ora-0076

4.14 Developmental composition of the core systems

The six core systems need not become globally integrated at once. Let \(I_d\subseteq I\) denote the fragments and overlaps available at developmental context \(d\). An arrow \(d\to d'\) may enlarge this cover, increase observer resolution, or introduce a new cross-fragment warrant. The partial core world at \(d\) is a compatible family

\[ (M_i,M_{ij},M_{ijk},\ldots )_{i,j,k\in I_d}. \]

This gives a concrete interpretation of developmental construction. Learning a word for a persisting object adds a language–object overlap. Learning that another agent can use the word to redirect attention adds a language–object–social coherence. Learning a numeral adds an object–number–language interface. The growth is simplicial: new higher compatibilities witness that previously separate capacities now participate in one compositional world.

Theorem 4.12 Developmental gluing under conservative growth

Assume each developmental cover has effective core descent. If every transition \(d\to d'\) conservatively transports the identified fragments and overlap comparisons, then the corresponding global realizations form a functor \(M:\mathbb D\to \mathsf{Mod}(\mathbb T_{\mathrm{core}})\) up to the declared coherence.

Proof ▼

At each \(d\), effective descent supplies a global realization unique up to equivalence. Conservative transport defines a morphism between the local descent data. Functoriality and uniqueness of descent induce the global comparison, and the same uniqueness supplies identity and composition coherence.

The theorem states the clean case. Development becomes scientifically interesting when the transport fails: the defect then localizes whether an old fragment, a newly introduced overlap, or the doctrine of gluing requires repair.