lin-0077
5.5 Invariance and dependence
If \(F:\mathcal M\to \mathcal N\) is a diffeomorphism, naturality of the Lie bracket gives
Consequently, involutivity and the zero-residual property are invariant when the fields, span, and metric are transported together. Replacing one smooth frame by another frame of the same distribution also preserves vanishing.
By contrast, the numerical residual norm, its tangential coefficients, and finite-sample thresholds depend on choices:
the intervention protocols and their scaling;
the visible span and the stratum on which its rank is constant;
the metric used to define normal projection; and
the estimation and separation assumptions.
A constant-rank stratum is \(\gamma \)-separated for a chosen metric, frame, and protocol normalization when the frame Gram matrix has smallest eigenvalue at least \(\gamma {\gt}0\), and each tested residual has norm either zero or at least \(\gamma \).
Separation is an estimation margin, not a consequence of statistical regularity.