Grothendieck's Constant Defected from His Six Functors: Enriched Kan Extensions, the √2 Plateau, and the Archimedean Wall

I introduce the Grothendieck defect K(ι) of an isometric inclusion ι: C ↪ D of normed operator categories — the supremum of the ratio between the factorization norm through C and the operator norm in D. This invariant satisfies K(ι) ≥ 1 and unifies all known Grothendieck-type inequalities. Eight results are established: (1) monotonicity for nested isometric inclusions; (2) a universal rank-r bound K(ι)|_r ≤ √r in three lines; (3) a cofactor identity B_A² = ‖A‖²_F I₄ − 2B_{cof(A)}; (4) the plateau K_G^ℝ(n) = √2 for n ≤ 18 via computer-assisted LP certificates with constraint generation, advancing the Fishburn–Reeds problem from n₀ ≥ 4 (1994) to n₀ ≥ 19; (5) K_G = K(ι) for ι: Hilb ↪ Ban^{comm} (Pisier's factorization form); (6) the six-functor identification K_G = sup ι*/ι! via enriched Kan extensions (Kelly), where ι! = ‖−‖ (left Kan = operator norm) and ι* = γ₂ (right Kan = factorization norm); (7) a summand rank bound; (8) K_G^{nc} = 2 from the summand bound plus Haagerup's GNS decomposition. The paper connects for the first time Grothendieck's 1953 Résumé to his 1960s six-functor formalism via enriched category theory, without requiring a Grothendieck topology on Ban. Companion repository with LP certificates (n=3..18), solver scripts, and reproducibility instructions available on GitHub.