Exact sum-of-squares thresholds for the DDVV inequality on real symmetric and Hermitian matrices

Unrefereed research candidate. In one sentence: the true constant in the DDVV commutator inequality for real symmetric matrices is 1, but the simplest (quadratic) sum-of-squares proof needs the constant 9/8 for 3×3 matrices and up to 3 in general; this paper computes the exact least constant for every matrix size and number of matrices, for real symmetric and for Hermitian matrices. The DDVV inequality, which began as a conjecture in submanifold geometry and is now a theorem, says that for real symmetric n×n matrices B1, …, Bm the sum of the squared Frobenius norms of all commutators [Bp, Bq] is at most (∑p ‖Bp‖2)2. Ge, Li and Zhou asked whether inequalities of this type have sum-of-squares proofs. For all n, m ≥ 2 this paper computes the least coefficient c for which c(∑p ‖Bp‖2)2 − ∑p,q ‖[Bp, Bq]‖2 is a sum of squares of quadratic forms in the matrix entries. For real symmetric matrices this least coefficient σ(n, m) equals the true constant 1 exactly when min(n, m) = 2; otherwise it is an explicit three-branch rational function of n and m, equal to 9/8 for 3×3 matrices whenever m ≥ 3, nondecreasing in both parameters, with supremum 3 (not attained). For Hermitian matrices, whose sharp constant is 1 for m = 2 and 4/3 for m ≥ 3, the paper proves a four-branch formula σH(n, m): the sharp Hermitian inequality has a quadratic sum-of-squares proof exactly when n = 2, m = 2, or m = 3 and n ≤ 7, and the supremum of σH is 4. Both proofs average Gram matrices over the symmetry group, which reduces every certificate to a cone with ten explicit generators, and then match explicit sum-of-squares identities with explicit dual functionals; the thresholds are also the exact values of the degree-four moment relaxation. Consequences follow for skew-Hermitian, complex symmetric and general real and complex tuples. The archive contains the paper (PDF and LaTeX source), the bibliography, the figure script, a one-command exact-arithmetic replay suite (SymPy 1.14.0; eight checkers plus regeneration of three certificate files that must be byte-identical to the shipped ones, valid under python -O), retained logs, a claims table, a verification guide, the bounded prior-art search, a rating-free internal review summary and independent algebra audit, the previous version, the research-gate record, CI and SHA-256 sums. Assurance: written proofs, with exact-arithmetic certificate scripts complementing them; internal AI-assisted review only (not human peer review, independent reproduction or proof-assistant formalisation). Not claimed: higher-degree (multiplier) certificates, other matrix types, absolute priority. Creator: Anonymous. Text and data CC0-1.0; code MIT.