ifc-0049

3.6 Double-infinitesimal repair geometry

Weil profiles describe families of probes, but they do not yet distinguish two repair processes or explain how those processes interact. For that purpose, let

be a double vector bundle. Its horizontal and vertical vector-bundle structures commute. The side bundle \(A_{\mathrm{asm}}\to M\) represents assimilatory directions; \(B_{\mathrm{acc}}\to M\) represents proto-accommodative directions. The application, not the double-vector-bundle axioms, supplies these meanings.

The canonical projection

\[ \mathbb D_{\mathrm{rep}}\longrightarrow A_{\mathrm{asm}}\times _M B_{\mathrm{acc}} \]

forgets joint information. Its kernel over the zero sections is the core \(K\to M\). Core elements project trivially to both side bundles. The core is therefore a candidate carrier of mixed corrections: under an application semantics, an assimilatory repair may alter the meaning of an accommodative probe even when neither side records that effect alone. This interpretation is additional structure, not a consequence of the double-vector-bundle axioms.

In a classical smooth model, equip each direction with a VB-algebroid structure.

This gives horizontal and vertical differentials on Meinrenken–Pike’s bigraded cochain Weil algebra

\[ \mathcal W_{\mathrm{Lie}}^{\bullet ,\bullet }(\mathbb D_{\mathrm{rep}}), \qquad d_{\mathrm{asm}}:(p,q)\mapsto (p+1,q), \quad d_{\mathrm{acc}}:(p,q)\mapsto (p,q+1). \]

Because both differentials have odd total degree, their graded commutator is

\[ \Omega _{\mathrm{cre}} :=[d_{\mathrm{asm}},d_{\mathrm{acc}}]_{\mathrm{gr}} =d_{\mathrm{asm}}d_{\mathrm{acc}} +d_{\mathrm{acc}}d_{\mathrm{asm}}. \]

Theorem 3.4 Meinrenken–Pike Weil-algebra criterion, cited

Let \(\mathbb D_{\mathrm{rep}}\) be a double vector bundle equipped with VB-algebroid structures over both side bundles. The two structures make \(\mathbb D_{\mathrm{rep}}\) a double Lie algebroid if and only if their horizontal and vertical differentials on the bigraded Weil algebra supercommute, equivalently \(\Omega _{\mathrm{cre}}=0\) [ Meinrenken and Pike , 2021 , Theorems II and 8.2 ] .

The theorem supplies a clean diagnostic interpretation in the classical realization. If \(\Omega _{\mathrm{cre}}\neq 0\), the candidate pair is not yet a double Lie algebroid. The two local repair semantics are incompatible. Once a representation and localization rule have been fixed, the value and support of \(\Omega _{\mathrm{cre}}\) can identify an algebraic locus where the two processes fail to supercommute. Treating that locus as a repair frontier requires a declared observer connecting the residual to task semantics. A model can observe this typed defect through a scalar norm or statistical test, but the obstruction itself is the bidegree-\((1,1)\) operator. At the internal tangent-categorical level, the corresponding diagnostic is \(\Theta _{\mathrm{int}}\); identifying it with \(\Omega _{\mathrm{cre}}\) requires the cochain realization described above.

A coordinate model of a missing mixed direction.

The smallest useful picture lives on \(M=\mathbb R^3\). Let two locally available motions be

\[ X=\frac{\partial }{\partial x}, \qquad Y=\frac{\partial }{\partial y} +x\frac{\partial }{\partial z}. \]

At the origin, the registered side directions span only the \(x\)- and \(y\)-axes. Their bracket is

\[ [X,Y]=\frac{\partial }{\partial z}, \]

so an infinitesimal loop that follows \(X\), then \(Y\), then reverses the two motions produces a second-order displacement in a direction absent from the original side span. The \(z\)-direction is the elementary coordinate shadow of mixed information that neither side reports alone.

This example is not itself a DIAL object, and a nonzero bracket is not yet a creative event. It shows only the geometric question that the double repair construction types more carefully: does an order effect close inside the declared representation, disappear under the registered observer, or expose a persistent direction for which the current theory has no expression?

Definition 3.5 Double repair geometry, proposed

A double repair geometry consists of a candidate DIAL object and, when available, a classical double-vector-bundle realization \(\mathbb D_{\mathrm{rep}}\) with candidate horizontal and vertical VB-algebroid structures. It also includes a chosen finite family of tangent-probe observables and an admission semantics for deciding whether an observed interchange defect is persistent.

This definition intentionally allows incompatible candidates. Internally, a compatible DIAL object is the zero-\(\Theta _{\mathrm{int}}\) case. In the classical cochain realization, a compatible double Lie algebroid is the zero-\(\Omega _{\mathrm{cre}}\) case. Either defect may motivate repair of the coupling, enrichment of the core, or a larger theory change.

The canonical example is the tangent prolongation of a Lie algebroid \(A\to M\):

Its double Weil algebra recovers the Weil algebra of \(A\) [ Meinrenken and Pike , 2021 ] . This makes the construction more than a formal analogy: a Lie algebroid of creative skills can be tangent-lifted, so one direction represents available skills and the other their variation across theory states. Chapter 4 develops that connection.

. Use the double structure to compare local repair orders, not to smuggle a finite theory change into tangent notation. The creative interchange obstruction localizes what the current coupled geometry cannot reconcile. A candidate repair must first persist and integrate into a coherent finite realization. If the old theory cannot support such a realization, a proposal engine may express a replacement as a finite sketch extension \(J:\mathbb S\to \mathbb S^+\); admission must then test its transport and new consequences.