10 DIAL as an Algorithmic Family

From Double Repair Geometry to DIAL-X

Part I described creative modes and a geometry of interacting repairs; Part II must turn those distinctions into executable and falsifiable methods. A geometry is not yet an algorithm. To obtain one, DIAL must be specialized to a base learning system \(X\). The specialization must state what \(X\) is trying to construct, which two repair processes are being compared, what makes their interaction observable, which structural changes may be proposed, which less disruptive alternatives must first be controlled, whether a local repair has a coherent finite realization, what would make such a proposal creative, what evidence can admit one of them, and what must persist after the verdict.

We write the resulting family as DIAL-X. Examples include DIAL-GIRL, based on Gradient Infinitesimal Reinforcement Learning; DIAL-ALLORA, based on A Lie-Algebraic Low-Rank Adaptation; and DIAL-SKFM, based on Spectral Kernel Flow Matching. The hyphen is important: DIAL does not replace the domain algorithm. It supplies a typed diagnostic, routing, and admission layer around domain operations whose meanings must already be declared.

Those specialization names also preserve the provenance of experiments run before the integrated systems received shorter names. The following chapters use CLIC for causal construction, OPTIC for skill construction, and RELIC for sequential decision structure; their experiment cards retain DIAL-SKFM, DIAL-SkillOpt, and DIAL-GIRL where those are the registered artifact names.

Throughout the book, a hyphen joins DIAL to its domain specialization, while an en dash introduces a registered experimental rung. Thus DIAL-SKFM is the system family and DIAL-SKFM–1C is one experiment; DIAL-URL is the coalgebraic construction and DIAL-URL–3 is one calibration within it.

This chapter inherits the formal-status convention stated at the opening of Chapter 3. The DIAL-X contract is a book-level algorithmic specification. Instantiating its fields yields an auditable method, but does not prove that the proposed internal category of DIAL objects exists, that a mixed witness has a unique interpretation, or that the method will improve a particular domain system. Those are separate mathematical and empirical obligations.

¹1. Chapter roadmap. The first half turns DIAL’s geometry into a four-level implementation invariant, target card, artifact dossier, evidential statuses, and generic algorithm. The second half specializes that contract to decision, adapter, causal, and skill construction before stating the boundary of the algorithmic claim. ↩