lin-0177
14.1 Why a Lie algebroid?
A tangent bundle gives every direction available at a state. An agent rarely has arbitrary access. Its possible changes are constrained by tools, permissions, schemas, and workflow state. A Lie algebroid separates the space of controlled directions from the changes they realize [ Mackenzie , 2005 ] .
A Lie algebroid over a manifold \(M\) consists of a vector bundle
a Lie bracket \([\cdot ,\cdot ]_A\) on sections \(\Gamma (A)\), and an anchor
satisfying
for sections \(s,t\) and smooth functions \(f\).
The anchor answers “what visible edit occurs?” The bracket answers “how do two controlled edits interact?” The kernel \(\ker \rho \) contains directions with no immediate visible displacement. Such directions can still alter provenance, tool state, routing assumptions, or future compositions.
|
Geometric object |
Agent interpretation |
Observable proxy |
Failure |
|---|---|---|---|
|
Section \(s\) |
typed edit or skill policy |
named repair operator |
not executable in current state |
|
Anchor \(\rho (s)\) |
realized artifact or workflow change |
diff, tool action, state transition |
anchor mismatch |
|
Bracket \([s,t]_A\) |
order-sensitive interaction |
\(s\! \to t\) versus \(t\! \to s\) contrast |
unstable composition |
|
Kernel direction |
latent procedural change |
trace difference with similar artifact |
future hidden regression |