lin-0029

0.15 A compact worked example

Consider a representation space \(H\), two learned transformations \(A_\theta ,B_\phi :H\to H\), and a decision map \(d:H\to Y\). Suppose the declaration says that the two transformations may be reordered without changing the decision:

\[ d\circ B_\phi \circ A_\theta = d\circ A_\theta \circ B_\phi . \]

Category and diagram.

The objects are \(H\) and \(Y\); the arrows are the adapters, their composites, and \(d\). The two sides are parallel paths from \(H\) to \(Y\).

Sketch.

The graph contains generators \(A,B,d\), and \(\mathcal D\) declares the two decision paths equal. The quotient of the path category identifies them.

Candidate model.

The learned maps \(A_\theta ,B_\phi ,d\) define \(D:\operatorname {Path}(S)\to \mathsf{Smooth}\). They may fail to factor through the quotient.

Observation.

At a probe \(h\in H\), one possible representative is

\[ e(h)= d(B_\phi (A_\theta (h))) - d(A_\theta (B_\phi (h))). \]

This subtraction uses the additional assumption that \(Y\) has suitable linear or metric structure.

Tangent observation.

For \(v\in T_hH\), the derivative \(De(h)[v]\) measures first-order change in the decision-level order discrepancy. Parameter probes \(\dot\theta ,\dot\phi \) answer a different question and must be labeled separately.

Quotient and localization.

Directions that alter hidden coordinates but leave \(d\) invariant may be decision-null. The remaining discrepancy can be localized to layers, adapter blocks, data regimes, or token positions.

Repair and admission.

A repair can adjust \(A_\theta \), \(B_\phi \), their routing, or the declared order relation. It is admitted only if the registered structural discrepancy improves on held-out probes while capability and safety constraints remain satisfied.

Admission contract

The equation above does not imply that all adapter orders ought to commute. It is a declared, decision-relative promise for a specified pair, regime, and support. If order carries intended semantics, forcing commutativity would be the wrong repair.

Application	Objects and arrows	Declaration	Obstruction	Typical repair
BRIDGE/SKFM	states and intervention fields	candidate distribution closes appropriately	bracket nonclosure or unstable causal extraction	revise fields, support, or downstream graph
LINCS-KET	indexed representations and attention routes	direct and extended routes agree	base or tangent route discrepancy	change architecture, route, or training rule
ALLORA	hidden states and adapters	declared orders preserve composition and safety	order-sensitive low-rank interaction	route, merge, separate, or retrain adapters
LASKO	task states and executable skills	sections compose and close	anchor, bracket, or feasibility failure	repair a skill or workflow edge
GIRL	states, values, and Bellman maps	value structure factors across computation	base or tangent Bellman failure	localized policy or representation update
LINCS-RLHF	alternatives and preference arrows	preference data support the chosen representation	cycle or nonintegrable preference field	relational fallback or guarded reward update
RADAR	relational views, joins, and charts	local charts descend to a shared geometry	incidence, cocycle, or apex failure	sparse decision-preserving chart repair
SID	contexts, sections, and restrictions	local predictive states glue effectively	compatibility or global-realization failure	tangent preservation or transverse correction
LINCS-Toulmin	claims, grounds, warrants, and sources	typed argument roles restrict and glue	role, source, qualifier, or rebuttal failure	localized typed argument edit
Trustworthy foundation models	agent roles, evidence, claims, and obligations	typed foundry workflows pass local and global checks	unsupported synthesis or unsafe cross-role composition	repair evidence, role routing, or admission policy

Table 4 The formal language stays fixed while the application semantics change. Later chapters make each row precise.