. The same zeta regularisation appears in two independent physical contexts: the Casimir effect between parallel plates (measured to 1% accuracy) and the critical dimension of bosonic string theory (D=26). The value 1/12 is linked to the second Bernoulli number B₂=1/6 via ζ(−1)=−B₂/2, whose generating function x/(eˣ−1) is the Planck distribution. The opposition is not a fitted parameter but a consequence of a single testable hypothesis applied to the spectral geometry of S³. One axiom, one hypothesis, two predictions, zero fitted parameters.