ifc-0063

4.4 Assimilation and accommodation in skill geometry

Chapter 3 described assimilation as change inside a maintained differential universe and accommodation as a typed change of that universe. The Lie algebroid makes this distinction operational. Fix the creative skill geometry

\[ A\xrightarrow {\pi }M,\qquad \rho :A\longrightarrow TM,\qquad [\cdot ,\cdot ]_A, \]

together with its Markdown language and compiler. A possibly time-dependent family of compiled local sections \(s_t\) generates an assimilatory trajectory when

\[ \dot x_t=\rho (s_t(x_t)) \]

and the trajectory, its skill compositions, and its admission tests remain expressible in this fixed package.

Definition 4.1 Lie algebroidal assimilation

An assimilatory skill update is an admitted rollout generated by sections of the maintained Lie algebroid, using the current anchor, bracket, base-state representation, and semantic compiler. It changes the theory state \(x\in M\) without changing the geometry that specifies which skills are locally available or how their interactions act on \(M\).

This definition is geometric rather than metric. A long rollout or a large finite displacement can still be assimilation if it is assembled from admissible flows in the existing algebroid. Conversely, a small textual edit can be proto-accommodative if it proposes a new state variable, skill type, or interaction law; it becomes accommodation only after finite realization and admission.

Pressure for proto-accommodation becomes credible when persistent evidence cannot be represented by the current anchored skill system. Even then, the diagnostics motivate an extension; they do not specify or admit it. Typical diagnostics include:

  • the required theory motion lies outside \(\operatorname {im}\rho \);

  • available sections fail to generate a needed direction under brackets;

  • a learned anchor–bracket residual persists after estimation error and observer misspecification have been checked;

  • distinct mechanisms collapse to the same base state or anchor image; or

  • the Markdown compiler cannot type a skill required by the proposed repair.

The candidate extension is represented by an extended algebroid

\[ A^+\xrightarrow {\pi ^+}M^+,\qquad \rho ^+:A^+\longrightarrow TM^+, \]

and transport data consisting at least of a semantic base map \(\bar J:M\to M^+\) and a bundle map \(\phi _A:A\to A^+\) over \(\bar J\).


Here \(J:\mathbb S\to \mathbb S^+\) is the presentation map and \(\bar J\) its induced or supplied semantic transport; they are not the same typed object. The anchor comparison

\[ \rho ^+\circ \phi _A \; \stackrel{?}{=}\; T\bar J\circ \rho \]

tests whether old skills retain their visible meaning under the proposed extension. Bracket preservation supplies a second comparison whenever the relevant sections are \(\bar J\)-related. If the standard compatibility conditions hold, \((\phi _A,\bar J)\) is a Lie algebroid morphism. Requiring a morphism is appropriate for conservative transport, but not every creative extension should preserve all old interactions strictly; any failure must instead be recorded and justified.

Definition 4.2 Admitted Lie algebroidal accommodation

An admitted accommodative skill update is a typed revision of the base \(M\), skill bundle \(A\), anchor \(\rho \), bracket, Markdown language, or compiler, motivated either by a persistent obstruction that no admitted rollout in the maintained algebroid accounts for, or by a registered opportunity whose value cannot be expressed in that algebroid. Its audit record includes the extended algebroid, the transport pair \((\phi _A,\bar J)\), the comparison defects, and an independent admission result that warrants retaining the revision.

Piaget’s conservation example has a particularly simple anchored form. Once quantity is represented by an observer \(q:M\to Q\), pouring skills \(s\) should satisfy

\[ Tq\bigl(\rho (s(x))\bigr)=0. \]

Before accommodation, the state space or observer family may encode only height and vessel shape, so this invariant direction is not expressible. Accommodation refines the base or its observers so that quantity-preserving motions become identifiable; later pours are then assimilated as trajectories tangent to the level sets of \(q\). For object permanence, accommodation similarly enlarges \(M\) with a persistent latent object state and lifts visible and occlusion-handling skills into \(A^+\). Subsequent occlusions can then be assimilated within the enlarged geometry.

. Search first for an assimilatory rollout in the current Lie algebroid. Propose a finite extension only after testing whether an apparent nonclosure, missing direction, or anchor defect is caused by finite data, an inadequate observer, or a compilation error. When extension is necessary, preserve the old skill geometry through explicit transport wherever the evidence warrants it.