ifc-0032
2.1 Three different achievements
Let a reduced view of a conceptual space be represented by
where \(\mathbb S\) declares the relevant types and compositional laws, \(D\) is the current realization, \(\Lambda \) is a family of observers, and \(\mathcal U\) is the language of admissible moves. This notation is not intended as a complete theory of concepts. It supplies a boundary among three levels of achievement. It is also a deliberately reduced view of the fuller representational package \(\mathfrak R\) introduced in Chapter 0: \(\mathcal K\) suppresses the preservation doctrine, proposal machinery, and admission procedure when they are not needed for the problem-solving/theory-extension distinction.
A problem-solving transition changes the realization or its state while preserving the maintained package:
The result may be extremely difficult or surprising, but its objects and criteria of success remain available in the inherited theory.
A candidate package change proposes a registered comparison
identifying which presentation, doctrine, semantics, probes, observers, generative operations, or admission procedures change, and how established content is intended to transport. At this stage coherence, transport, and usefulness remain to be tested.
A candidate theory extension is a candidate package change whose presentation or generative language changes through a registered map, span, or zigzag containing
The presentation component may add, split, identify, or reorganize objects, arrows, equations, or primitive operations. A change only to an observer, estimator, probe family, or admission threshold is a package change but not, by itself, a theory extension.
An admitted package change is a candidate package change whose componentwise coherence, claimed transport, and novel consequences have passed a registered test independent of proposal construction.
An admitted theory extension is an admitted package change whose package comparison contains a non-equivalent presentation or generative-language change \(J\). An unqualified claim that a system constructed a theory extension refers to this status; earlier stages are proposals or candidate extensions.
This refines the primer’s presentation-level datum by including a realization and its transport. The displayed map is still the simplest case. An inclusion-like map naturally records added generators or equations, but a revision that forgets, splits, or identifies earlier structure may require a span or zigzag of presentations, together with an explicit account of information lost in each direction. Subsequent uses of \(J\) should therefore be read as registered theory maps, not as a claim that every conceptual change is literally an inclusion.
A mere relabeling or invertible change of coordinates is not a theory extension: it may improve tractability or reveal structure, but it does not by itself enlarge what the theory can express. Nor is an arbitrary incompatible replacement an extension, because it supplies no accountable relation to the earlier theory. Between these cases lies the creative target: a candidate must expose new consequences, state what survives, and earn admission. This requirement turns creativity from unconstrained novelty into a problem of typed theory surgery.
Six statuses therefore recur: a model update keeps \(\mathfrak R\) fixed; an equivalent presentation changes notation without changing expressivity; a package proposal registers \(\Upsilon \); a theory-extension proposal records its presentation-changing component \(J\); an admitted package change adds verified transport and independent evidence; and an admitted theory extension also establishes the claimed non-equivalent language change. An equivalent presentation may be mathematically or computationally valuable, but it is not evidence of theory construction merely because it is unfamiliar.