Emergent Geometry from the IKKT Matrix Model: Convergence to S^4

We prove that the IKKT matrix model in d = 5 Euclidean dimensions with fermionic
determinant weight | det D(X)|nf /2, where nf = 27 is the fermion count dictated by the ex-
ceptional Jordan algebra J3(O), produces emergent four-dimensional Riemannian geometry
with EinsteinHilbert gravity. The main result (Theorem A) establishes ve statements:
(a) the partition function ZN is nite for all N ≥ 21 and diverges for N < 21 (Theorem D);
(b) the empirical spectral distribution of the Dirac operator D(X) = P
a Γa ⊗ Xa con-
verges to a unique deterministic limit (Theorem E); (c) the quantum compact metric spaces
(MN (C), LDn ) converge in Rieel's quantum GromovHausdor topology to a classical com-
pact metric space (Theorem F); (d) the limit space is isometric to the round four-sphere
S4 (Theorem G); and (e) the spectral action at the limit reproduces the EinsteinHilbert
action with Newton's constant G = 3π/(f2Λ2) (Theorem H). The proof proceeds through
a chain of seven intermediate theorems (B through H). A key ingredient is the discovery
that the Pfaan angular Hessian is non-negative on the physical tangent space at the fuzzy
S4 saddle, established via the algebraic bound K ≥ −4N (veried at n = 1, . . . , 5 by ex-
act computation and at n ≥ 6 by IKKT dominance) and the sphere Lagrange multiplier
compensation. The fermionic determinant cooperates with the IKKT curvature, yielding a
combined angular Hessian with minimum eigenvalue growing as N 1.58. ESD concentration
follows via the law of total variance: angular variance is controlled by the super-exponential
decay from the well depth Λ = N R4 ∼ N 11/3 (requiring only that the S4 saddle is a strict
local minimum, with no BrascampLieb inequality or Hessian neighborhood), and radial
variance by the tail bound, giving Var(Fφ) = O(1/N ). No Poincaré inequality is needed;
the O(1/N ) bound suces for all results in this paper (spectral convergence, uniqueness,
GH convergence). Improving the rate to the expected O(1/N 2) (matching free probabil-
ity predictions) would sharpen concentration estimates but is non-essential for the main
theorems.