1. The problem is to determine if it is possible to find exact rational values for the five coefficients $r,s,t,\alpha,\beta,\gamma$ in the closed-form expression $$ P = \pi^{r} A^{s} \exp\big(t \zeta'(-1)\big) \frac{G(3/4)^{\alpha} G(1/2)^{\beta}}{G(1/4)^{\gamma}} $$ where $P = \prod_{k=1}^\infty \zeta(2k)^{(-1)^{k+1}}$. 2. The derivation shows that $r=\frac{1}{2}$ is already known exactly. 3. The logarithm of $P$ is expressed as a sum of four blocks: Bernoulli block $S_B$, elementary sums $S_{2\pi}+S_2$, and factorial block $S_{\mathrm{fact}}$. 4. The Bernoulli block $S_B$ can be expressed as a rational linear combination of $\ln A$, $\zeta'(-1)$, and a rational constant, with rational coefficients $s_0,t_0,c_0$. 5. The factorial block $S_{\mathrm{fact}}$ is expressed via Gauss multiplication and Legendre duplication formulas, then telescoped using Barnes $G$ function identities, resulting in a finite product with rational exponents $p_0,p_1,\alpha,\beta,\gamma$. 6. Barnes multiplication theorems provide two independent rational linear relations among $\ln G(1/4), \ln G(1/2), \ln G(3/4)$ and $\ln \pi, \ln A, \zeta'(-1)$. 7. Using these relations, two of the three Barnes logarithms can be eliminated, rewriting $\log P$ as a rational linear combination of $\ln \pi, \ln A, \zeta'(-1), 1$. 8. Collecting all terms yields a finite rational linear system $$ M x = v $$ where $x = (s,t,\alpha,\beta,\gamma)^T$, $M$ is a rational $5 \times 5$ matrix, and $v$ is a rational vector. 9. The matrix $M$ and vector $v$ are explicitly computable by finite algebraic steps involving only rational arithmetic. 10. Solving this system by exact Gaussian elimination over $\mathbb{Q}$ produces exact rational values for $s,t,\alpha,\beta,\gamma$. 11. Therefore, it is indeed possible to find exact rational values for all five coefficients $r,s,t,\alpha,\beta,\gamma$ by following the finite, symbolic, deterministic procedure described. 12. The document also offers an algorithmic recipe and an appendix with explicit finite sums and matrix construction to mechanically compute all rational entries and solve the system. \boxed{\text{Yes, the rational values for all five coefficients can be found exactly by the given finite symbolic derivation.}}