nf.classical.1nf

Atomic Domain First Normal Form (1NF)

Contract ID: nf.classical.1nf | Family: shape | Index Status: active | AuthorityEffect: None

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1. Executive Summary & Metadata

Atomic Domain First Normal Form (1NF) enforces structural normalization and explicit defect routing in the shape regime.

Property	Value
Canonical ID	nf.classical.1nf
Short Name	1NF
Full Name	Atomic Domain First Normal Form
Family	shape (Relational, Algebraic & Cochain Shape Normal Forms (Data at Rest & Structural Decompositions))
Qualification	Extended Form Contract (Normal Form Contract)
Data / Math Posture	at_rest
Index Status	active
Theory Status	established
Verification Status	absent
Contract Version	v0
Governance Fence	AuthorityEffect: None (certified_implies_authorized: false)

Foundational Treatise Specification

• Author: Codd
• Year: 1970
• Structural Invariant: Scalar domains, a set of tuples, and a declared candidate key.
• Anomaly Excluded: Positional identity and repeating groups.
• Lossless Join Property: Identity.
• Dependency Preservation: Preserved.
• Qualifies As True Normal Form: True

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2. Visual Architecture & Normalization Diagram

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3. Concrete Worked Example & Mechanism

Visual Transition Summary

Stage	Carrier / Operator	Concrete State / Invariant
1. Before / Candidate Input	Violation: Non-Atomic Attribute	Student(id=1, name='Alice', ; courses='CS101, MATH202') ; Defect: Set/CSV valued cell
2. Normalization / Gate	1NF Atomic Flatten	forall A in H: dom(A) scalar (certified_implies_authorized: false)
3. After / Certified Output	Normalized 1NF Relation	(id=1, name='Alice', course='CS101') ; (id=1, name='Alice', course='MATH202') ; Key: (id, course)

Contract-Declared Worked Cases

- ref: coda_all_universe/1_the_tower/normal_forms/contracts/_book_09_full_normal_form_atlas.md
  section: 2.1 Classical Relational Normal Forms

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4. The Four-Pillar Quadrilateral & Non-Claims

Every contract in simplex-normal-forms is evaluated against the Four-Pillar Quadrilateral defined in contracts/_golden_nfs/standard.yaml:

Pillar	Requirement	Contract Specification for nf.classical.1nf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	hidden nested structure / repeated groups / non-atomic attribute domains
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: the body is a set of tuples, each attribute value lies in a declared scalar domain, and tuple identity is a declared candidate key Protected Skeleton: ['heading', 'domains', 'tuple_atomicity', 'candidate_keys']
Pillar III: Lossless Reconstruction	State the non-loss reconstruction operator (join, direct sum, pullback, or replay).	Lossless reconstruction under declared projection, direct sum, or pullback operators.
Pillar IV: Syntactic Decidability	State how conformance is decided from declared constraints without unbudgeted scans.	Decidable from declared schema and witness metadata within bounded evaluation budgets.

Explicit Non-Claims (certified_implies_authorized: false)

• does not certify runtime repair
• does not imply authorization
• does not resolve functional dependency or join dependency anomalies

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• heading
• domains
• candidate_keys
• relation_predicate

5.2 Required Evidence

None declared.

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	every attribute value belongs to declared atomic domain
criterion	Check	no column encodes repeated values or multi-attribute structures (e.g. unparsed CSV, nested lists) without explicit domain contract
criterion	Check	tuple identity derives from declared candidate key, not physical row position

5.4 Terminal Route Map

Every evaluation terminates in a declared local route mapped to a canonical terminal in standard.yaml:

Local Terminal Route	Canonical Terminal
Accept	Preserve
RejectNestedCarrier	Reject
RejectUntypedRepeatedGroup	Reject

5.5 Declared Failure Modes

• symptom: invoice line items stored as CSV in one column, diagnosis: non-atomic carrier violating 1NF, terminal: RejectNestedCarrier

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Deep-Dive Level Treatise

AuthorityEffect: None.
Level: Level 01 (Atomic Domains & Scalar Heading Separation)
Evidential Grade: NATIVE IMPLEMENTED
Contract Reference: nf.classical.1nf
Mathematical Posture: Classical Relational Algebra (Codd 1970)
Data Posture: At Rest

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1. Schematic & Conceptual Architecture

First Normal Form (1NF) establishes the foundational discipline of the relational model:

• All domain values are strictly atomic (indivisible scalars).
• Repeating groups and array-valued attributes are eliminated via unnesting or relation splitting.
• Each relation has an explicit heading of distinct attribute names, and tuple ordering is unordered (set semantics).

       0NF Relation (Repeated Groups)
  ┌────┬──────────────┬────────────────────────┐
  │ ID │ Name         │ Phone Numbers          │
  ├────┼──────────────┼────────────────────────┤
  │ 1  │ Alice        │ ["555-0100", "555-0101"]│
  └────┴──────────────┴────────────────────────┘
                       │
             Atomic Unnesting (1NF)
                       ▼
  ┌────┬──────────────┬────────────────────────┐
  │ ID │ Name         │ Phone Number           │
  ├────┼──────────────┼────────────────────────┤
  │ 1  │ Alice        │ "555-0100"             │
  │ 1  │ Alice        │ "555-0101"             │
  └────┴──────────────┴────────────────────────┘

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2. Mathematical Invariants & Quadrilateral Properties

1. Relvar Formulation: A relation $R$ over heading $H = {A_1, \dots, A_n}$ is in 1NF if and only if for every tuple $t \in R$ and every attribute $A_i$: $$t[A_i] \in \operatorname{dom}(A_i)$$ where $\operatorname{dom}(A_i)$ is a declared set of atomic scalar values, and $R$ contains no duplicate tuples.

2. The Quadrilateral Invariants:

   • Anomaly Excluded: Attribute positional dependence, repeating group update anomalies, and nested collection search failure.
   • Protected Skeleton: Candidate keys and scalar attribute values.
   • Erased Structure: Intra-tuple collection ordering and nested group boundaries.
   • Reconstruction: Projection and aggregation over candidate keys.

3. Non-Claims:

   • 1NF does not eliminate functional dependency redundancies (reserved for 2NF/3NF/BCNF).
   • 1NF does not ensure under declared assumptions single-attribute primary keys.

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3. Terminal Routes & Verification Logic

Relation Schema & Table Instance
      │
      ├── All values scalar & no duplicates ──► Preserve (1NF Satisfied)
      │
      ├── Attribute contains non-scalar bag ──► Quarantine (NonAtomicAttribute)
      │
      ├── Duplicate tuples detected ──────────► Refuse (DuplicateTuples)
      │
      └── Missing candidate key definition ──► Refuse (NoCandidateKey)

• Preserve: All attributes belong to atomic scalar domains and candidate key is unique.
• Quarantine: Nested collections detected; requires unnesting transformation.
• Refuse: Duplicate rows or missing key integrity.

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4. Test Evidence & Local Execution

Verified in test_tranche3_advanced_relational_nfs.py and test_normal_forms.py:

• check_1nf(schema, instance): Validates that all attribute values are primitive Python types (int, str, Fraction, bool) and no nested list/dict structures exist.
• test_1nf_duplicate_tuple_rejection: Asserts that multiset bag relations fail 1NF verification.

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5. Epistemic Claim Quadrant

1. What it supports:
   • Strict scalar relational algebra and first-order predicate calculus querying.
   • Elimination of positional and repeating-group anomalies.
2. What it does not support:
   • Prevention of partial or transitive functional dependency anomalies.
3. The narrowest defensible claim:
   • 1NF ensures under declared assumptions that every cell of the relation contains exactly one value from the underlying domain.
4. The next test needed:
   • Property tests asserting that unnesting preserves candidate key functional dependencies under Cartesian expansion.

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6. Epistemic Claim Hygiene Quadrant

In accordance with the repository's epistemic claim discipline:

1. What it supports:
   • Supports syntactic and operational checking of nf.classical.1nf (1NF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (Accept, RejectNestedCarrier, RejectUntypedRepeatedGroup) when required declarations or invariants fail.
2. What it does not support:
   • Does not confer operational authority (AuthorityEffect: None, certified_implies_authorized: false).
   • Does not claim semantic truth of undeclared external processes or unverified upstream inputs beyond the declaredprotected skeleton.
3. The narrowest defensible claim:
   • Within the declared substrate at_rest, this contract enforces that hidden nested structure / repeated groups / non-atomic attribute domains is either excluded under the body is a set of tuples, each attribute value lies in a declared scalar domain, and tuple identity is a declared candidate key or routed to an explicit refusal/recovery terminal.
4. The next test needed:
   • Property-based falsification and mutation testing across boundary edge cases and composed multi-step pipelines for nf.classical.1nf.

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7. Canonical YAML Contract (contracts/nf.classical.1nf.yaml)

Click to expand full canonical YAML contract (contracts/nf.classical.1nf.yaml)

schema_version: smplx.nf.contract.v0
schema_ref: ./normal_form_contract_schema.yaml
contract_kind: normal_form_contract
identity:
  id: nf.classical.1nf
  name: Atomic Domain First Normal Form
  short_name: 1NF
  version: v0
  theory_status: established
  contract_status: active
  verification_status: absent
  family: shape
  status: active
classification:
  hierarchy_level: normal_form
  subject_kind:
  - relvar
  data_posture: at_rest
  mathematical_posture: operational
purpose:
  excludes_anomaly: hidden nested structure / repeated groups / non-atomic attribute
    domains
  preserves_invariant: the body is a set of tuples, each attribute value lies in a
    declared scalar domain, and tuple identity is a declared candidate key
  protected_skeleton:
  - heading
  - domains
  - tuple_atomicity
  - candidate_keys
  erased_structure: []
  non_claims:
  - does not certify runtime repair
  - does not imply authorization
  - does not resolve functional dependency or join dependency anomalies
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  citation:
    author: Edgar F. Codd
    year: 1970
    work: A Relational Model of Data for Large Shared Data Banks
  formal_statement: 'A relvar is in first normal form when every attribute takes values
    in a declared scalar domain, the body is a set of tuples, and tuple identity is
    a declared candidate key rather than storage position. A domain is scalar when
    the schema does not give it relation-valued or repeating-group operators. Atomicity
    is fixed by those declared operators. A string that the predicate treats as one
    scalar is scalar. A string that the predicate stipulates to encode several values
    is a repeating group.

    '
  quadrilateral:
    qualifies_as_true_normal_form: true
    anomaly_exclusion: positional identity, repeating groups, and relation-valued
      cells
    lossless_join:
      status: identity
      statement: 'No decomposition is performed. Reconstruction is the identity on
        the relation value.

        '
    dependency_preservation:
      status: not_applicable
      statement: 'No dependency is moved to another relvar. Every declared dependency
        remains on this heading.

        '
    syntactic_decision:
      status: schema_decidable
      statement: 'Decided from the heading, the domain declarations, and the candidate
        keys. A particular table body is not the decider.

        '
  chain:
    implied_by:
    - 2NF
    - 3NF
    - EKNF
    - BCNF
    - 4NF
    - ETNF
    - 5NF
    - 6NF
declarations_required:
- heading
- domains
- candidate_keys
- relation_predicate
evidence_required: []
validation_criteria:
- every attribute value belongs to declared atomic domain
- no column encodes repeated values or multi-attribute structures (e.g. unparsed CSV,
  nested lists) without explicit domain contract
- tuple identity derives from declared candidate key, not physical row position
proof_obligations:
- id: 1nf.set_semantics_and_domain_admissibility
  status: theorem_shaped
  ref: coda_all_universe/1_the_tower/normal_forms/contracts/_book_09_full_normal_form_atlas.md
  statement: 'A 1NF relvar is a set of tuples over the declared scalar domains, with
    identity given by a candidate key. Domain scalarity is the schema''s operator
    declaration.

    '
terminal_routes:
- Accept
- RejectNestedCarrier
- RejectUntypedRepeatedGroup
terminal_route_map:
  Accept: Preserve
  RejectNestedCarrier: Reject
  RejectUntypedRepeatedGroup: Reject
certification:
  certificate_required: false
  certificate_kind: null
  obstruction_class_required: false
  loss_ledger_required: false
  verifier_required: false
  loss_ledger_trigger_id: null
authorization:
  authorized_by: null
  certified_implies_authorized: false
failure_modes:
- symptom: invoice line items stored as CSV in one column
  diagnosis: non-atomic carrier violating 1NF
  terminal: RejectNestedCarrier
examples:
- ref: coda_all_universe/1_the_tower/normal_forms/contracts/_book_09_full_normal_form_atlas.md
  section: 2.1 Classical Relational Normal Forms
references:
- coda_all_universe/1_the_tower/normal_forms/contracts/_book_09_full_normal_form_atlas.md
- coda_all_universe/1_the_tower/normal_forms/contracts/_paper_0_relvar_relation_table_and_boundary_relvar.md
- ./normal_form_contract_schema.yaml