Modern physics relies on calculus, but it never explains why calculus works.
We routinely describe reality using smooth curves, derivatives, and continuous change, yet these mathematical tools are simply assumed to apply. Time is treated as a continuous parameter, and physical quantities are expected to vary smoothly with respect to it. But at the deepest levels we know this picture breaks down: quantum events happen discretely, measurements produce jumps, and information is created irreversibly. The problem is not that calculus is “wrong,” but that it quietly assumes conditions—continuous change and dense information—that are not always met. This paper starts from that gap and asks a more basic question: under what conditions is calculus actually valid as a description of physical reality?
Dimensional Emergence Calculus (DEC) is a framework designed to answer that question.
Instead of assuming time and space from the outset, DEC builds change from two more primitive ingredients: ordering and commitment. Ordering tracks how events are sequenced, while commitment tracks when irreversible distinctions are made. These two processes are independent, and together they form a two-dimensional “ordering space.” DEC defines new differential operators on this space and shows how ordinary derivatives—like d/dt—only emerge when commitments happen densely and ordering varies slowly. When those conditions hold, standard calculus appears as an excellent approximation. When they don’t, the familiar mathematical language of smooth change simply stops applying, and DEC provides the replacement.
The deeper implication is that calculus itself becomes a physical phenomenon, not just a mathematical convenience.
In this view, smooth time and continuous motion are not fundamental features of the universe—they are emergent patterns that arise when information accumulates in the right way. DEC makes this precise by identifying clear, operational criteria for when calculus is valid and when it must fail. This reframes long-standing puzzles about quantum jumps, measurement, and breakdowns near extreme conditions as failures of an inappropriate mathematical tool, rather than failures of physics itself. Whether or not nature ultimately uses this structure, the paper shows that a self-consistent calculus can be built without assuming continuity—and that alone opens a new way of thinking about how reality is organised.