4.3 Brackets expose interaction

Creative skills are rarely independent coordinates. Applying analogy before counterexample search can yield a different candidate theory from reversing the order. The Lie bracket records the infinitesimal interaction between two skill sections. If \(m\) and \(n\) compile to \(s_m\) and \(s_n\), then

is the leading infinitesimal order effect obtained from a small commutator loop; the finite loop discrepancy is second order in its side lengths. In an exact Lie algebroid,

This bracket is not a textual difference between two scripts. It compares the state-dependent vector fields produced by their compiled semantics. Two very different scripts can therefore have a negligible bracket when their realized actions commute. Conversely, two small edits can have a large bracket when their order changes what later operations become available. The coordinate construction in Section 3.6 gives the minimal picture: two registered motions generate a third direction only through their order of composition.

The exact identity above belongs to the declared Lie-algebroid model. In a learned implementation, it must itself be checked. The residual

measures failure of the estimated bracket and anchor to satisfy this exact law. It is a model-diagnostic residual, not an additional Lie-algebroid axiom.

Suppose a declared distribution \(\mathcal D\subseteq TM\) contains the visible theory changes currently regarded as admissible. On a constant-rank region, choose a local splitting or metric and hence a projection \(P_{\mathcal D}\). The closure residual

tests whether interacting legal skills generate a direction outside the present declaration. Its numerical representative depends on the chosen projection, whereas vanishing of the quotient class modulo \(\mathcal D\) does not. A persistent residual may indicate a missing composite skill, an omitted state variable, an inadequate observer, or a genuine need to extend the theory. It does not by itself choose among these explanations.

Brackets also provide a practical screening rule. Many inexpensive script pairs can be tested with short rollouts; pairs with large order effects or closure defects receive the expensive proof, simulation, or empirical validation budget. The bracket therefore shapes exploration without reducing admission to a scalar objective.