Part IV

A catalog can contain descriptions that its current filters do not know how to use. Adding a predicate gives later queries a way to ask another question of those records. The addition may be as simple as naming a conjunction already expressible, or may require a new evaluator, evidence and decisions about its scope. Its importance lies in the uses it makes available, not in how dramatic the edit looks.

Part IV follows that addition through proposal, checking and admission. In the constructed fashion example, “puffy” acquires an evaluation procedure and a declared population. A receiving team must then determine which comparisons support the filter and whether existing queries retain their meaning. Adding a column or training a classifier may perform part of the work; the receiving contract determines what else is required.

Vol I, Chapters 5 and 7 develop related questions about categories, transformations and reliance; the puffy-dress construction here is a worked example of the formal discipline.

What This Part Formalizes

Predicate invention (A17) specifies admission obligations: a definition must be evaluable, its required comparisons supported and the receiving invariants satisfied. Exact overlap agreement can invoke A13 when the sheaf hypotheses hold. A statistical comparison has its own procedure and reach. Neither comparison alone establishes the other admission conditions.

Conservative extension (A17b) preserves exactly the consequences in the old language. That logical result is distinct from preserving the behavior of a deployed query or meeting a contract after its evidence changes. A new definition can be deductively conservative while altering execution, and a justified correction can require abandoning an old conclusion. The relevant preservation claim must name what survives; a breaking revision must account for what changes.

The invariant set (A18) distinguishes hard from soft constraints. Hard invariants must hold; violation means rejection. Soft constraints are penalized; violation means cost. The distinction matters because real systems have both: the constructed catalog treats referential integrity as hard and its ranking preferences as soft. The coherence budget is allocated across both kinds.

The proposal operator (A19) supplies candidates; the certification contract (A19b) asks what can be established about them for the intended use. A proposal may be informative while still uncertified. A checker can examine a candidate supplied by a person or another service without enumerating every alternative. The distinction matters at the point of reliance: successful certification, a demonstrated violation and an unfinished check must not become interchangeable answers.

Predicate search space (A20) makes the candidate grammar and search procedure explicit. Bounds can make enumeration finite; pruning can make it useful while leaving some candidates unexplored. A failed bounded search therefore needs a record of its coverage, not a claim that no acceptable predicate exists.

The coherence cost model (A21) is Part IV's culminating construction. Every coherence operation has a cost: local check cost, overlap check cost, witness verification cost, version migration cost. The coherence budget declares the maximum cost the system will pay per operation. Its purpose is to expose what a checking regime funds and what happens when the budget binds. It is an accounting framework, not a consequence of the CAP theorem or a universal ordering of verification costs.

Key Anchors

Anchor	Name	Status	Chapter
A17	Predicate Invention	formal	Ch. 15
A17b	Conservative Extension	formal	Ch. 16
A18	Invariant Set	formal	Ch. 16
A19	Proposal Operator	formal	Ch. 17
A19b	Certification Contract	formal	Ch. 17
A20	Predicate Search Space	formal	Ch. 18
A21	Coherence Cost Model	formal	Ch. 19

What the Formalism Does Not Capture

The cost model measures coherence work. It does not measure political acceptability. Two signature extensions may have identical formal cost but vastly different institutional consequences — one extends a vocabulary that a single team controls, the other touches a shared ontology governed by committee. Its organizational category can include time spent securing agreement. That expenditure does not determine whether the committee's decision is acceptable to the people subject to it.

A declared certification task can exceed the available evidence or checking resources. The contract must expose that unfinished work. Choosing the procedure also has consequences: someone determines the property, the tolerated error and the authority needed for use. Supplying an evaluator does not settle those choices.

Part V brings these obligations into a receiving operation: the records it needs, the checks it performs and the grounds it must carry forward.