ifc-0298
19.5 The Weil profile of an obstruction
A single bracket estimate can spike because of minibatch noise, a chart boundary, a transient optimization state, or an unreliable observer. Theory extension therefore requires a profile across infinitesimal probes. Let \(W\) range over a registered family \(\mathcal W\) of Weil algebras, and let \(T_W\) denote the corresponding prolongation. The lifted residual has the form
For a morphism of probe shapes \(W\to W'\), the induced comparison must carry the residuals compatibly. The profile
should therefore behave as a natural family rather than as an unrelated list of scalar scores [ Leung , 2017 , 2018 ] .
The bracket appears through an order-\(\epsilon ^2\) commutator. First-order dual numbers alone do not expose this mixed coefficient. The probe family must include a double infinitesimal such as \(D\otimes D\), together with the flips and comparison maps needed to exchange its two infinitesimal directions. This is precisely where the double geometry earns its role: the mixed coefficient is not a second copy of either side derivative.
A candidate obstruction is Weil-persistent relative to \(\mathcal W\) when it is natural under the registered probe maps, remains separated from the zero class across admitted scales, retains a coherently transported direction, localizes to the same deficient part of the sketch, and survives independent observer and data resampling.
This definition contains both structural and statistical obligations. The residual itself remains typed; a scalar persistence statistic may summarize evidence, but it cannot erase where the failure lives or which direction it selects. The finite-sample DIAL-CAN and DILATE experiments described below illustrate why this matters: an estimated bracket may carry a stable zero boundary even when a learned calibrator has never observed one side of that boundary.