# 3. Continuity Chapter 2 named the continuous joining flow $\mathbf F$. For a particular ordered pair of registrations, we now write the local bookkeeping of that same flow as $$ \mathbf{S}(\mathbf{r};1,2). $$ This does not introduce a second flow beside $\mathbf F$. It is the registration-to-registration accounting of that one continuous flow for the ordered pair $(1,2)$. So transport is not something that happens after reconfiguration; it is the local structure of that reconfiguration. Continuity now makes a local claim: a region changes only through exchange with neighboring regions across its boundary. The difference $$ u_2(\mathbf{r})-u_1(\mathbf{r}) $$ is understood as the result of a redistribution of the same energy within itself, described by a flow connecting the two registrations. Energy in a region changes only by crossing its boundary to a neighboring region. In one direction, say the x-direction, the statement is $$ u_2-u_1+\partial_x S_{12}=0. $$ Here $S_{12}$ refers to the redistribution flow connecting registrations $1$ and $2$, that is, the bookkeeping summary of the continuous flow $\mathbf F$ across that ordered pair. The equation does not say that change is small. It says that the difference between registrations is locally accountable by transport. The statement is local, but it is imposed at once across the whole extent of $u$. It constrains how the whole registered distribution can change while remaining one continuous reconfiguration of $u$. This is accounting of energy, not yet its dynamics. It is like accounting for the brightness of the pixels on a screen without yet recognizing the image they compose.[^platos-allegory] Continuity is therefore the statement that an ordered difference between registrations is a redistribution of the same energy. The scalar change in the energy field $u$, the continuous joining flow $\mathbf F$, and the ordered bookkeeping by $\mathbf S$ are three writings of the same continuous event. Together they give closed bookkeeping. We now turn to exploring the implied consequences of this energy accounting in free space. [^platos-allegory]: As Plato and many others observed, one can become skilled at recognizing images, patterns, and regular sequences of appearance while remaining ignorant of what produces them. Physics can encode repeatable regularities without thereby laying hold of the underlying causes. Even so, such encoding is far better than treating the screen as uniform brightness alone.