Fourier Shadows and Tracial Free Infinite Divisibility

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.