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A crucial step in the history of General Relativity was Einstein's adoptionof the principle of general covariance which demands a coordinate independentformulation for our spacetime theories. General covariance helps us todisentangle a theory's substantive content from its merely representationalartifacts. It is an indispensable tool for a modern understanding of spacetimetheories. Motivated by quantum gravity, one may wish to extend these notions toquantum spacetime theories (whatever those are). Relatedly, one might want toextend these notions to discrete spacetime theories (i.e., lattice theories).This paper delivers such an extension with surprising consequences.  One's first intuition regarding discrete spacetime theories may be that theyintroduce a great deal of fixed background structure (i.e., a lattice) andthereby limit our theory's possible symmetries down to those which preservethis fixed structure (i.e., discrete symmetries). However, as I will discuss,these intuitions are doubly wrong and overhasty. Discrete spacetime theoriescan and do have continuous translation and rotation symmetries. Moreover, theexact same theory can be given a wide variety of lattice structures and caneven be described with no lattice at all. As my discrete analog of generalcovariance will reveal: lattice structure is rather less like a fixedbackground structure or part of an underlying manifold and rather more like acoordinate system, i.e., merely a representational artifact. Thus, the worldcannot be "fundamentally set on a square lattice" (or any other lattice) anymore than it could be "fundamentally set in a certain coordinate system". Likecoordinate systems, lattice structures are just not the sort of thing that canbe fundamental; they are both thoroughly representational. Spacetime cannot bediscrete (even when it might be representable as such).

A Discrete Analog of General Covariance -- Part 1: Could the world be fundamentally set on a lattice?