ifc-0125

9.1 What can be transformed?

Let the current representation, with its differential semantics exposed, be

\[ \mathfrak R_{\mathrm{diff}} =(\mathbb S,\mathfrak D,\mathcal M,F,\Lambda ,\mathcal G,\mathcal A). \]

A transformation can act on any component:

Presentation.: Add or split a sort, introduce an operation, weaken or add an equation, or designate a new universal construction.
Semantic universe.: Move to models that support distinctions or internal constructions unavailable in \(\mathcal M\).
Differential semantics.: Add probe shapes, alter their interpretation, or replace the tangent action \(F\) by \(F^+\).
Observers.: Introduce a measurement or invariant, or revise a quotient, thereby changing which failures can be detected.
Generative operations.: Add, remove, or retype legal moves, constructors, proposal operators, or rules of composition.
Admission.: Revise the tests, evidence types, or preservation demands by which proposals are retained.

The audit record must identify the level. Otherwise ordinary parameter repair can be advertised as ontology change, while a genuine change in observability can be hidden as preprocessing. A proposed change to admission procedures must also be assessed under frozen prior criteria or an independent external procedure; it cannot validate itself merely by redefining success.

Definition 9.1 Transformational proposal

A transformational proposal is a versioned comparison

\[ \Upsilon :\mathfrak R_{\mathrm{diff}} \longrightarrow \mathfrak R^+_{\mathrm{diff}} \]

that changes at least one component governing what may be represented, varied, observed, composed, or admitted. It specifies the motivating obstruction and the intended interpretation of the new structure. The arrow is package-level notation: no category of representational packages, and hence no formal class of morphisms between all such packages, is assumed here.

A package transformation need not alter the theory presentation. Changing an observer, probe semantics, inference policy, or admission interface can change the conceptual space in Boden’s operational sense while leaving \(\mathbb S\) fixed. This book reserves theory extension for the stronger admitted case in which the presentation or its generative language changes. Before admission, the same presentation-level change is a candidate theory extension; the audit record must say which claim is being made.

The proposal is not yet an achievement. A random edit to a grammar is transformational in a syntactic sense but not creative in the evaluative sense used here. Admission requires explanatory or constructive gain.