This paper studies the isomonodromic geometry of the Painlevé III equation of type D₆ and constructs a new explicitly described entire function, the isomonodromic cosine F(s) = cos(2√(s(1−s))), whose zeros all lie on the critical line Re(s)=1/2. The function is proved to be entire of order 1, symmetric under F(s)=F(1−s), and to have explicitly computable zeros with asymptotic spacing π/2, interpreted as the WKB semiclassical spacing of the associated Painlevé III-D₆ oper. The construction is placed inside a de Branges positivity framework for wild isomonodromy, using decorated character varieties, positivity conditions, Herglotz/Weyl–Titchmarsh functions, and a finite-order tau-function condition on an explicit integrality sublocus. The paper does not claim a proof of the Riemann Hypothesis. Instead, it shows that the direct bridge from this de Branges/isomonodromic construction to the completed Riemann zeta function fails: the resulting zero-counting law lacks the Riemann–von Mangoldt logarithmic factor. Thus the constructed critical-line function is provably distinct from the Riemann zeros. The significance of the work is that it provides a rigorous critical-line entire function arising from Painlevé/isomonodromic geometry, identifies a precise obstruction to a direct zeta bridge, and formulates a sharper open problem — the Monodromy Selection Conjecture — for where arithmetic input would have to enter.