How can continuous space and smooth motion emerge from discrete records?

A smooth temperature field can describe many individual molecular events without making temperature a separate substance. OPH proposes an analogous relation between finite observer records and effective fields, geometry and clocks. Larger-scale comparisons can approach smooth laws when the construction supplies an appropriate refinement and controls the error in the observables. Merely drawing a graph finely is insufficient: the limit must preserve causal behavior, quantum responses and compatible ruler and clock readings. Different discrete models can have different limits, including ones with unwanted preferred directions. Continuum mathematics therefore describes collective behavior where the stated bounds support it. It need not be a literal inventory of infinitely many elementary points traversed by each moving object.

See also Is reality digital?, How does OPH answer Zeno’s paradox?, Is there a smallest length in nature?

Sources: Observer spacetime and the conditional Einstein reconstruction

Also in Chapter 15: Why Does Space Have Three Directions?, Chapter 17: Why Is There a Speed Limit?, Chapter 25: Why Does Gravity Look Like Geometry?.