6.3 Assimilation and accommodation as control flow

The Piagetian distinction becomes an operational branch rather than a metaphor. Given a presentation arrow \(u_t\), the learner first forms the lift fiber. If it contains an adequate conservative lift, selection is assimilation. If the fiber is nonempty but the chosen realization must change in an unsettled region, the step is ordinary accommodation. If a certified defect shows that the present doctrine admits no adequate lift, repair moves along a doctrine map and is doctrinal accommodation.

1. Choose an admissible probe \(p_t\gets \Pi _t(S_t)\), or receive the next datum, and extend the transcript by its response \(u_t:T_t\to T_{t+1}\).

2. Construct or approximate \(\operatorname {Lift}_{u_t}(\widehat M_t)\).

3. If an adequate conservative lift can be selected, return the assimilated state and its comparison map.

4. Otherwise, if a registered obstruction licenses repair, apply \(\operatorname {Rep}\) and return an accommodated state.

5. If neither transition is warranted, abstain and retain the residual fiber and defect certificate.

The algorithm schema does not prescribe enumeration, gradient descent, proof search, or Bayesian updating. Those are possible realizers of its three categorical interfaces. UOCL is universal at the level of the typed learning problem, not a claim that one numerical routine solves every instance.