# Category Theory Foundation for the Civic Exchange Protocol (CEP)

> Mathematical definition of the category **CEP**

## Abstract: A Categorical Data Model for Verifiable Information Exchange

Government and institutional information today is fragmented across heterogeneous systems that lack shared identity, provenance, and attestation structures.  
The **Civic Exchange Protocol (CEP)** provides a mathematically principled interoperability layer defined as a category **CEP**, whose objects are attested CEP entities and whose morphisms are typed CEP relationships and exchanges.

The core of CEP rests on six structural invariants:

1. Distributed trust / decentralized verification
2. Immutable provenance chains
3. Jurisdictional autonomy
4. Verifiable ID as a universal construction
5. Cryptographic attestation
6. Non-destructive, hash-anchored revision chains

These invariants guarantee identity consistency, provenance integrity, and long-term trustworthiness across civic systems while preserving local control.

CEP's categorical semantics allow data from heterogeneous sources to interoperate, compose, and verify in a predictable, inspectable, and audit-ready manner.

---

# 1. The Category CEP

## 1.1 Objects: Attested CEP Entities

Objects in **CEP** are attested entity records conforming to the CEP schema.  
An object represents a legal or administrative actor (e.g., agency, vendor, nonprofit, district) at a specific revision.

Formally:

```
Ob(CEP) = { E | E is a valid, attested CEP entity record }
```

Each entity object must satisfy the following invariants:

### Invariant: Cryptographic Attestation

Every entity includes a verifiable attestation signed with independently resolvable keys.

### Invariant: Distributed Trust / Decentralized Verification

Verification must not rely on a single authority.  
Keys and verification endpoints must support independent validation across jurisdictions.

---

## 1.2 Morphisms: CEP Relationships and Exchanges

Morphisms are directed civic flows between entities.

### Relationship morphisms

Structural, legal, or organizational links:

-   grant_award: FederalAgency → StateAgency
-   contract: Agency → Vendor
-   license: Regulator → Licensee

### Exchange morphisms

Operational transfers of value, obligation, or authority:

-   disbursement: Grantor → Grantee
-   payment: ContractingOffice → Vendor
-   allocation: BudgetDept → Division

Morphisms carry:

-   type
-   jurisdiction
-   attestation
-   parent references
-   revision metadata

### Invariant: Immutable Provenance Chains

Every morphism participates in a complete, immutable provenance chain.  
No selective omission, truncation, or overwriting is permitted.

---

## 1.3 Identity Morphisms

Each entity E has an identity morphism:

```
id_E : E → E
```

This corresponds to the entity's inaugural self-attested revision.

---

# 2. Composition and Provenance

Given:

```
f: A → B
g: B → C
```

their composition is:

```
g ∘ f : A → C
```

Composition expresses derived provenance, allowing a downstream recipient to understand the full lineage of a flow.

### Invariant: Provenance Chains Must Be Immutable and Complete

Composition must preserve the entire provenance chain, with no truncation or mutation.

### Associativity

As required by category theory:

```
(h ∘ g) ∘ f = h ∘ (g ∘ f)
```

CEP guarantees associativity through:

-   parent relationship and exchange references
-   hash-linked revision chains
-   cryptographic attestations

---

# 3. Verifiable Identity as a Universal Construction

Government systems often assign different identifiers to the same real-world entity.  
CEP resolves this through a single canonical **Verifiable ID**, defined mathematically as a **Limit** over the diagram of source identifiers.

## 3.1 Source Identifier Diagram

An entity may appear in:

-   SAM.gov
-   state vendor systems
-   LEI databases
-   county procurement files
-   legacy systems

These form a diagram of partial mappings.

## 3.2 The Limit

The Verifiable ID is the unique object V equipped with projections to all identifier sources such that any other object mapping to the same sources factors uniquely through V.

### Invariant: Verifiable ID as Universal Construction

Once assigned, the Verifiable ID is stable across revisions and cannot be overwritten.  
Identity is extended only by adding evidence, never by destructively modifying past identity claims.

---

# 4. Revision Chains and Hash Anchoring

Real-world data evolves over time.  
CEP models this evolution through a non-destructive revision sequence:

```
E₁ → E₂ → E₃ → ...
```

Each revision is an object in CEP, linked by an amendment morphism.

### Invariant: Non-Destructive Revision Chain

Revisions must accumulate.  
No revision may be deleted or overwritten.

### Structure of Revisions

Each revision includes:

-   a `revisionNumber`
-   a `previousRecordHash` linking cryptographically to the prior revision
-   a new attestation verifying content and timestamp

These structures guarantee:

-   tamper-evident change history
-   reproducibility of past states
-   future-verifiable evidence of updates

---

# 5. Jurisdictional Structure as Slice Categories

Government entities operate within political or administrative boundaries.  
CEP models jurisdictional scoping using **slice categories**.

For a jurisdiction J:

```
CEP / J
```

contains:

-   all entities whose jurisdiction aligns with J
-   all morphisms where both endpoints lie in J

### Invariant: Jurisdictional Autonomy

Each jurisdiction governs its own objects and attestations.  
Higher-level contexts may compose with these records but may not override them.

This supports:

-   local control
-   independent verification
-   aggregation across jurisdictions

without centralizing authority.

---

# 6. The Verified Subcategory

Define:

```
CEP_verified ⊆ CEP
```

as the full subcategory where:

-   attestations validate
-   verification methods resolve
-   hash links match content
-   provenance chains are intact

### Invariant: Cryptographic Attestation

Verification depends on cryptographic evidence, not on trust in a hosting party.

Morphisms in the verified subcategory remain verified under composition.

---

# 7. Summary of Embedded Invariants

CEP embeds six structural invariants directly into the mathematical definition of **CEP**:

1. Distributed verification
2. Immutable provenance chains
3. Jurisdictional autonomy
4. Universal-construction Verifiable ID
5. Cryptographic attestation
6. Non-destructive revision chains

These invariants ensure that CEP provides a verifiable, interoperable, tamper-evident, and jurisdictionally respectful foundation for verifiable information exchange across jurisdictions and domains.
