Three papers. In the first, an Arctic ecosystem's time series contains all the information needed to predict a regime shift, but statistical analysis of that time series fails because noise overwhelms the signal. The same trajectory, read as a position in phase space relative to the stochastic separatrix, succeeds — because noise helps define the geometry instead of obscuring the statistics.
In the second, a single coarse observation of a two-phase fluid is consistent with many fine-scale states — the inverse problem is ill-posed. A stream of coarse observations over time resolves the ambiguity — because different fine-scale states evolve differently, and temporal continuity breaks the degeneracy that spatial snapshots cannot.
In the third, a robot's SLAM optimization produces sparse matrices identical to those arising from elastic bar finite element problems. The matrices are the same. The sparsity patterns are the same. But recognizing this equivalence imports forty years of domain decomposition theory, making a hard problem tractable with existing tools.
In each case, the data doesn't change. The Arctic trajectory is the same whether read as a time series or as a phase-space position. The fluid observations are the same whether treated as independent snapshots or as a temporal stream. The SLAM matrices are the same whether viewed as robot-specific optimization or as structural engineering. What changes is the frame placed around the data — the representation that determines which mathematical tools apply.
The through-claim: the representation is load-bearing. It is not a neutral container for the data but a structural element that determines tractability. A time series that fails as statistics succeeds as geometry. A snapshot that fails as an inverse problem succeeds as a temporal stream. A matrix that is hard as SLAM is easy as elasticity. The same information, differently framed, produces different problems — and different problems have different solutions.
This means the choice of representation is as consequential as the data it represents. The representation determines which methods apply, which estimates converge, which existing solutions transfer. Getting more data is one way to solve a hard problem. Re-seeing the same data is another.