ifc-0297

19.4 From noncommutativity to structural obstruction

The commutator of the two realized flows measures the lowest-order difference between “assimilate then accommodate” and “accommodate then assimilate.” Under the usual smoothness and local-flow assumptions,

\[ \Phi _{-\epsilon }^{B} \Phi _{-\epsilon }^{A} \Phi _{\epsilon }^{B} \Phi _{\epsilon }^{A} =\operatorname {id} +\epsilon ^2 [\rho _A(a),\rho _B(b)] +O(\epsilon ^3). \]

The bracket is consequently an order-sensitive mixed witness. It is not yet a creative obstruction. Neural vector fields commonly fail to commute, and a large bracket can arise inside a perfectly adequate representation.

To ask whether the current artistic language is closed under this interaction, combine the anchors:

\[ \rho _A\oplus \rho _B: A\oplus B\longrightarrow TM. \]

Assume first that its image has locally constant rank. Define the artistic obstruction bundle by the exact sequence

\[ A\oplus B \xrightarrow {\rho _A\oplus \rho _B} TM \xrightarrow {q_{\mathrm{art}}} Q_{\mathrm{art}} \longrightarrow 0, \]

where

\[ Q_{\mathrm{art}} =\operatorname {coker}(\rho _A\oplus \rho _B). \]
Definition 19.1 Artistic closure obstruction

For registered local sections \(a\) and \(b\), the DILATE obstruction is

\[ \Omega _{\mathrm{art}}(a,b) =q_{\mathrm{art}}\! \left([\rho _A(a),\rho _B(b)]\right) \in Q_{\mathrm{art}}. \]

It is the component of the mixed bracket that cannot be represented by the currently registered assimilatory and proto-accommodative directions.

Proposition 19.2 Tensoriality of the quotient obstruction

On a constant-rank stratum, \(\Omega _{\mathrm{art}}\) is \(C^\infty (M)\)-bilinear in \(a\) and \(b\). It therefore defines a local bundle morphism

\[ A\otimes _M B\longrightarrow Q_{\mathrm{art}}. \]
Proof ▼

Write \(X=\rho _A(a)\) and \(Y=\rho _B(b)\). For \(f,g\in C^\infty (M)\),

\[ [fX,gY]=fg[X,Y]+fX(g)Y-gY(f)X. \]

The last two terms lie in \(\operatorname {im}\rho _B\) and \(\operatorname {im}\rho _A\), respectively, so \(q_{\mathrm{art}}\) annihilates them. Hence \(q_{\mathrm{art}}([fX,gY])=fg\, q_{\mathrm{art}}([X,Y])\).

If \(\Omega _{\mathrm{art}}(a,b)=0\), the order effect closes inside the current language. It may still support valuable combinational or exploratory behavior, but it does not force a new basis element. If the quotient class is nonzero, the interaction exposes a direction for which the present artistic theory has no registered expression.

When the anchor image changes rank, \(Q_{\mathrm{art}}\) need not be a vector bundle. One must work stratum by stratum or replace the ordinary cokernel by an appropriate quotient sheaf, stacky quotient, or derived cokernel. Singular rank changes may themselves mark creative frontiers, but the elementary bundle formula must not be applied across them without justification.

. DILATE does not equate creativity with a nonzero Lie bracket. It looks for the persistent component of that bracket that survives projection away from everything the current visual theory can already express.