nf.derivation.irnf

Integer Ray Normal Form (IRNF)

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

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1. Executive Summary & Metadata

Integer Ray Normal Form (IRNF) enforces structural normalization and explicit defect routing in the derivation regime.

Property	Value
Canonical ID	nf.derivation.irnf
Short Name	IRNF
Full Name	Integer Ray Normal Form
Family	derivation (MSS-N Tower Algebraic Derivation, Split Extension-Retraction & Obstruction Contracts)
Qualification	Extended Form Contract (Normal Form Contract)
Data / Math Posture	at_rest
Index Status	draft
Theory Status	established
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 (IRNF Candidate)	Anomaly risk: floating-point ; rounding drift anomaly and ; redundant arbitrary scalar...
2. Normalization / Gate	IRNF Verification Gate	weight profiles are... (certified_implies_authorized: false)
3. After / Certified Output	Certified IRNF Output	Invariant: ['carrier', ; 'primitive_weights', 'total_mass', ; 'gcd_factor']

Contract-Declared Worked Cases

- ref: https://github.com/dbccpage/19L57G1/blob/f1d56c010e6ba10a3b9418bd7cc5953c067250f2/lean4/Derivation/WeightProfile.lean
  section: lean4/Derivation/WeightProfile.lean
  note: ray equivalence without float division

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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.irnf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	floating-point rounding drift anomaly and redundant arbitrary scalar representations of equivalent projective rays
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: weight profiles are represented as coprime integer vectors with zero division, and ray equivalence is decided via exact integer cross-multiplication Protected Skeleton: ['carrier', 'primitive_weights', 'total_mass', 'gcd_factor']
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 assert continuous measure theory or infinite probability spaces
• does not claim physical probability or Born rule uniqueness
• does not allow division by zero or negative weight components
• does not imply operational authority

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• carrier
• weight_vector
• primitivity_certificate

5.2 Required Evidence

• primitive_integer_weights
• gcd_reduction_factor
• total_mass_sum

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	all weight entries are non-negative integers
criterion	Check	at least one entry is strictly positive
criterion	Check	greatest common divisor across all entries is exactly 1
criterion	Check	ray equivalence between two profiles satisfies w(p) * total(v) == v(p) * total(w)
criterion	Check	same_ray may accept two zero vectors, but this primitive-representative contract rejects zero mass before gcd reduction

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
AcceptPrimitiveRay	Preserve
RejectFloatingPointContamination	Reject
RejectNonPrimitiveScaling	Recompute
RejectZeroVector	Reject
Undetermined	Abstain

5.5 Declared Failure Modes

• symptom: weights are provided with floating-point values or rounding epsilon, diagnosis: floating-point contamination violates exact rational arithmetic, terminal: RejectFloatingPointContamination
• symptom: weights share a common non-trivial integer divisor greater than 1, diagnosis: representation is not primitive within its projective ray, terminal: RejectNonPrimitiveScaling
• symptom: weight vector contains only zeros, diagnosis: zero vector does not define a projective ray, terminal: RejectZeroVector

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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.irnf (IRNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (AcceptPrimitiveRay, RejectFloatingPointContamination, RejectNonPrimitiveScaling, RejectZeroVector) 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 floating-point rounding drift anomaly and redundant arbitrary scalar representations of equivalent projective rays is either excluded under weight profiles are represented as coprime integer vectors with zero division, and ray equivalence is decided via exact integer cross-multiplication 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.irnf.

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

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

# Draft declaration: no bound local contract evaluator is supplied; upstream mathematics and schema checks are separate evidence.
schema_version: smplx.nf.contract.v0
contract_kind: form_contract
identity:
  id: nf.derivation.irnf
  name: Integer Ray Normal Form
  short_name: IRNF
  version: v0
  status: draft
  family: derivation
  theory_status: established
  contract_status: draft
  verification_status: absent
classification:
  hierarchy_level: form
  subject_kind:
  - weight_profile
  - projective_premeasure
  - integer_carrier
  data_posture: at_rest
  mathematical_posture: formal_algebraic
purpose:
  excludes_anomaly: floating-point rounding drift anomaly and redundant arbitrary scalar representations
    of equivalent projective rays
  preserves_invariant: weight profiles are represented as coprime integer vectors with zero division,
    and ray equivalence is decided via exact integer cross-multiplication
  protected_skeleton:
  - carrier
  - primitive_weights
  - total_mass
  - gcd_factor
  erased_structure:
  - arbitrary_positive_scalar_multiples
  - floating_point_approximations
  non_claims:
  - does not assert continuous measure theory or infinite probability spaces
  - does not claim physical probability or Born rule uniqueness
  - does not allow division by zero or negative weight components
  - does not imply operational authority
quadrilateral:
  qualifies_as_true_normal_form: false
  qualification_gap: no_project_join_theorem
  anomaly_exclusion: numerical rounding drift and redundant representation of identical relative weight
    profiles
  lossless_join:
    status: projective_scaling_reconstruction
    statement: the unreduced weight vector is reconstructed losslessly by multiplying the primitive representative
      by the scalar factor gcd(w)
  dependency_preservation:
    status: exact_ratios_preserved
    statement: pairwise ratios between all carrier components are preserved identically without precision
      loss
  syntactic_decision:
    status: declaration_decidable
    statement: decided via integer greatest common divisor and cross-multiplication identities over Z
declarations_required:
- carrier
- weight_vector
- primitivity_certificate
evidence_required:
- primitive_integer_weights
- gcd_reduction_factor
- total_mass_sum
validation_criteria:
- all weight entries are non-negative integers
- at least one entry is strictly positive
- greatest common divisor across all entries is exactly 1
- ray equivalence between two profiles satisfies w(p) * total(v) == v(p) * total(w)
- same_ray may accept two zero vectors, but this primitive-representative contract rejects zero mass before
  gcd reduction
terminal_routes:
- AcceptPrimitiveRay
- RejectFloatingPointContamination
- RejectNonPrimitiveScaling
- RejectZeroVector
- Undetermined
terminal_route_map:
  AcceptPrimitiveRay: Preserve
  RejectFloatingPointContamination: Reject
  RejectNonPrimitiveScaling: Recompute
  RejectZeroVector: Reject
  Undetermined: Abstain
certification:
  certificate_required: true
  certificate_kind: integer_ray_certificate
  obstruction_class_required: false
  loss_ledger_required: false
  verifier_required: true
authorization:
  authorized_by: null
  certified_implies_authorized: false
failure_modes:
- symptom: weights are provided with floating-point values or rounding epsilon
  diagnosis: floating-point contamination violates exact rational arithmetic
  terminal: RejectFloatingPointContamination
- symptom: weights share a common non-trivial integer divisor greater than 1
  diagnosis: representation is not primitive within its projective ray
  terminal: RejectNonPrimitiveScaling
- symptom: weight vector contains only zeros
  diagnosis: zero vector does not define a projective ray
  terminal: RejectZeroVector
examples:
- ref: https://github.com/dbccpage/19L57G1/blob/f1d56c010e6ba10a3b9418bd7cc5953c067250f2/lean4/Derivation/WeightProfile.lean
  section: lean4/Derivation/WeightProfile.lean
  note: ray equivalence without float division
references:
- https://github.com/dbccpage/19L57G1/blob/f1d56c010e6ba10a3b9418bd7cc5953c067250f2/docs/GENESIS_CLOSEOUT.md
- ./README.md