No. The state of affairs along the slice of space passing through Earth's equator, for example, does not uniquely determine the state of affairs at 1° north latitude. But the state of affairs now, does determine the state of affairs one second in the future. (Relativistic motion can tilt the axes somewhat, but not enough to interchange space and time.)
All our physical models are described by local partial differential equations. Given the data on an (n-1) dimensional slice (including derivatives, of course), we can propagate that to cover the whole space. (there are complications once GR is in the picture making the notion of global slices questionable, but the same result holds "locally".)
If the data at the slice doesn't include derivatives, you can't propagate in time either.
You know what to do.
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