ifc-0115
8.1 The boundary of a conceptual space
Use the differential refinement of the representational package introduced in Chapter 0 and held fixed in Chapter 7:
The sketch \(\mathbb S\) declares the available types and operations; \(\mathfrak D\) states what models preserve; \(\mathcal M\) interprets them; the Weil action \(F\) specifies local probes; \(\Lambda \) makes selected behavior visible; \(\mathcal G\) supplies legal moves; and \(\mathcal A\) determines which results may be retained. Exploratory creativity varies a state, term, proof, program, or model while this package remains fixed. Adaptive selection may choose among registered probes or moves, but it may not silently add a probe shape, observer, or operation to the package. The newly reached state, theorem, or program may be retained as content of a model or repertoire; retaining it is not by itself a revision of the package that made the construction expressible.
One may represent legal moves by a directed graph, a transition category, or a space of skill rollouts. Let \(\mathcal E_{\mathfrak R_{\mathrm{diff}}}(X)\) denote the states reachable from \(X\) by legal finite compositions of those moves. The notation is deliberately schematic: reachability may mean proof derivability, execution, simulator evolution, or integration of a local field.
A candidate exploratory transition \(X\rightsquigarrow Y\) is a path from \(X\) to \(Y\) using only operations and search controls declared in \(\mathfrak R_{\mathrm{diff}}\). It becomes an admitted creative transition relative to an evaluator when the path is valid, the path or its consequence is novel relative to the registered repertoire, the result is valuable under a registered observer, and it passes the registered admission procedure. These are separate obligations.
The second sentence matters. Reachability gives legality, not interestingness. A proof system can derive countless trivial variants; a grammar can generate an enormous number of worthless expressions. Exploratory creativity needs both a generative boundary and an evaluative boundary. Compression progress, surprise, and search efficiency may be useful observers, but none substitutes for the novelty–value judgment or the admission record.