Part II

Two spreadsheets can encode the same customer list in different column orders and date formats. A recipient who only displays them has one problem; a program that substitutes one for the other has another. A date convention left implicit can survive an attractive rendering and still alter the next calculation. The mapping must say what survives the change.

Part I established the vocabulary: commitment sets, witnessed assertions, schemas, epistemic status. Part II equips that vocabulary with the structural machinery of equivalence. Vol I, Chapter 7 (The Witness Protocol) borrows Klein's discipline of specifying transformations before claiming invariance, while expressly declining to identify institutional arrangements with a mathematical group. The formal development here is self-contained.

What This Part Formalizes

Group actions and invariants (A7) formalize the Erlangen insight: invariants are properties that survive the declared action. A declared preservation contract identifies what must hold under specified changes; a group action is one mathematical account of such changes. A coordinate choice is not an invariant — it is a convention that different views may resolve differently.

Isomorphism and witness (A8) formalizes identity as behavior under probes. Two objects are isomorphic in the chosen category when inverse structure-preserving maps are supplied. The inverse is not optional — it is the witness that makes the identification auditable.

Adjunction (A9) supplies a natural correspondence between maps in two categories. In a preorder it can characterize a least or greatest approximation relative to a fixed map. A lossy translation need not form an adjunction; the unit, counit and their laws must be supplied. Those maps do not constitute a numerical measure of information or economic cost.

Witnessed sameness (A10) is Part II's culminating definition. Three levels of sameness — equality ($=$), isomorphism ($\cong$), equivalence ($\simeq$) — are distinguished, each with its own transport discipline. Transport is the formal mechanism by which a property proved for $A$ is carried to $B$ along a witnessed equivalence $e : A \simeq B$. All equivalences are scoped: $x \sim_S y$ holds in scope $S$, and using the equivalence outside that scope is a type error.

The vocabulary operator (A11) closes the loop. Vocabulary evolves: new predicates are added, old ones deprecated. The vocabulary operator $V \mapsto V'$ under invariant set $I$ formalizes this evolution as a controlled feedback loop, requiring conservative extension where possible — an extension claiming conservativity must add no consequences in the old language; a revision needs an explicit account of what changes.

Key Anchors

Anchor	Name	Status	Chapter
A7	Group Action & Invariant	formal	Ch. 6
A8	Isomorphism & Witness	formal	Ch. 7
A9	Adjunction	formal	Ch. 8
A10	Witnessed Sameness	formal	Ch. 9
A11	Vocabulary Operator	formal	Interlude

What the Formalism Does Not Capture

Equivalence is silent about trust. The witness for $A \simeq B$ says that the two objects are interchangeable for certain purposes; it does not say whether you should treat them as interchangeable in a given institutional context. A formally valid transport certificate can license a substitution that is structurally sound but politically unacceptable — merging two customer databases whose declared structural correspondence has been checked but whose data governance regimes are incompatible.

An adjunction can establish a universal property of a translation while leaving the choice of representations politically consequential. Which distinctions remain expressible, and who can change the chosen maps, are further questions. Neither a universal property nor a low conversion cost answers them.

Part III asks what happens when context itself varies: when truth is local, and coherence is the problem of gluing locals into globals.