The Algebra of Diff´erance

We begin with a single irreducible datum: a directed difference between a distictionand another. Making no further assumptions, we ask what algebraic structure ariseswhen such difference are collected and composed freely. The answer is the path algebraof a quiver. We develop its theory entirely from within: its grading, its idempotentdecomposition, its radical, the reachability structure that emerges from the Peirceblocks, the period that emerges from a canonical grading refinement, and the growththeory that emerges from the Hilbert series. All structure is derived; nothing is imposed.We treat the representation theory, the spectral theory of the adjacency operator, andthe limits of what this free algebra can express without enrichment by relations.