nf.derivation.scnf

Separation Cost Normal Form (SCNF)

Contract ID: nf.derivation.scnf | Family: derivation | Index Status: draft | AuthorityEffect: None

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1. Executive Summary & Metadata

Separation Cost Normal Form (SCNF) enforces structural normalization and explicit defect routing in the derivation regime.

Property	Value
Canonical ID	nf.derivation.scnf
Short Name	SCNF
Full Name	Separation Cost Normal Form
Family	derivation (MSS-N Tower Algebraic Derivation, Split Extension-Retraction & Obstruction Contracts)
Qualification	Extended Form Contract (Normal Form Contract)
Data / Math Posture	in_motion
Index Status	draft
Theory Status	established
Verification Status	executable
Contract Version	v0
Governance Fence	AuthorityEffect: None (certified_implies_authorized: false)

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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	Input Carrier (SCNF Candidate)	Anomaly risk: a minimum reported ; without the measure that made it ; minimal, or a tie silently...
2. Normalization / Gate	SCNF Verification Gate	a cost claim names its... (certified_implies_authorized: false)
3. After / Certified Output	Certified SCNF Output	Invariant: ['delta', ; 'cost_measure', 'minimizer_set', ; 'isomorphism_relation',...

Contract-Declared Worked Cases

- ref: 19L57G1
  section: paper/paper.md
  note: the first separation and the tied minimum under the item-count measure

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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.derivation.scnf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	a minimum reported without the measure that made it minimal, or a tie silently resolved
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: a cost claim names its measure, and whether the minimizer is unique is a property of that measure rather than of the separation Protected Skeleton: ['delta', 'cost_measure', 'minimizer_set', 'isomorphism_relation', 'tie_record']
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 derive the cost measure; the measure is declared, and a different measure can change both the minimum and its uniqueness
• does not derive the grammar, the isomorphism relation, or the closure and continuation policies
• does not report a unique minimizer under a measure that admits several
• does not treat the count of tied classes as fixed across readings of admission
• does not imply authorization

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• cost_measure
• grammar
• isomorphism_relation
• closure_policy
• continuation_policy
• reading_of_admission

5.2 Required Evidence

• delta_contents
• measure_value_per_candidate
• minimizer_set
• isomorphism_classes_of_minimizers

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	every cost claim names the measure under which it was computed
criterion	Check	the minimizer set is reported in full, not by one representative
criterion	Check	minimizers are counted up to the declared isomorphism relation
criterion	Check	a minimizer set of size greater than one is recorded as a tie and no member is preferred
criterion	Check	the count of tied classes is reported together with the reading of admission that produced it
criterion	Check	a claim of uniqueness names the measure that makes it unique

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
AcceptUniqueMinimizer	Preserve
RecordTie	Preserve
RequireMeasureDeclaration	Refuse
RejectUnmeasuredMinimum	Reject
Escalate	Preserve

5.5 Declared Failure Modes

• symptom: a cheapest separation is reported with no measure named, diagnosis: minimality asserted outside any declared order, terminal: RejectUnmeasuredMinimum
• symptom: one of several tied minimizers is reported as the minimizer, diagnosis: a tie was resolved by enumeration order, terminal: RecordTie
• symptom: two reports of the same separation disagree on uniqueness, diagnosis: different measures were in force and only one was recorded, terminal: RequireMeasureDeclaration
• symptom: the number of minimizers changes when the grammar is restated, diagnosis: the isomorphism relation was left undeclared, terminal: RequireMeasureDeclaration

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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.derivation.scnf (SCNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (AcceptUniqueMinimizer, RecordTie, RequireMeasureDeclaration, RejectUnmeasuredMinimum) 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 in_motion, this contract enforces that a minimum reported without the measure that made it minimal, or a tie silently resolved is either excluded under a cost claim names its measure, and whether the minimizer is unique is a property of that measure rather than of the separation 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.derivation.scnf.

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7. Canonical YAML Contract (contracts/nf.derivation.scnf.yaml)

Click to expand full canonical YAML contract (contracts/nf.derivation.scnf.yaml)

schema_version: smplx.nf.contract.v0
contract_kind: form_contract
identity:
  id: nf.derivation.scnf
  name: Separation Cost Normal Form
  short_name: SCNF
  version: v0
  status: draft
  family: derivation
  theory_status: established
  contract_status: draft
  verification_status: executable
classification:
  hierarchy_level: form
  subject_kind:
  - delta
  - cost_measure
  - separation_claim
  data_posture: in_motion
  mathematical_posture: operational
purpose:
  excludes_anomaly: a minimum reported without the measure that made it minimal, or a tie
    silently resolved
  preserves_invariant: a cost claim names its measure, and whether the minimizer is unique
    is a property of that measure rather than of the separation
  protected_skeleton:
  - delta
  - cost_measure
  - minimizer_set
  - isomorphism_relation
  - tie_record
  erased_structure: []
  non_claims:
  - does not derive the cost measure; the measure is declared, and a different measure can
    change both the minimum and its uniqueness
  - does not derive the grammar, the isomorphism relation, or the closure and continuation
    policies
  - does not report a unique minimizer under a measure that admits several
  - does not treat the count of tied classes as fixed across readings of admission
  - does not imply authorization
quadrilateral:
  qualifies_as_true_normal_form: false
  qualification_gap: no_project_join_theorem
  anomaly_exclusion: a minimum reported without its measure, or a tie reduced to one member
    by silence
  lossless_join:
    status: not_a_relational_decomposition
    statement: the minimizer set is reported in full up to the declared isomorphism
  dependency_preservation:
    status: measure_stays_attached
    statement: a different declared measure is a different claim
  syntactic_decision:
    status: declaration_decidable
    statement: the check reads the measure, the values, and the isomorphism, not an unrecorded
      preference
declarations_required:
- cost_measure
- grammar
- isomorphism_relation
- closure_policy
- continuation_policy
- reading_of_admission
evidence_required:
- delta_contents
- measure_value_per_candidate
- minimizer_set
- isomorphism_classes_of_minimizers
validation_criteria:
- every cost claim names the measure under which it was computed
- the minimizer set is reported in full, not by one representative
- minimizers are counted up to the declared isomorphism relation
- a minimizer set of size greater than one is recorded as a tie and no member is preferred
- the count of tied classes is reported together with the reading of admission that produced
  it
- a claim of uniqueness names the measure that makes it unique
proof_obligations:
- id: scnf.first_separation_costs_two_items
  status: upstream_theorem
  ref: 19L57G1
  section: paper/paper.md
- id: scnf.item_count_measure_admits_tied_minimizers
  status: upstream_theorem
  ref: 19L57G1
  section: paper/paper.md
- id: scnf.lexicographic_measure_is_unique_up_to_isomorphism
  status: upstream_theorem
  ref: 19L57G1
  section: paper/paper.md
- id: scnf.measures_are_declared_not_derived
  status: declared_upstream
  ref: 19L57G1
  section: paper/paper.md
terminal_routes:
- AcceptUniqueMinimizer
- RecordTie
- RequireMeasureDeclaration
- RejectUnmeasuredMinimum
- Escalate
certification:
  certificate_required: true
  certificate_kind: cost_certificate
  obstruction_class_required: false
  loss_ledger_required: true
  verifier_required: true
  loss_ledger_trigger_id: null
authorization:
  authorized_by: null
  certified_implies_authorized: false
failure_modes:
- symptom: a cheapest separation is reported with no measure named
  diagnosis: minimality asserted outside any declared order
  terminal: RejectUnmeasuredMinimum
- symptom: one of several tied minimizers is reported as the minimizer
  diagnosis: a tie was resolved by enumeration order
  terminal: RecordTie
- symptom: two reports of the same separation disagree on uniqueness
  diagnosis: different measures were in force and only one was recorded
  terminal: RequireMeasureDeclaration
- symptom: the number of minimizers changes when the grammar is restated
  diagnosis: the isomorphism relation was left undeclared
  terminal: RequireMeasureDeclaration
examples:
- ref: 19L57G1
  section: paper/paper.md
  note: the first separation and the tied minimum under the item-count measure
references:
- 19L/PIN.md
- ./nf.derivation.aonf.yaml
- ./README.md
schema_ref: ./normal_form_contract_schema.yaml
terminal_route_map:
  AcceptUniqueMinimizer: Preserve
  RecordTie: Preserve
  RequireMeasureDeclaration: Refuse
  RejectUnmeasuredMinimum: Reject
  Escalate: Preserve