Regular Totally Nonnegative Loops Without Entire Matrix Logarithms

For positive a,b,c,d, put Q(a,b;t)=[[0,a],[bt,0]]. We prove that 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 proves the nonexistence direction; proportional generators prove the converse. An explicit noncommuting example has rational Taylor coefficients. The examples contradict the single-logarithm assertion of Lemma 8.4 in arXiv:0812.0840v3 and show 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. This English note is AI-assisted, originating-researcher self-audited and unrefereed. No independent peer review or proof-assistant verification is claimed; novelty remains undetermined.