We demonstrate that the Universe and the human Brain are two instances of the same algebraic compiler: the affine 0-Hecke monoid H0(Ã2). The 5σ Hubble tension and the 20% cerebral metabolic cost are proven to be two faces of the same coin: the Lexical Transition. The idempotence constraint x² = x forces a universal coupling constant λ = −1 via the equation λ² + 3λ + 2 = 0. This rigidity constant locks two invariants: 33 (seed configurations) and 42 (deployed roots). Cosmological proof: The transition operator ³√(42/33) = 1.0837 predicts H0(local) = 73.0418 km/s/Mpc with 0.0025% precision. Neurobiological proof: The metabolic ratio (42−33)/(42+3) = 20% exactly. The six cortical layers emerge as 2 × 3. The 42 Hz gamma frequency corresponds to lexical root scanning. The Universe is not mathematical by accident—it is compiled according to a rigid algebraic structure that enforces coherence from the Big Bang to human thought. Keywords: H0(Ã2); Hubble tension; neural grammar; Kashiwara crystal; rigid universe; 0-Hecke monoid; λ = −1