The cosmic dipole anomaly — the observation that the matter dipole exceeds the CMB kinematic expectation by a factor of approximately 2.7 at >5σ significance — poses the most fundamental challenge to FLRW cosmology. No framework within ΛCDM predicts either the existence or the amplitude of this excess. We show that the Bootstrap Universe framework, in which the cosmos crystallises on a rotating Zoll 3-sphere (S³), naturally predicts both a preferred cosmic direction and a specific matter-to-CMB dipole ratio of e = 2.718... The prediction rests on two proven mathematical properties of S³: (i) it is a Lie group (SU(2)) with a surjective exponential map, making exponential growth geometrically natural; (ii) it is a Zoll manifold with a unique characteristic timescale, forcing the growth exponent to unity. The CMB dipole is a fossil of the early crystallisation asymmetry; the matter dipole reflects the current asymmetry after exponential amplification over one natural timescale. Their ratio is exp(H₀ × 1/H₀) = exp(1) = e, independent of initial conditions. The CatWISE quasar measurement gives 2.7 ± 0.2, matching the prediction to 99.3%. This is the only framework that predicts the specific numerical value of the anomaly.