Thakur's Conjecture on Carlitz-Wieferich Primes

Independently reproduced. We recomputed the claim from the definitions in arXiv:2607.15305, from scratch and with no computer-algebra dependency, so the check shares no code with the author's appendix. Confirmed: $x^3-8x^2-4x-11$ is irreducible over $\mathbb{F}_{19}$, so $\mathbb{F}_{19^3}$ is a field; $P$ is monic of degree $5$, irreducible over $\mathbb{F}_{19^3}$, and genuinely uses the cubic extension; and $\rho_P(1) \equiv 1 \bmod P^2$, which is the definition of a $c$-Wieferich prime, computed through the Carlitz recursion inside $\mathbb{F}_q[T]/(P^2)$. The Bamunoba-Bergstrom criterion the paper cites, $M_5(\theta) = 0$, was computed by a separate route and agrees. We also confirmed that $\mu(X)$ divides $X + X^q + \cdots + X^{q^4}$, which is what makes $G = \mu(T^q - T)$ divide $[5]$. Since $19 \nmid 5$, the counterexample stands. This matters more than usual here because the author states he cannot verify the mathematics directly. arXiv preprint (v2), not peer-reviewed.