ifc-0142

10.5 The DIAL-X contract

An algorithmic specialization is the package

\begin{equation} \begin{aligned} \mathfrak C_X=\bigl(& \mathsf{Target}_X,\mathsf{Cr}_X,M_X, \mathbb {A}_X,\mathbb {B}_X,H_X,V_X,\Lambda _X,\\ & \Theta _X,\mathsf{Loc}_X,\gamma _X,\mathcal L_X,\mathsf{Ctl}_X,\Upsilon _X, \mathsf{Adm}_X,\mathsf{Mem}_X\bigr). \end{aligned} \end{equation}

10.1

Here \(\mathsf{Cr}_X\) may be empty for a purely adaptive DIAL application; when a creativity claim is made, it is a required part of the specification. The object \(M_X\models \mathbb {S}_X\) is the maintained realization. The proposed categorical layer uses enriched sketches \(\mathbb {A}_X\) and \(\mathbb {B}_X\) to present the two repair theories; an executable specialization realizes their selected operations computationally as \(H_X\) and \(V_X\). The observer family \(\Lambda _X\) makes their effects empirically or formally visible. The interchange witness \(\Theta _X\) compares their two orders, and \(\mathsf{Loc}_X\) records the localized mixed support when such a localization is meaningful. The integration observer \(\gamma _X\) tests finite realization, and \(\mathcal L_X\) contains well-typed proposals. The control suite \(\mathsf{Ctl}_X\) registers less disruptive in-language, measurement, hidden-state, no-change, and intervention-only alternatives. Writing \(\mathfrak R_X\) for the representational package specialized to \(X\), \(\Upsilon _X:\mathfrak R_X\to \mathfrak R_X^+\) records its change, with presentation component \(J_X:\mathbb S_X\to \mathbb S_X^+\) when applicable; \(\Upsilon _X\) is the identity for no-change, combinational, and exploratory dossiers. \(\mathsf{Adm}_X\) assigns epistemic status from independent evidence, and \(\mathsf{Mem}_X\) stores admitted declarations, states, failures, and transport maps.

The two sides need not literally be named assimilation and accommodation. They may be task and safety adapters, critic learning and policy transport, or local field estimation and global causal assembly. Assimilation and accommodation describe the eventual structural verdict. The application must say what its two sides mean before their interaction is interpreted.

When subtraction is meaningful, a computational proxy for interchange is

\[ \Theta _X(M) =V_X\! \left(H_X(M)\right)-H_X^{+}\! \left(V_X(M)\right). \]

In a general categorical realization this expression denotes a comparison map rather than vector subtraction. The superscript on \(H_X^{+}\) records that the horizontal operation may itself require transport after a vertical change. Only an admitted observer may turn the comparison into a numerical residual.