The constraint ∇·E = 0 is non-local in character, relating values across a region, so counting degrees of freedom requires boundary conditions and gauge choices that are globally, not pointwise, determined.
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Non-local(as used in physics and philosophy of science)
Something that depends on what's happening at distant places, not just what's right next to it; for example, a domino at one end of a line can affect dominoes far away, rather than only affecting its immediate neighbor.
Pointwise(as used in mathematics and physics)
One point or location at a time, without considering the bigger picture of the whole system.
∇·E (nabla dot E)(as used in physics and mathematics)
A mathematical notation that measures how much an electric field is spreading out or converging at different points in space; think of it like checking whether water is gushing out of a faucet (positive) or draining down a sink (negative).