The quality or state of being frayed; the condition of having edges that are worn, torn, or unraveled.
From 'fray' plus '-ed' adjective suffix plus '-ness' noun suffix. This creates an abstract noun representing the quality of being in a frayed state.
The suffix '-ness' is one of English's most productive word-building tools—you can add it to almost any adjective to turn a quality into a countable concept, which is why 'frayedness' immediately makes sense even if you've never seen it before.
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