A Gluing Attempt

Consider a constructed catalog example. The records, identifiers and evaluations below are schematic; they do not describe a company's deployment. A device has a recorded FDA 510(k) clearance, while a separate record establishes that it is not CE marked. The catalog asks whether it satisfies a stated condition in both markets.

There are two local answers. Before attempting to glue them, we have to establish whether they answer the same question.

The Views

The US view records product identity, the relevant clearance and its scope. The EU view records product identity and CE-marking status under the applicable regime. These are different regulatory acts. Their records may share a product identifier without sharing a predicate about authorization. FDA clearance through the 510(k) route concerns a finding of substantial equivalence; an exemption from that submission requirement is a different status, subject to its own conditions.¹

For this example, both views distinguish an explicit negative record from a failed search. Missing information remains unknown. The EU value for device X is stipulated by an explicit status record, not inferred from absence in EUDAMED. In particular, the Commission's January 2024 account described the available EUDAMED modules as being in voluntary use. That database could not warrant the blanket inference that an absent entry meant an unmarked device.²

The Predicate

Let market_status(x, m) be a simplified catalog test: recorded clearance in the stipulated US case, recorded CE marking in the stipulated EU case. This is not a complete determination of whether a device may legally be marketed. The intended use, version, applicable procedure and other requirements cannot be discharged by our two fields.

The market argument matters. market_status(x, US) and market_status(x, EU) are distinct propositions, even when the product argument is identical. A context-indexed representation keeps this difference visible.

The catalog defines its aggregate test:

Conjunction is an aggregation rule. Calling the two views a cover does not make conjunction a sheaf-gluing operation. Nor does a shared identifier authorize replacing a statement indexed by one market with a statement indexed by another.

The predicate package therefore records the two evaluations, the rule combining them, and the product and date to which the records apply. It claims no invariant under which a status, once acquired, persists forever. Continuing validity has to be established under the relevant terms.

The Query

The query first obtains the local evaluations and their grounds, then applies the declared rule. Its three-valued contract distinguishes a supported negative result from an incomplete inquiry. In this example, true conjoined with false is false; true conjoined with unknown is unknown.

QueryContract {
  expression: both(device_X)
  arguments: [market_status(device_X, US), market_status(device_X, EU)]
  aggregation: conjunction
  missing_evidence: Unknown
  status_scope: [product_version, intended_use, applicable_date]
}

A catalog may decline to list a product while the result is unknown. That is an action policy under uncertainty. It must not turn the missing premise into a discovered negative fact.

The Restriction

For device X, the stipulated source evaluations are:

ClaimReceipt {
  id: example-US-X
  claim: market_status(device_X, US) = true
  evidence: example-clearance-record-X
  derivation: documented-status
}

ClaimReceipt {
  id: example-EU-X
  claim: market_status(device_X, EU) = false
  evidence: example-explicit-unmarked-status-X
  derivation: documented-status
}

The documentary status of each ground remains open to inspection. A label such as DOCUMENTARY, or a confidence field set to one, would add no verification of its contents. The example assumes the two scoped records support the stated local values so that we can examine what follows from them.

The Gluing Condition

The pair {US: true, EU: false} is perfectly consistent. The catalog's aggregate answer is false. Neither result has been suppressed, and no contradiction has been proved.

To construct a genuine gluing question, one would need a site, a presheaf and restriction maps that place local data on a common overlap. If both records carry the same product identity, their identity restrictions can agree while their market-indexed status fields differ. A sheaf could assemble those fields into a global record preserving both values. The aggregate test would still be false.

Conversely, erasing the market argument and demanding equality of the remaining truth values creates a different problem. The resulting conflict comes from identifying unlike propositions. It does not show that the original information had no global section. The purported obstruction would be evidence against the translation imposed by the catalog, not against the coexistence of its sources.

This is where the formal discipline earns its keep: it makes us exhibit the restrictions before treating disagreement as a mathematical finding. A shared object does not make every statement about it interchangeable.

The Logic Mismatch

Now let device Y have the stipulated US clearance record but no usable EU status evidence. Its local values are true and unknown. The aggregate remains unknown.

We can record why the inquiry stopped: the EU premise has not been established. We cannot describe this as a conflict between US testimony and EU rejection. There is no EU rejection in the record. Even if an institution adopted a defensible closed-world rule for some complete, bounded register, it would have to establish that rule's coverage before deriving a negative from absence. Declaring the policy is not evidence that its conditions hold.

The distinction changes the next task. Device X fails the stated two-market test on the assumed facts. Device Y requires further evidence before the test can be decided. A single red indicator might lead to the same immediate listing decision, but it would conceal two different reasons and two different routes forward.

The Transport Temptation

The business might ask whether material supporting the US determination could assist a European assessment. That is a legitimate evidentiary question. Different procedures do not establish that none of the underlying testing or documentation can be useful elsewhere.

It is also different from the proposed implication cleared_US(x) ⇒ CE_marked(x). No ground for that status substitution has been supplied in this example. Product identity alone cannot supply it. The receiving assessment must determine which evidence bears on its requirements and what additional work remains.

A transport certificate, in the sense of Chapter 14, has to identify the property and the map under which it is preserved. Calling a request an approximation does not prove even a weaker implication. Where no adequate evidence is available, the system records an unsupported transfer request; it does not manufacture a counterexample to every possible use of the source material.

The Receipt

The appropriate record for device X is therefore an evaluated negative, with a separate disposition of the attempted substitution:

QueryResult {
  query: both(device_X)
  local_values: { US: true, EU: false }
  premises: [example-US-X, example-EU-X]
  aggregation: conjunction
  value: false
  status_transfer: {
    request: cleared_US(device_X) implies CE_marked(device_X)
    disposition: unsupported-by-supplied-evidence
  }
}

For device Y, value would be Unknown, with the missing EU premise identified. An obstruction witness belongs only where an actual incompatibility has been established under the declared representation. Neither a negative conjunction nor missing evidence supplies one by itself.

What the Machinery Bought

Ordinary typed records and a careful query contract can already preserve these distinctions. The categorical account helps specify what a more ambitious gluing claim would require; it does not make every aggregation an application of the sheaf theorem.

The practical achievement is to preserve which question was answered, on whose grounds, and under which rule its answer was used. The company can change the catalog's market coverage, obtain further evidence, or pursue an additional assessment. Changing the coverage changes the promise of the listing. It does not alter the device's status in the market omitted.

The Point

The first obligation of a gluing attempt is to establish what is supposed to agree. Here the local information can coexist, the chosen aggregate can be false, and the grounds for a different regulatory claim can still be missing. Treating all three as one obstruction would impoverish the decision it purports to explain.

Part IV turns to constructing predicates. It inherits a requirement already visible in this small example: a new expression must not gain an answer by forgetting which question its evidence answered.

Footnotes

1. FDA, “Premarket Notification 510(k)”, “Introduction,” “What is Substantial Equivalence,” and “When a 510(k) is Not Required,” inspected September 11, 2026. The source distinguishes clearance from identity with the predicate device and describes exemptions with limitations. The catalog records here are constructed, not records retrieved from that database. ↩

2. European Commission, COM(2024) 43 final, January 23, 2024, explanatory memorandum §1(b), “European database on medical devices (Eudamed).” This is the proposal's dated account of deployment, not a claim about the database's later coverage or the proposal's enactment. ↩