Read-back: Davenport.pnt_dlvp

The theorem is a single closed statement with no parameters, no hypotheses of its own, and no typeclass assumptions: it asserts the existence of a pair of real numbers $c$ and $C$, both strictly positive ($0 < c$ and $0 < C$), such that a certain inequality holds for every real number $x$ satisfying $2 \le x$. In full, it says:

Here $\psi$ is Mathlib's Chebyshev function, $\psi(x) = \sum_{0 < n \le \lfloor x \rfloor} \Lambda(n)$, the sum of the von Mangoldt function over the positive integers $n$ up to $x$; it is a real-valued, right-continuous step function of the real variable $x$ that depends on $x$ only through $\lfloor x \rfloor$. The quantity $\log x$ is the natural logarithm, $\sqrt{\cdot}$ is the real square root, and $\exp$ is the real exponential; the exponent is $-c\sqrt{\log x}$, i.e. the product of $-c$ with $\sqrt{\log x}$, so the exponential factor lies strictly between $0$ and $1$ whenever $x > 1$, and the right-hand side is therefore strictly smaller than $C x$.

The quantifier order is essential to the content: the two constants $c$ and $C$ are chosen once and for all, before $x$ is introduced. They are therefore absolute numerical constants — they may not depend on $x$, and a single pair must work simultaneously for every real $x \ge 2$, including arbitrarily large $x$. Nothing in the statement pins down, bounds, or otherwise constrains $c$ and $C$ beyond their positivity: $c$ may be arbitrarily small and $C$ arbitrarily large, and no relation between them is required. Conversely, the statement makes no claim of uniqueness or optimality — it is a plain $\exists$, not $\exists!$, and no "for all sufficiently small $c$" or "for all $c$ below some threshold" is asserted.

The bound is two-sided, because it is stated on the absolute value $|\psi(x) - x|$: it simultaneously asserts $\psi(x) - x \le C x \exp(-c\sqrt{\log x})$ and $x - \psi(x) \le C x \exp(-c\sqrt{\log x})$. The inequality is non-strict ($\le$, not $<$). It is an upper bound only; no matching lower bound on $|\psi(x) - x|$ is claimed.

On the range and degenerate cases: the hypothesis $2 \le x$ is satisfiable (so the inner universally quantified claim is not vacuous), and it is the only restriction on $x$ — there is no upper cutoff, so the assertion covers all $x \in [2, \infty)$. Real numbers $x < 2$ are excluded entirely, so the statement says nothing about $x \in [0,2)$ or about negative $x$, and in particular nothing about the degenerate values $\psi(x) = 0$ that occur for $x < 2$. On the retained range $x \ge 2$ one has $\log x \ge \log 2 > 0$, so $\sqrt{\log x}$ is a genuine positive square root and no junk value of the total functions $\log$ or $\sqrt{\cdot}$ (which would return $0$ on non-positive arguments) is invoked. The right-hand side is a product of the positive constant $C$, the positive number $x$, and a positive exponential, hence positive throughout the range. Because $x$ ranges over the reals rather than the naturals, the claim at a real $x$ compares the step value $\psi(x) = \psi(\lfloor x \rfloor)$ against the real number $x$ itself.

Finally, the statement invokes nothing from the accompanying definition files: the auxiliary notions available in the preamble — the arithmetic-progression Chebyshev sum $\psi_{N,q,a} = \sum_{n < N,\ n \equiv a \pmod q} \Lambda(n)$, the zero-free-region boundary $1 - c/\log(q(|\operatorname{Im} s| + 2))$ and its associated predicate, the exceptional-set predicate for Dirichlet $L$-functions, the Gauss sum, and the twisted von Mangoldt sum — do not appear in the assertion. No Dirichlet character, modulus $q$, residue class $a$, $L$-function, or zero-free region occurs anywhere in what is claimed; the theorem is entirely about the single unrestricted Chebyshev function $\psi$ on $[2,\infty)$.