ora-0186
16.6 The persistence core
Consider a lifetime path \(E_0\xrightarrow {f_0}E_1\xrightarrow {f_1}\cdots \), and let \(K_0\) be a knowledge package learned in \(E_0\). For a finite prefix, each composite transport returns a subobject of \(K_0\) on which coverage, semantic, observer, and admission defects vanish.
Assume the relevant subobject lattice of \(K_0\) admits finite meets. The persistence core through horizon \(n\) is
where \(\mathsf{Good}(f,K_0)\hookrightarrow K_0\) is the largest registered subobject on which \(f\) admits structural transport. An infinite-horizon core is the corresponding meet when it exists.
The definition makes forgetting observable. If \(\mathsf{Core}_{n+1}\subsetneq \mathsf{Core}_n\), the lost portion comes with the first environment morphism and defect type that excluded it. Conversely, accommodation can enlarge the storehouse by adjoining a repaired replacement rather than overwriting the failed structure.
Let \(K_0\) be a structural knowledge package and \(E_0\to \cdots \to E_n\) a finite environment path. Suppose each step admits structural transport on a common subobject \(C\hookrightarrow K_0\), the transport comparisons satisfy the pasting law, and settled observers and safety admission are conservative at every step. Then \(C\) is contained in \(\mathsf{Core}_n(K_0)\). Every universal witness and repair certificate in \(C\) transports coherently to every \(E_j\), and every future problem whose factorization lies in \(C\) retains an admitted factorization after transport.
By Theorem 16.3, the one-step hypotheses give
Hence \(C\) lies in their meet. Naturality transports universal witnesses and repair certificates. Transporting a factorization diagram and pasting its comparison cells gives the final claim; observer conservativity and safety preservation keep the witness admitted.
The theorem is finite because infinite persistence needs extra compactness, continuity, or accessibility assumptions. Even if every finite meet is nonempty, the infinite meet can vanish. A lifelong theorem must state what prevents this limit failure.