Pyramid Head is rather...unique when compared to the other killers.
For the most part, the Entity has recruited those with a defined past; be it innocent people who were twisted into the monsters they are now, deranged souls who acquired power to further spread their evil, or even creatures originating from dimension far beyond human comprehension.
Pyramid Head is different though. He has no past, no former alias to shed light on what he is now. He is a concept brought to life. The physical personification of pain, torture and punishment.
When James Sunderland stepped into Silent Hills, the guilt and remorse of his dark crimes took form in the Executioner wielding the Great Knife. Even when James finally left, Pyramid Head's influence remained, permanently staining Silent Hill. So much so that future visitors of the lost town would claim sightings of a grotesque figure with a large geometric head slowly dragging what appeared to be a large rusted knife.
And thus became the legacy of Pyramid Head. A pastless monster endlessly wandering the streets of Silent Hill in search of new victims, inflicting great amounts of pain and death so that they may find atonement.
But now he is in the Realm of the Entity. Death is not an escape and thus atonement will never be gained.
The Executioner's punishment will last for eternity