ifc-0162

12.2 From vector-field brackets to involution algebroids

CLIC and OPTIC share an algebraic intuition but occupy different semantic levels. CLIC begins with intervention-induced vector fields in a smooth realization and estimates Lie brackets such as \([v_i,v_j]\). Their support helps localize a missing causal mechanism. OPTIC begins one level earlier, with an anchored differential bundle internal to a tangent category. Its primitive coherence is the involution on the appropriate prolongation; after passage to sections in a classical realization, that structure induces the familiar Lie bracket.

Let \((\mathcal M,T)\) be a tangent category supporting the required differential bundles, and let

\[ A_{\mathrm{skill}}\longrightarrow M_{\mathrm{skill}} \]

be an involution algebroid of locally available skill directions. As in Chapter 4, two partial compilers must remain distinct:

\[ \begin{aligned} \operatorname {compile}^{\mathrm{sec}}_E& : \operatorname {Ob}(\mathsf{Md}_{\mathrm{sec}}) \rightharpoonup \Gamma _{\mathrm{loc}}(A_{\mathrm{skill}}),\\ \operatorname {compile}^{\mathrm{pol}}_E& : \operatorname {Ob}(\mathsf{Md}_{\mathrm{pol}}) \rightharpoonup \mathsf{Prog}_E(\mathbb S_{\mathrm{skill}}). \end{aligned} \]

Here \(\mathsf{Prog}_E(\mathbb S_{\mathrm{skill}})\) denotes the typed executable programs authorized by environment \(E\) for the current workflow sketch. The first turns a local skill artifact into a section \(s_m\); the second turns a workflow artifact into the executable target \(P_m\). The anchor realizes \(s_m\) as an infinitesimal change of the maintained skill state. The involution coherence controls how infinitesimal skill operations are exchanged; in a smooth realization, its section-level shadow is \([s_m,s_n]\).

When the observer \(O\) takes values in a space with subtraction, a normalized finite paired-order experiment can estimate this shadow:

\[ \widehat\Delta _\epsilon (m,n;x) =\frac{1}{\epsilon ^2}\! \left[ O\! \left(\Phi _n^\epsilon \Phi _m^\epsilon (x)\right) -O\! \left(\Phi _m^\epsilon \Phi _n^\epsilon (x)\right)\right]. \]

Under the usual smooth-flow hypotheses, its small-\(\epsilon \) limit is the derivative of \(O\) along the anchored section bracket, up to the order convention. In a general tangent-category realization, the paired routes require a comparison map rather than literal subtraction. A nonzero value may reveal order interference, a hidden shared resource, an incorrect interface, or an unavailable composite. It does not by itself say which explanation is correct. As in CLIC, the bracket is a localization signal rather than an admission certificate.

OPTIC adds the second DIAL direction. One side describes changes available inside the current skill theory; the other describes locally represented changes to its interfaces or workflow organization. When both sides admit involution-algebroid semantics, their mixed interaction belongs to the candidate double involution structure introduced in Chapter 3. We write \(\Theta _{\mathrm{skill}}\) for its internal interchange defect and require a declared realization to relate it to paired executions or section brackets. A finite grammar edit remains a sketch map, not an infinitesimal vector merely because local probes helped propose it.

. The tangent-category theory of one involution algebroid is established. The complete internal double structure used by DIAL remains a research target. OPTIC experiments therefore report the concrete realization being tested; they do not treat an observed execution-order difference as a proof of an abstract double-involution theorem.