4.1 From a script to a creative skill

For one fixed language and interface version, let \(\mathsf{Md}_{\mathrm{adm}}\) denote a category of admissible, typed Markdown artifacts. Its objects include front matter, ordered workflow steps, tool declarations, invariant checks, probe requests, and admission rules. Its arrows are controlled revisions that preserve that version’s required interface. A revision of the grammar, type system, or interface instead requires a registered map between versioned artifact categories; it is not silently treated as another arrow of the old category. A typical contract might read:

We distinguish two typed subcollections. A local skill artifact belongs to \(\mathsf{Md}_{\mathrm{sec}}\) and compiles to a section. A discovery-controller artifact belongs to \(\mathsf{Md}_{\mathrm{pol}}\) and compiles to a policy over admissible actions. Both sit inside \(\mathsf{Md}_{\mathrm{adm}}\) and may share syntax, but their semantic codomains differ.

---
skill: analogical_transfer
reads: [theory.objects, obstruction.local]
writes: [candidate.interfaces]
invariants: [source_provenance, admitted_results]
probes: [first_order, paired_order]
admission: held_out_counterexample
---
1. Select a source construction with a typed structural match.
2. Propose the smallest target interface needed by the analogy.
3. Record transported laws and laws that are not preserved.
4. Generate a discriminating consequence and send it to admission.

The header declares what the skill may read and change. The body describes a proposal policy. Neither is executable until interpreted relative to an environment \(E\), which includes the parser, model, tools, theory-store version, and authorization policy. We write the resulting partial semantic map as

Here \(A\to M\) is a Lie algebroid over a smooth realization \(M\) of theory states, and \(s_m\) is the local skill section compiled from script \(m\). The map is partial because a script may be ill typed, request an unavailable tool, or fail its authorization contract.

A local section is a state-dependent choice of one skill direction in each fiber over the region where the skill is defined. Thus \(s_m\) is not merely the Markdown text or a fixed parameter vector: it says which typed operation the compiled program makes available at each compatible theory state. The anchor then converts that internal choice into an observable motion of the base state.

This notation does not assert that every tangent category is already a smooth manifold or possesses a Lie algebroid. It selects a smooth or suitably synthetic realization in which the LASKO construction is meaningful. Other realizations may require an internal or generalized analogue of the same anchored structure.