A drug trial. The control group’s outcomes form a bell curve centered at 50. The treatment group’s outcomes form two peaks — one at 30, one at 70 — centered at the same mean of 50. The average treatment effect is zero. Every standard statistical test says the drug does nothing.

But the drug split the population in half. Some patients improved dramatically; others declined equally. The treatment didn’t shift the distribution. It changed its shape from one hill to two. This is the failure case that motivates topological causal inference (arXiv:2603.14169): when the effect is structural rather than locational, mean-based estimands are blind.

Persistent homology detects these shape changes. It counts connected components, loops, and voids in the distribution across multiple scales, producing a summary (the persistence diagram) that captures distributional geometry. The topological causal effect measures whether treatment changes the number or persistence of these features.

The complication: persistent homology doesn’t commute with covariate adjustment. You can’t just take the topological summary of the marginal distribution and call it causal, because confounders create spurious topological features. The paper proves identification under a conditional version of the ignorability assumption — you have to compute topological effects within strata, not marginally.

The through-claim: the tools you use to measure an effect determine which effects you can see. Mean-based tools see shifts. Variance-based tools see spreads. Only topological tools see bifurcations. The absence of evidence was the evidence of an absent tool.