# Verification report

The accepted theorem concerns bounded tracial tuples. For every k>=2,
Fourier shadows obtainable from free sums of jointly freely infinitely
divisible laws form a proper subset of products of such shadows. Two
axis semicircles provide the witness. Every tracial lift of their product
shadow fails positivity at formal powers 0<t<1/4: the fixed sum of
Hermitian squares 3(X^2-Y^2)^2+[X,Y]^*[X,Y] has expectation 2t(4t-1).
At a putative fifth root this is -2/25. No endpoint sufficiency is asserted.

The additional commutator identity, commuting free-root rank-one
classification and bounded Pauli compound-Poisson Fourier collision
are proved in the source. The general identity retains Re(c^2) in
nontracial states; it must not be replaced by |c|^2. Bounded Fock
operators give actual positive laws and all roots in the Pauli example.

check_exact.py passed 2,973 exact assertions using Gaussian rationals,
noncrossing partitions, matrix states, and direct Fock actions in both
Python 3.13 and Python 3.12. This is one suite replayed in two runtimes.
check_fourier_sos.mjs independently derives the lift-obstruction
polynomial over Z[t,a] and passed 97 assertions, including negative-root
certificates. Finite controls supplement the written general proof.

The native editor compiler accepted main.tex. An already-installed,
cached Tectonic exported the publication PDF. Its five rendered pages
were visually inspected; no clipping, overlap, missing references or
broken mathematics was observed. The retained build log has no overfull
or underfull boxes, undefined references or warnings.

The AIM source paragraph and relevant prior theorems were read. The
source does not explicitly impose traciality in the target paragraph.
The original record is therefore not counted as fully resolved. Novelty
remains undetermined after bounded primary-source searches. Standard
conditional positivity, cumulants and compound-Poisson/Fock machinery
are credited. This is originating-researcher self-audit, not independent
human review or proof-assistant verification. Publication does not change
those qualifications.