nf.physics.rcnf

Regge Calculus Normal Form (RCNF)

Contract ID: nf.physics.rcnf | Family: physics | Index Status: draft | AuthorityEffect: None

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1. Executive Summary & Metadata

Regge Calculus Normal Form (RCNF) enforces structural normalization and explicit defect routing in the physics regime.

Property	Value
Canonical ID	nf.physics.rcnf
Short Name	RCNF
Full Name	Regge Calculus Normal Form
Family	physics (Discrete Differential Geometry, Regge Calculus, Gauge Theory, Lattice Fermions & Quantum Gravity)
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 (RCNF Candidate)	Anomaly risk: non-manifold vertex ; and hinge singularities, metric ; degeneracy, Gram matrix...
2. Normalization / Gate	RCNF Verification Gate	piecewise-flat simplicial... (certified_implies_authorized: false)
3. After / Certified Output	Certified RCNF Output	Invariant: ; ['edge_lengths_squared', ; 'signed_cayley_menger_positivit...

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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.physics.rcnf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	non-manifold vertex and hinge singularities, metric degeneracy, Gram matrix indefiniteness, negative dimension-signed Cayley-Menger determinants, and violated triangle inequalities
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: piecewise-flat simplicial Riemannian metric with dimension-signed Cayley-Menger determinants (-1)^{n+1} det(CM) > 0, positive-definite Euclidean Gram matrices G > 0 on every simplex, and spherical link complexes Lk(v) ~= S^{n-1} Protected Skeleton: ['edge_lengths_squared', 'signed_cayley_menger_positivity', 'gram_matrix_positive_definiteness', 'vertex_link_topology', 'hinge_deficit_angles', 'simplex_volumes']
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 force smooth Riemannian manifolds in the continuum limit
• does not eliminate topological singularities or coordinate gauge redundancies
• hinge star checks alone do not establish manifoldhood; vertex links must be explicitly verified against spherical topologies to exclude suspensions of non-spherical manifolds

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• vertices: list of discrete vertex ids
• edges: list of distinct pairs (u, v) with squared lengths l_e^2 > 0
• simplices: list of oriented n-simplices
• cayley_menger_matrix: matrix of squared distances
• gram_matrix: simplex Gram matrix G_{ij} = (l_{0i}^2 + l_{0j}^2 - l_{ij}^2) / 2
• manifold_class: declared manifold dimension n and boundary specification (closed or manifold-with-boundary)

5.2 Required Evidence

• vertices: list
• edges: list
• simplices: list
• squared_edge_lengths: dict
• gram_matrices: list
• gram_principal_minors_positive: boolean
• signed_cm_determinants_positive: boolean
• vertex_link_spherical_witnesses: list
• hinge_link_circle_witnesses: list

5.3 Validation Criteria

Criterion ID	Name	Condition
``	****
``	****
``	****
``	****
``	****
``	****

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
SimplicialManifoldMetricPreserved	Preserve
MetricDegenerate	Reject
GramIndefinite	Reject
NonManifoldLinkSingularity	Reject
Unsupported	Unsupported

5.5 Declared Failure Modes

All defects route explicitly through the Terminal Route Map above.

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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.physics.rcnf (RCNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (SimplicialManifoldMetricPreserved, MetricDegenerate, GramIndefinite, NonManifoldLinkSingularity) 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 non-manifold vertex and hinge singularities, metric degeneracy, Gram matrix indefiniteness, negative dimension-signed Cayley-Menger determinants, and violated triangle inequalities is either excluded under piecewise-flat simplicial Riemannian metric with dimension-signed Cayley-Menger determinants (-1)^{n+1} det(CM) > 0, positive-definite Euclidean Gram matrices G > 0 on every simplex, and spherical link complexes Lk(v) ~= S^{n-1} 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.physics.rcnf.

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7. Canonical YAML Contract (contracts/nf.physics.rcnf.yaml)

Click to expand full canonical YAML contract (contracts/nf.physics.rcnf.yaml)

schema_version: smplx.nf.contract.v0
contract_kind: normal_form_contract
identity:
  id: nf.physics.rcnf
  name: Regge Calculus Normal Form
  short_name: RCNF
  version: v0
  theory_status: established
  contract_status: draft
  verification_status: absent
  family: physics
  status: draft
classification:
  hierarchy_level: normal_form
  subject_kind:
  - simplicial_complex
  - metric_geometry
  data_posture: at_rest
  mathematical_posture: formal_algebraic
purpose:
  excludes_anomaly: non-manifold vertex and hinge singularities, metric degeneracy,
    Gram matrix indefiniteness, negative dimension-signed Cayley-Menger determinants,
    and violated triangle inequalities
  preserves_invariant: piecewise-flat simplicial Riemannian metric with dimension-signed
    Cayley-Menger determinants (-1)^{n+1} det(CM) > 0, positive-definite Euclidean
    Gram matrices G > 0 on every simplex, and spherical link complexes Lk(v) ~= S^{n-1}
  protected_skeleton:
  - edge_lengths_squared
  - signed_cayley_menger_positivity
  - gram_matrix_positive_definiteness
  - vertex_link_topology
  - hinge_deficit_angles
  - simplex_volumes
  erased_structure:
  - continuous_smooth_coordinates
  non_claims:
  - does not force smooth Riemannian manifolds in the continuum limit
  - does not eliminate topological singularities or coordinate gauge redundancies
  - hinge star checks alone do not establish manifoldhood; vertex links must be explicitly
    verified against spherical topologies to exclude suspensions of non-spherical manifolds
declarations_required:
- vertices: list of discrete vertex ids
- edges: list of distinct pairs (u, v) with squared lengths l_e^2 > 0
- simplices: list of oriented n-simplices
- cayley_menger_matrix: matrix of squared distances
- gram_matrix: simplex Gram matrix G_{ij} = (l_{0i}^2 + l_{0j}^2 - l_{ij}^2) / 2
- manifold_class: declared manifold dimension n and boundary specification (closed or manifold-with-boundary)
validation_criteria:
- metric_positivity: every edge length l_e is strictly positive
- triangle_inequality: for every triangle (e1, e2, e3), l1 + l2 > l3 and cyclic permutations hold
- volume_positivity: Cayley-Menger determinant det(CM) satisfies (-1)^{n+1} det(CM) > 0 for every n-simplex
- gram_positive_definiteness: for every n-simplex, the Gram matrix G relative to vertex v_0 is strictly positive-definite (all principal minors strictly positive)
- vertex_link_manifoldhood: for every vertex v, the link complex Lk(v) is piecewise-linearly homeomorphic to S^{n-1} (or B^{n-1} if boundary)
- hinge_link_regularity: every codimension-2 hinge has a link PL homeomorphic to S^1 (or I if boundary) without branch cuts
terminal_routes:
- SimplicialManifoldMetricPreserved
- MetricDegenerate
- GramIndefinite
- NonManifoldLinkSingularity
- Unsupported
terminal_route_map:
  SimplicialManifoldMetricPreserved: Preserve
  MetricDegenerate: Reject
  GramIndefinite: Reject
  NonManifoldLinkSingularity: Reject
  Unsupported: Unsupported
extensions:
  condition_to_local_route_map:
    all_valid: SimplicialManifoldMetricPreserved
    negative_volume: MetricDegenerate
    gram_indefinite: GramIndefinite
    link_singularity: NonManifoldLinkSingularity
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  formal_statement: Piecewise-flat simplicial Riemannian metric with dimension-signed
    Cayley-Menger determinants (-1)^{n+1} det(CM) > 0, strictly positive-definite
    Gram matrices G > 0, and spherical vertex links Lk(v) preserving PL manifoldhood.
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_not_a_relational_project_join
    anomaly_exclusion: non-manifold vertex and hinge singularities, metric degeneracy,
      Gram matrix indefiniteness, and negative dimension-signed Cayley-Menger determinants
    lossless_join:
      status: declared_reconstruction
      statement: piecewise-flat simplicial Riemannian metric with dimension-signed
        Cayley-Menger determinants and positive-definite Gram matrices
    dependency_preservation:
      status: declared_with_the_invariant
      statement: The preserved invariant stays on the declared schema. This form does
        not state a functional-dependency projection of a flat relvar decomposition.
    syntactic_decision:
      status: declaration_decidable
      statement: The check reads the declared constraints and validation criteria.
        One unrecorded runtime image is not the decider.
canonical_terminal_map:
  SimplicialManifoldMetricPreserved: Preserve
  MetricDegenerate: Reject
  GramIndefinite: Reject
  NonManifoldLinkSingularity: Reject
  Unsupported: Unsupported
authorization:
  authorized_by: null
  certified_implies_authorized: false
certification:
  certificate_required: true
  certificate_kind: nf.physics.rcnf_certificate
  obstruction_class_required: false
  loss_ledger_required: false
  loss_ledger_trigger_id: null
  verifier_required: true
evidence_required:
- vertices: list
- edges: list
- simplices: list
- squared_edge_lengths: dict
- gram_matrices: list
- gram_principal_minors_positive: boolean
- signed_cm_determinants_positive: boolean
- vertex_link_spherical_witnesses: list
- hinge_link_circle_witnesses: list
schema_ref: ./normal_form_contract_schema.yaml