MathOverflow

The primes $ p = u^2 + 71 v^2 $ (other than $71$ itself), up to $12345,$ are precisely the primes $q$ up to that bound for which$$ x^7 - 2 x^6 - x^5 + x^4 + x^3 + x^2 - x - 1 \equiv 0 \pmod q $$
has seven distinct roots.
In Cox, Primes of the Form $x^2 + n y^2$, Theorem 9.2 says that there is a polynomial of degree $h(-4n)$ that does the job. Indeed, class numbers 3 and 4 are dealt with in two papers by K. S. Williams. Oh, Cox refers to Weber....

Consider the following theories:
$T_1$: $\mathsf{ZFC+PD}$ where $\mathsf{PD}$ is stated as a schema.
$T_2$: $\mathsf{ZFC+PD}$ where $\mathsf{PD}$ is a single sentence in the language of set theory.

I posted this question on MSE first but it seems I'm not getting an answer.I'm reading the paper on the generalized sphere theorem by Grove and Shiohama, and there is an observation they made that I'm struggling to prove.
We have two unit geodesics $\tau,\sigma:[0,L]\to M$, where $M$ has sectional curvatures $sec\geq\delta>0$, and let $c_t:[0,1]\to M$ be the geodesic joining $\tau(t)$ to $\sigma(t)$, for $t\in[0,L]$. Here $\tau$ and $\sigma$ are close, in the sense...

Say that a forcing notion $\mathbb{P}$ is slow iff there is some $f:\mathbb{R}\rightarrow\mathbb{R}$ (in $V$) such that for every $\mathbb{P}$-name for a real, $\nu$, we have $\Vdash_\mathbb{P}\exists r\in V(\nu\oplus \check{r}\not\ge_Tf(r))$. For example...