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Are you clever enough? In memory of Avigrail

This riddle is dedicated to Avigrail, a brilliant riddler who was so affected by the politicking and anti-riddle sentiment on the site that he became cynical and abusive and ended up being chased into exile.

There are four parts to this puzzle. Each of them points towards the same solution, and you can get the solution from any one on its own, but I'll only accept an answer that includes explanations of how to solve all four.

This is the interesting one (also probably the easiest):

for all Lambda unique e L for all unique r

In this riddle, all the clues point towards the concept of the answer:

I might be metal, or you might
Find me in woods or at the beach.
I can answer your questions -
Always truly, but oft unwelcome.

In this riddle, all the clues point towards the English word for the answer:

I might be paper.
Half of me has it;
Or, with repeats,
Only one lacks it.

In this riddle, all the clues point towards the German word for the answer:

I might be paper.
I rhyme with a predator.
Bond a fork and then
A twisted limb frozen.