We compare tensor and free addition of bounded tracial tuples whose joint laws are freely infinitely divisible, after projecting both laws to the scalar Fourier series tau(exp(i sum u_j X_j)). The Fourier shadows obtainable from free sums form a proper subset of products of such shadows in every dimension at least two. A witness is the product of two variance-one semicircle Fourier transforms on separate coordinate axes. Every tracial joint lift of this product fails to have a free fifth root: the sum of Hermitian squares 3(X^2-Y^2)^2+[X,Y]^*[X,Y] has formal expectation -2/25. More generally, its expectation at free exponent t is 2t(4t-1), so positivity requires t >= 1/4. No endpoint sufficiency is asserted. We also prove commuting free-root rank-one rigidity and give two distinct bounded tracial free compound Poisson laws with identical Fourier shadows.

Source context: AIM Free Analysis workshop problems, section 0.10, first question; frozen Hugging Face record AIM-PROBABILITY-0131 in ulamai/UnsolvedMath v1.6.0. The result uses an explicit bounded-tracial common-coordinate formulation. It does not classify all obtainable series, settle the scalar k=1 comparison, or exclude arbitrary nontracial lifts. The original source record is not counted as fully resolved. Standard cumulant, conditional-positivity and Fock-space machinery is credited. Exact standard-library Python and Node verification code accompanies the analytic proofs. This English preprint is AI-assisted, originating-researcher self-audited and unrefereed. Novelty remains undetermined after a bounded primary-source search; no independent human review, proof-assistant verification or absolute-priority claim is made.
