ifc-0299

19.6 Constructing a candidate technique

Suppose a persistent residual identifies a missing direction. DILATE then proposes a candidate generator bundle \(C\to M\), an anchor \(\rho _C:C\to TM\), and a local section \(c\) whose quotient action accounts for it:

\[ q_{\mathrm{art}}\! \left(\rho _C(c)\right) \simeq \Omega _{\mathrm{art}}(a,b). \]

The equation is in \(Q_{\mathrm{art}}\): it does not select a canonical lift of the obstruction back into \(TM\). The declared type, localization, and regularization of that lift are therefore part of the proposal record. Depending on the localization, \(c\) may be a new visual token, a relation, a spatial operation, a token-interaction law, a geometric invariant, or a multistage rendering procedure. The corresponding finite creative act is a sketch map

\[ i:S\hookrightarrow S^+=S[c]. \]

The notation \(S[c]\) suppresses the relations required to type \(c\). A bare symbol with no domain, codomain, equations, or observable consequences does not constitute a theory extension.

The first test of the proposal is explanatory closure. Let \(q_{\mathrm{art}}^+\) project away the operations registered in the extended theory. On held-out probes, DILATE asks whether

\[ q_{\mathrm{art}}^+\! \left([\rho _A(a),\rho _B(b)]\right) \simeq 0. \]

An extension that adds capacity but leaves the localized quotient residual unchanged has not explained the frontier. Conversely, forcing the residual to zero by an unrestricted memorizing generator fails irreducibility and productivity.