Part III

A merchant's catalog and a regulatory filing can describe the same product while answering different questions. Before combining their answers, a recipient must establish what each record asserts and how the proposed comparison relates them. Agreement is useful only after those questions have been made comparable; a difference between unlike claims is not yet a contradiction.

Part III develops one exact account of this work. Local data are assigned to views, comparison maps say what reaches their overlaps, and the sheaf condition states when a matching family has a unique amalgamation. The choice of views and maps is consequential: it determines which differences the construction preserves and which comparisons it asks anyone to make.

For the courtroom analogy, the hybrid seam, and the custody problem, see Vol I, Chapters 5–6. What follows assumes only Parts I and II.

What This Part Formalizes

Cover and restriction (A12) and context site structure (A12b) formalize the space of views. Contexts form a category (or poset) with refinement morphisms; a cover $\{U_i \to U\}$ declares that a collection of local views jointly exhausts a domain. Restriction maps translate claims from coarse views to fine views. The Grothendieck topology specifies which covers the system recognizes — a governance decision, not a mathematical given.

The sheaf condition (A13) is the mathematical center of this part. For a specified sheaf, an exactly matching family on a cover has a unique global section. Locality makes a section distinguishable through its restrictions; gluing establishes that compatible restrictions can be assembled. This gives A5 a precise model for exact composition. A recipient using statistical estimation, an independent investigation or a justified decision rule must establish that result on its own grounds. It need not turn the result into an exact amalgamation to make it useful.

Fibration (A14) formalizes context-dependent predicates. "Best" is not a flat predicate; it lives in a fiber over the context (diet, budget, taste). The fibration $p : E \to B$ captures this: the fiber $E_b$ over context $b$ contains the types and predicates valid in that context, and reindexing maps specialize predicates as context gains parameters. There is no context-free "best." There is no context-free "affordable." The predicate exists only relative to a base.

Logic selection (A15) distinguishes reasoning rules from the treatment of absent information. Classical logic permits double-negation elimination generally; intuitionistic logic does not. A closed-world policy can license a negative conclusion where an open-world policy leaves the matter unestablished. Combining the records requires preserving the grounds of those different conclusions. A common status field cannot settle the difference merely by giving them the same label.

Transport discipline (A16) specifies the evidence required for a substitution. The recipient checks the relation, the property map and their applicability to its scope; a record merely asserting that a certificate exists does not complete those checks. Type theory supplies a precise account of transport along identifications. A16 states a related engineering contract whose implementation must be demonstrated. A missing certificate leaves the proposed substitution unsupported, rather than proving that no other grounds could support it.

Key Anchors

Anchor	Name	Status	Chapter
A12	Cover & Restriction	formal	Ch. 10
A12b	Context Site Structure	formal	Ch. 10
A13	Sheaf Condition	formal	Ch. 11
A14	Fibration	formal	Ch. 12
A15	Logic Selection	formal	Ch. 13
A16	Transport Discipline	engineering	Ch. 14

What the Formalism Does Not Capture

The exact result concerns sections of the specified sheaf. It does not establish the truth of the observations represented by them or require jurisdictions to adopt identical rules. Two different permissions can be recorded together without forcing them to become one permission. Where an actual comparison fails, its record can identify what differs; deciding whether either rule should change requires an authority and an argument the sheaf condition does not supply.

Fibrations capture context-dependence but not the politics of context-creation. Who decides which contexts exist? Who decides which covers are recognized? The chosen site records contexts and recognized covers; it does not by itself record who had authority to choose them or justify that choice. The formal reader should understand: the site structure is a parameter of the system, not a theorem about the world.

Part IV asks what happens when the vocabulary itself changes: which new questions can be admitted, and what the new use owes to results already in circulation.