nf.ontology.ainf

Adjoint-Invariant Normal Form (AINF)

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

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1. Executive Summary & Metadata

Adjoint-Invariant Normal Form (AINF) enforces structural normalization and explicit defect routing in the derivation regime.

Property	Value
Canonical ID	nf.ontology.ainf
Short Name	AINF
Full Name	Adjoint-Invariant 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	proposed
Verification Status	absent
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 (AINF Candidate)	Anomaly risk: symmetry breaking ; and invariant leakage during ; structural lift
2. Normalization / Gate	AINF Verification Gate	natural bijection of the... (certified_implies_authorized: false)
3. After / Certified Output	Certified AINF Output	Invariant: ['up_functor_F', ; 'forgetful_functor_U', ; 'unit_witness_eta',...

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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.ontology.ainf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	symmetry breaking and invariant leakage during structural lift
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: natural bijection of the adjunction Hom(F(X), Y) ≅ Hom(X, U(Y)) Protected Skeleton: ['up_functor_F', 'forgetful_functor_U', 'unit_witness_eta', 'counit_witness_epsilon']
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 claim the intermediate lifted space is observable by the base carrier
• does not imply authorization
• does not claim general adjunction lift F and descent U are invertible or form an equivalence of categories

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• up_functor_lift: F: C -> D
• forgetful_functor_descent: U: D -> C
• adjunction_isomorphism
• witness_unit: eta: Id_C -> U F
• witness_counit: epsilon: F U -> Id_D

5.2 Required Evidence

• base_carrier_state_before
• base_carrier_state_after
• local_gauge_repair_trace

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	every structural lift must be paired with a corresponding forgetful descent
``	****
criterion	Check	non-satisfaction of triangle identities triggers immediate invariant obstruction

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
Conserve	Preserve
RadiateResidual	Decompose
RecomputeGauge	Recompute
EscalateSymmetryViolation	Escalate
Lift	Lift

5.5 Declared Failure Modes

• symptom: counit-unit composition fails identity check, diagnosis: triangle identity violated; functorial lift leaked invariants, terminal: EscalateSymmetryViolation

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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.ontology.ainf (AINF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (Conserve, RadiateResidual, RecomputeGauge, EscalateSymmetryViolation) 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 symmetry breaking and invariant leakage during structural lift is either excluded under natural bijection of the adjunction Hom(F(X), Y) ≅ Hom(X, U(Y)) 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.ontology.ainf.

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7. Canonical YAML Contract (contracts/nf.ontology.ainf.yaml)

Click to expand full canonical YAML contract (contracts/nf.ontology.ainf.yaml)

schema_version: smplx.nf.contract.v0
schema_ref: ./normal_form_contract_schema.yaml
contract_kind: normal_form_contract

identity:
  id: nf.ontology.ainf
  name: Adjoint-Invariant Normal Form
  short_name: AINF
  version: v0
  status: draft
  family: derivation
  theory_status: proposed
  contract_status: draft
  verification_status: absent

classification:
  hierarchy_level: normal_form
  subject_kind:
    - adjunction_witness
    - functorial_lift
    - invariant_carrier
  data_posture: in_motion
  mathematical_posture: operational

purpose:
  excludes_anomaly: symmetry breaking and invariant leakage during structural lift
  preserves_invariant: natural bijection of the adjunction Hom(F(X), Y) ≅ Hom(X, U(Y))
  protected_skeleton:
    - up_functor_F
    - forgetful_functor_U
    - unit_witness_eta
    - counit_witness_epsilon
  erased_structure:
    - transient_carrier_complexity
    - continuous_pde_approximations
  non_claims:
    - does not claim the intermediate lifted space is observable by the base carrier
    - does not imply authorization
    - does not claim general adjunction lift F and descent U are invertible or form an equivalence of categories

obstruction_substrate:
  finite_carrier:
    description: discrete state space representing conserved quantities under functorial transport
    max_rank: null
  cochain_complex:
    C0: base_carrier_states
    C1: functorial_lifts_and_descents
    C2: symmetry_defects
    d0: maps state transition to its functorial action
    d1: evaluates Noether invariant conservation
    quotient: "Q1 = C1 / im(d0)"
  obstruction_class:
    notation: "[omega(e)]"
    exact: "[omega] == 0"
    residual_closed: "d1(omega) == 0 and [omega] != 0"
    open: "d1(omega) != 0"

declarations_required:
  - up_functor_lift: "F: C -> D"
  - forgetful_functor_descent: "U: D -> C"
  - adjunction_isomorphism
  - witness_unit: "eta: Id_C -> U F"
  - witness_counit: "epsilon: F U -> Id_D"

evidence_required:
  - base_carrier_state_before
  - base_carrier_state_after
  - local_gauge_repair_trace

validation_criteria:
  - every structural lift must be paired with a corresponding forgetful descent
  - the unit and counit witnesses must strictly satisfy triangle identities: U(epsilon) o eta(U) = Id_U and epsilon(F) o F(eta) = Id_F
  - non-satisfaction of triangle identities triggers immediate invariant obstruction

proof_obligations: []

terminal_routes:
  - Conserve
  - RadiateResidual
  - RecomputeGauge
  - EscalateSymmetryViolation
  - Lift

terminal_route_map:
  Conserve: Preserve
  RadiateResidual: Decompose
  RecomputeGauge: Recompute
  EscalateSymmetryViolation: Escalate
  Lift: Lift

certification:
  certificate_required: true
  certificate_kind: adjunction_certificate
  obstruction_class_required: true
  loss_ledger_required: false
  loss_ledger_trigger_id: null
  verifier_required: true

authorization:
  authorized_by: null
  certified_implies_authorized: false

failure_modes:
  - symptom: counit-unit composition fails identity check
    diagnosis: triangle identity violated; functorial lift leaked invariants
    terminal: EscalateSymmetryViolation

definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  section: "3.9"
  formal_statement: >
    A structural lift F: C -> D and forgetful descent U: D -> C form a certified adjunction
    witness F -| U if and only if natural transformations eta: Id_C -> U F and epsilon: F U -> Id_D
    satisfy the triangle identities U(epsilon) o eta(U) = Id_U and epsilon(F) o F(eta) = Id_F.
    Any defect in the triangle commutations represents an obstruction class in Q1 that forbids
    lossless adjunction transport.
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_categorical_adjunction_isomorphism_not_relational_join
    qualifies_as_extended_form: true
    anomaly_exclusion: symmetry breaking and invariant leakage during structural lift
    lossless_join:
      status: adjunction_hom_isomorphism
      statement: unit and counit natural transformations establish the adjunction hom-set bijection Hom(F(X), Y) ≅ Hom(X, U(Y)); invertible object or morphism reconstruction does not hold in general and requires the strictly stronger hypothesis of an adjoint equivalence (invertible unit and counit)
    dependency_preservation:
      status: triangle_identities_preserved
      statement: adjunction naturality and functorial composition are preserved on components
    syntactic_decision:
      status: schema_decidable
      statement: triangle identity satisfaction is verifiable from declared functorial mappings and witness morphisms