ora-0158
14.1 Registered repair problems
Let \(p:E\to X\) be the simplicial family of decorated decisions used in Equation 9.6. The base simplex in \(X\) records the compositional history; its lift to \(E\) records evidence, action, consistency, consequence, and warrant data.
A registered repair problem is a tuple
where \(e_\Lambda :\Lambda _i^n\to E\) is the observed decorated boundary, \(\sigma :\Delta ^n\to X\) is its proposed base completion, \(\mathcal O_{\mathrm{set}}\) is a family of obligations that must be preserved, \(\mathcal O_{\mathrm{open}}\) is a family still permitted to change, and \(\Omega \) is an observer of comparison defects. Its admissible repair space is the subspace
cut out by the settled obligations and the declared safety invariant.
The division between settled and open obligations is the formal analogue of Piagetian assimilation. It says what may be revised without pretending that the rest of the learner’s world has been reconsidered. Accommodation occurs when no filler exists for the current base simplex or when every filler violates a settled obligation.
|
Repair profile |
Homotopy type |
Meaning |
Required response |
|---|---|---|---|
|
Impossible |
\(\varnothing \) |
No admissible completion has the declared boundary and base semantics |
Change evidence, safety constraints, or the declaration |
|
Coherently unique |
Contractible |
All admissible completions are joined by a coherent system of identifications |
Assimilate without arbitrary tie-breaking |
|
Discrete ambiguity |
Several components |
Qualitatively different repair families survive |
Probe for evidence that separates components |
|
Higher ambiguity |
Nontrivial loops or higher cells |
Repairs agree at the object level but their witnesses do not cohere uniquely |
Preserve the mapping space or request higher coherence data |