10.6 The generic DIAL-X algorithm

DIAL-X structural diagnosis, repair, and admission

1. Input: A DIAL-X contract \(\mathfrak C_X\), maintained realization \(M\), evidence \(E\), a creative mandate when creativity is claimed, and a registered resource budget.

2. Initialize the candidate-dossier set \(\mathcal P\leftarrow \varnothing \).

3. Validate the declaration, observer semantics, and intervention contract for the current regime.

4. Estimate the horizontal and vertical side operations separately, including uncertainty and side-coherence certificates.

5. If either side certificate fails:

6. reject its proposed semantics; repair the estimator, gather a typed probe, or abstain before interpreting mixed structure.

7. persist the failed side certificate and return an evidence request or auditable abstention.

8. Estimate the interchange comparison \(\Theta _X\) from the two routes; record its observable localization \(\mathsf{Loc}_X\) when possible.

9. If the mixed witness is practically equivalent to zero:

10. close the structural-repair branch and allow \(X\) to continue ordinary learning or search inside the current declaration.

11. add any independently proposed in-declaration dossiers to \(\mathcal P\); the zero witness supplies no novelty claim.

12. Else if the witness is persistently nonzero and localized:

13. construct the typed in-declaration candidate set \(\mathcal L_X^{\mathrm{in}}\subseteq \mathcal L_X\) and execute the applicable registered controls in \(\mathsf{Ctl}_X\).

14. Test each candidate with the registered finite-realization observer \(\gamma _X\) over a finite path, rollout, or persistence horizon.

15. If at least one candidate has a coherent finite realization inside the current declaration:

16. add the realized candidate dossiers to \(\mathcal P\) for independent admission.

17. mark them as in-declaration proposals; do not infer theory change from the original defect.

18. if creativity is claimed, classify them separately as combinational or exploratory relative to \(\mathcal R_X\).
    

19. Else if the registered control suite has ruled out its specified less disruptive alternatives and an expressivity result, benchmark construction, or exhaustive finite analysis certifies that the relevant in-declaration repair class cannot realize the target:

20. propose typed package comparisons \(\Upsilon _X:\mathfrak R_X\to \mathfrak R_X^{+}\), with a presentation map \(J_X:\mathbb {S}_X\to \mathbb {S}_X^{+}\) when applicable, including the no-change and intervention-only alternatives.

21. add every well-typed extension dossier to \(\mathcal P\), with its transport obligations and transformational status still pending.

22. Else:

23. persist the unresolved integration record and return an evidence request or abstention without naming an accommodation.

24. Else:

25. select a discriminating experiment and return its request, or persist an auditable abstention when no such experiment is available.

26. If \(\mathcal P=\varnothing \):

27. persist a no-change record and return the current admitted state or an auditable no-op.

28. For each candidate presented as creative, record its Boden mode and evaluate novelty against \(\mathcal R_X\); do not infer novelty from obstruction magnitude.

29. Include current, no-change, and previously admitted candidates whenever their types permit comparison.

30. Evaluate proposals using evidence not used to generate or rank them.

31. Admit, reject, defer, or request information; do not force a binary choice when the certificate is unresolved.

32. Persist the repair declaration, mixed and integration records, package comparison, learner-state transport, control results, evidence, rejected alternatives, and epistemic status.

33. Output: A status-bearing terminal record: an admitted, rejected, or deferred candidate dossier; a discriminating-experiment request; an auditable abstention; or a no-change result.

The ordering of the gates is part of the algorithm. A mixed residual computed after a malformed side has no DIAL semantics. Failure to enumerate a repair under a finite search budget is not proof that the language is inadequate; only the registered nonrealizability gate opens the declaration-extension branch. Stronger-search controls may expose proposal failure, but their failure alone does not establish expressive failure. Likewise, a repair selected on the same finite data that generated it has not undergone independent admission.

A zero mixed witness closes only DIAL’s structural-repair route. It does not imply that the domain has no combinational or exploratory opportunities: the base system may still construct a new composite or traverse a new path inside the fixed declaration. Such an artifact receives its own novelty and value tests, but its creativity cannot be attributed to a nonexistent obstruction.