nf.logic.qanf

Quantifier Alternation Normal Form (QANF)

Contract ID: nf.logic.qanf | Family: experimental | Index Status: draft | AuthorityEffect: None

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1. Executive Summary & Metadata

Quantifier Alternation Normal Form (QANF) enforces structural normalization and explicit defect routing in the experimental regime.

Property	Value
Canonical ID	nf.logic.qanf
Short Name	QANF
Full Name	Quantifier Alternation Normal Form
Family	experimental (Bounded Quantum-Classical, Contextuality, Quantifier Alternation & Agent-State Contracts)
Qualification	Extended Form Contract (Normal Form Contract)
Data / Math Posture	at_rest
Index Status	draft
Theory Status	proposed
Verification Status	absent
Contract Version	v0
Governance Fence	AuthorityEffect: None (certified_implies_authorized: false)

Foundational Treatise Specification

• Levels: Level 13 - 14
• Spec Files: ['13_grammar.md', '14_math.md']
• Anomaly Excluded: Undecidable looping and unbounded search explosions in higher-order logical specifications.
• Preserved Invariant: Quantifier alternation depth bounded by Sigma_k^0 / Pi_k^0 with strict induction budget B.
• Lossless Reconstruction: Primitive recursive evaluators decide bounded statements; budget overflow yields BudgetUnknown.

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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 (QANF Candidate)	Anomaly risk: a search whose ; quantifier alternations or step ; count exceed the declared bound
2. Normalization / Gate	QANF Verification Gate	the sentence is prenex, i... (certified_implies_authorized: false)
3. After / Certified Output	Certified QANF Output	Invariant: ['prenex_prefix', ; 'alternation_bound', ; 'step_budget']

Contract-Declared Worked Cases

- ref: coda_all_universe/mssn_sn_tower/definitions.md
  section: '3.4'

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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.logic.qanf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	a search whose quantifier alternations or step count exceed the declared bound
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: the sentence is prenex, its alternation depth is the declared Sigma or Pi bound, and the evaluator halts inside the declared budget Protected Skeleton: ['prenex_prefix', 'alternation_bound', 'step_budget']
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 decide arithmetic sentences that sit outside the declared alternation bound
• does not remove the bound by an appeal to an unbounded search
• does not imply authorization

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• prenex_sentence
• alternation_class
• step_budget

5.2 Required Evidence

None declared.

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	the sentence is prenex
criterion	Check	the alternation count is at most the declared bound
criterion	Check	the evaluator halts within the declared step budget

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
RejectExcessAlternation	Reject
RejectBudgetExceeded	BudgetUnknown

5.5 Declared Failure Modes

• symptom: the prenex prefix alternates more times than the declared bound, diagnosis: alternation depth exceeds the contract, terminal: RejectExcessAlternation
• symptom: evaluation has not halted at step B, diagnosis: the declared budget is exhausted, terminal: RejectBudgetExceeded

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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.logic.qanf (QANF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (Accept, RejectExcessAlternation, RejectBudgetExceeded) 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 a search whose quantifier alternations or step count exceed the declared bound is either excluded under the sentence is prenex, its alternation depth is the declared Sigma or Pi bound, and the evaluator halts inside the declared budget 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.logic.qanf.

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7. Canonical YAML Contract (contracts/nf.logic.qanf.yaml)

Click to expand full canonical YAML contract (contracts/nf.logic.qanf.yaml)

schema_version: smplx.nf.contract.v0
schema_ref: ./normal_form_contract_schema.yaml
contract_kind: normal_form_contract
identity:
  id: nf.logic.qanf
  name: Quantifier Alternation Normal Form
  short_name: QANF
  version: v0
  theory_status: proposed
  contract_status: draft
  verification_status: absent
  family: experimental
  status: draft
classification:
  hierarchy_level: normal_form
  subject_kind:
  - prenex_sentence
  - evaluation_budget
  data_posture: at_rest
  mathematical_posture: formal_algebraic
purpose:
  excludes_anomaly: a search whose quantifier alternations or step count exceed the
    declared bound
  preserves_invariant: the sentence is prenex, its alternation depth is the declared
    Sigma or Pi bound, and the evaluator halts inside the declared budget
  protected_skeleton:
  - prenex_prefix
  - alternation_bound
  - step_budget
  erased_structure: []
  non_claims:
  - does not decide arithmetic sentences that sit outside the declared alternation
    bound
  - does not remove the bound by an appeal to an unbounded search
  - does not imply authorization
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  section: '3.4'
  formal_statement: 'A sentence in this form is written in prenex normal form. The
    number of alternating quantifier blocks is at most the declared bound, recorded
    as a Sigma or Pi class. Evaluation runs inside a declared step budget B and halts.
    A sentence that cannot be placed inside that bound is refused.

    '
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_bounded_evaluation_not_a_relational_join
    qualifies_as_extended_form: true
    anomaly_exclusion: unbounded search and an alternation depth above the declaration
    lossless_join:
      status: bounded_evaluation
      statement: the prenex sentence and the budget determine the evaluator's halt
        result
    dependency_preservation:
      status: bound_retained
      statement: transformations keep the sentence inside the declared alternation
        bound and the budget B
    syntactic_decision:
      status: schema_decidable
      statement: the prefix, the alternation count, and the budget are read from the
        declared sentence
declarations_required:
- prenex_sentence
- alternation_class
- step_budget
evidence_required: []
validation_criteria:
- the sentence is prenex
- the alternation count is at most the declared bound
- the evaluator halts within the declared step budget
terminal_routes:
- Accept
- RejectExcessAlternation
- RejectBudgetExceeded
terminal_route_map:
  Accept: Preserve
  RejectExcessAlternation: Reject
  RejectBudgetExceeded: BudgetUnknown
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: the prenex prefix alternates more times than the declared bound
  diagnosis: alternation depth exceeds the contract
  terminal: RejectExcessAlternation
- symptom: evaluation has not halted at step B
  diagnosis: the declared budget is exhausted
  terminal: RejectBudgetExceeded
examples:
- ref: coda_all_universe/mssn_sn_tower/definitions.md
  section: '3.4'
references:
- coda_all_universe/mssn_sn_tower/definitions.md