Are there any known special properties of a number located between twin primes?
The question came up in the discussion. (The expression below has been rephrased to a weaker form for clarity)
In studying admissibility of k-tuples I observed that
$$\varphi( \lambda ) \le \lim_{\lambda < n \le \infty} \inf \varphi( n ) $$
where we use the set theoretical representation of a twin prime pair as $\{\lambda-1, \lambda+1\} \subset \mathbb{P}$ and the definition of $\lim \inf x_n$ is given by $\lim_{N\rightarrow \infty} \inf x_k: k \in [N, \infty]$, and I'm not quite sure how to prove the statement above on the totient of a "lumenal" (?) number beyond conjecture. I think amWhy is curious as well.
This post may be related, because this is a special case of the same problem. Now, of course, with the explicit formula for Euler totient, $$ \varphi(n) = n \prod_{p|n} \bigg( 1 - \frac{1}{p} \bigg) $$ we do, in fact, end up with $\frac{ \varphi(n)}{n}$ having the product form in terms of the prime divisors of $n$ giving the value $n-1$ when $n \in \mathbb{P}$. If we were not prime, then there would be another factor less than one in the formula for totient. Similar logic applies for the number $n+1$, because $n+1$ cannot be prime and as a result $\varphi(n+1) < n$. But this doesn't entirely explain the conjectured expression.
For context, we arrive at a similar expression for the totient of a prime number $$\varphi( p ) > \sup_{0 < n < p} \varphi( n ) $$
