For positive a,b,c,d, put Q(a,b;t)=[[0,a],[bt,0]]. We prove that the folded matrix A(t)=exp Q(a,b;t) exp Q(c,d;t) has an infinitely supported regular totally nonnegative periodic unfolding, yet admits an entire complex matrix logarithm if and only if ad=bc. A Jordan trace-parity obstruction supplies the nonexistence direction; proportional generators supply the converse. An explicit noncommuting instance has rational Taylor coefficients and exactly certified opposite-sign nonscalar Jordan values.

The examples contradict the single-logarithm assertion of Lemma 8.4 in the specific preprint arXiv:0812.0840v3, and demonstrate sharpness within regular TN foldings of the known Kutzschebauch--Studer two-exponential bound. That bound and general single-logarithm failures are credited prior work. The journal-version passage was not verified. This is not a full solution of the general AESW-type factorization request in AIM-LINEAR_ALGEBRA-0007, nor an absolute-priority claim.

Exact Python and Node regression code supplements the written analytic proof. The English note is AI-assisted, originating-researcher self-audited and unrefereed; no independent review or proof-assistant verification is claimed. Novelty of the regular two-phase refinement remains undetermined.
