nf.physics.schlafli_nf

Schläfli Differential Identity Normal Form (SIDNF)

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

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1. Executive Summary & Metadata

Schläfli Differential Identity Normal Form (SIDNF) enforces structural normalization and explicit defect routing in the physics regime.

Property	Value
Canonical ID	nf.physics.schlafli_nf
Short Name	SIDNF
Full Name	Schläfli Differential Identity 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	in_motion
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 (SIDNF Candidate)	Anomaly risk: spurious interior ; curvature variations violating the ; geometric Schläfli identity sum...
2. Normalization / Gate	SIDNF Verification Gate	exact vanishing of the... (certified_implies_authorized: false)
3. After / Certified Output	Certified SIDNF Output	Invariant: ['hinge_simplices', ; 'hinge_volumes_A_h', ; 'metric_edge_lengths',...

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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.schlafli_nf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	spurious interior curvature variations violating the geometric Schläfli identity sum_{h in int} A_h d_epsilon_h = 0, unhandled boundary angle terms, and unqualified gauge-orthogonality claims on curved simplicial geometries
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: exact vanishing of the interior deficit-angle contraction sum_{h in int} A_h * (partial epsilon_h / partial l_e) = 0 across the entire admissible tangent space of interior edge length variations, with explicit boundary variation accounting Protected Skeleton: ['hinge_simplices', 'hinge_volumes_A_h', 'metric_edge_lengths', 'analytical_angle_jacobian', 'admissible_variation_subspace', 'boundary_hinge_terms']
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)

• the Schläfli identity is a variational geometric identity, not an algebraic nilpotent operator (d^2 = 0)
• curved Regge configurations do not generally retain the exact discrete gauge symmetries of flat configurations; gauge-orthogonality holds only when an explicit discrete gauge zero-mode witness is supplied
• evaluating a single numerical perturbation does not establish validity for all perturbations; the contracted gradient vector A^T J_epsilon must vanish across all interior variations
• exact identity satisfaction must be distinguished from bounded numerical tolerance residuals

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• simplicial_complex: dimension n, interior hinges H_int, boundary hinges H_bdry
• background_metric: edge length vector l in R_{>0}^E
• hinge_volume_vector: A_h(l) (lengths in 3D, triangle areas in 4D)
• deficit_angle_vector: epsilon_h(l)
• analytical_angle_jacobian: matrix J_{he} = partial epsilon_h / partial l_e derived from simplex dihedral angles
• variation_subspace: tangent space T_l M of admissible interior edge variations dl

5.2 Required Evidence

• simplicial_complex_dimension: integer
• interior_hinges: list
• boundary_hinges: list
• edge_lengths: list
• hinge_volumes: list
• analytical_angle_jacobian: list
• schlafli_contraction_vector: list
• contraction_infinity_norm: float
• boundary_variation_witness: dict
• gauge_symmetry_witness: {'present': 'boolean', 'zero_modes': '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
SchlafliExactIdentityCertified	Preserve
SchlafliBoundedResidualCertified	Repair
SchlafliIdentityBreach	Reject
BoundaryTermDefect	Refuse
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.schlafli_nf (SIDNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (SchlafliExactIdentityCertified, SchlafliBoundedResidualCertified, SchlafliIdentityBreach, BoundaryTermDefect) 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 spurious interior curvature variations violating the geometric Schläfli identity sum_{h in int} A_h d_epsilon_h = 0, unhandled boundary angle terms, and unqualified gauge-orthogonality claims on curved simplicial geometries is either excluded under exact vanishing of the interior deficit-angle contraction sum_{h in int} A_h * (partial epsilon_h / partial l_e) = 0 across the entire admissible tangent space of interior edge length variations, with explicit boundary variation accounting 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.schlafli_nf.

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

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

schema_version: smplx.nf.contract.v0
contract_kind: normal_form_contract
identity:
  id: nf.physics.schlafli_nf
  name: Schläfli Differential Identity Normal Form
  short_name: SIDNF
  version: v0
  theory_status: established
  contract_status: draft
  verification_status: absent
  family: physics
  status: draft
classification:
  hierarchy_level: normal_form
  subject_kind:
  - schlafli_differential_identity
  - metric_variation
  - regge_geometry
  data_posture: in_motion
  mathematical_posture: formal_algebraic
purpose:
  excludes_anomaly: spurious interior curvature variations violating the geometric
    Schläfli identity sum_{h in int} A_h d_epsilon_h = 0, unhandled boundary angle
    terms, and unqualified gauge-orthogonality claims on curved simplicial geometries
  preserves_invariant: exact vanishing of the interior deficit-angle contraction sum_{h
    in int} A_h * (partial epsilon_h / partial l_e) = 0 across the entire admissible
    tangent space of interior edge length variations, with explicit boundary variation
    accounting
  protected_skeleton:
  - hinge_simplices
  - hinge_volumes_A_h
  - metric_edge_lengths
  - analytical_angle_jacobian
  - admissible_variation_subspace
  - boundary_hinge_terms
  erased_structure:
  - arbitrary_unbounded_heuristic_perturbations
  non_claims:
  - the Schläfli identity is a variational geometric identity, not an algebraic nilpotent
    operator (d^2 = 0)
  - curved Regge configurations do not generally retain the exact discrete gauge symmetries
    of flat configurations; gauge-orthogonality holds only when an explicit discrete
    gauge zero-mode witness is supplied
  - evaluating a single numerical perturbation does not establish validity for all
    perturbations; the contracted gradient vector A^T J_epsilon must vanish across
    all interior variations
  - exact identity satisfaction must be distinguished from bounded numerical tolerance
    residuals
declarations_required:
- simplicial_complex: dimension n, interior hinges H_int, boundary hinges H_bdry
- background_metric: edge length vector l in R_{>0}^E
- hinge_volume_vector: A_h(l) (lengths in 3D, triangle areas in 4D)
- deficit_angle_vector: epsilon_h(l)
- analytical_angle_jacobian: matrix J_{he} = partial epsilon_h / partial l_e derived from simplex dihedral angles
- variation_subspace: tangent space T_l M of admissible interior edge variations dl
validation_criteria:
- exact_schlafli_contraction: interior contraction c_e = sum_{h in H_int} A_h * (partial epsilon_h / partial l_e) vanishes identically (c_e == 0) across all interior edges e
- bounded_numerical_residual: when evaluated under finite arithmetic, ||c||_infty <= tau_residual with declared tolerance tau_residual
- boundary_term_accounting: any non-vanishing total variation matches declared boundary hinge variations sum_{h in H_bdry} A_h d_theta_h
- gauge_symmetry_witness_check: if gauge invariance is asserted, verify explicit zero-mode kernel witness of the discrete Regge Hessian
terminal_routes:
- SchlafliExactIdentityCertified
- SchlafliBoundedResidualCertified
- SchlafliIdentityBreach
- BoundaryTermDefect
- Unsupported
terminal_route_map:
  SchlafliExactIdentityCertified: Preserve
  SchlafliBoundedResidualCertified: Repair
  SchlafliIdentityBreach: Reject
  BoundaryTermDefect: Refuse
  Unsupported: Unsupported
extensions:
  condition_to_local_route_map:
    exact_zero_contraction: SchlafliExactIdentityCertified
    bounded_tolerance_residual: SchlafliBoundedResidualCertified
    non_zero_unaccounted_contraction: SchlafliIdentityBreach
    unhandled_boundary_variations: BoundaryTermDefect
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  formal_statement: The Schläfli differential identity normal form asserts that the
    contraction of interior deficit angle differentials with hinge volumes vanishes
    sum_{h in int} A_h d_epsilon_h = 0 for any interior metric perturbation, ensuring
    the Einstein-Regge equations of motion depend purely on deficit angles without
    spurious variation terms.
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_not_a_relational_project_join
    anomaly_exclusion: spurious interior curvature variations violating the geometric
      Schläfli identity and unhandled boundary angle variations
    lossless_join:
      status: declared_reconstruction
      statement: vanishing of interior deficit-angle variations across the interior
        tangent space
    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:
  SchlafliExactIdentityCertified: Preserve
  SchlafliBoundedResidualCertified: Repair
  SchlafliIdentityBreach: Reject
  BoundaryTermDefect: Refuse
  Unsupported: Unsupported
authorization:
  authorized_by: null
  certified_implies_authorized: false
certification:
  certificate_required: true
  certificate_kind: nf.physics.schlafli_nf_certificate
  obstruction_class_required: false
  loss_ledger_required: false
  loss_ledger_trigger_id: null
  verifier_required: true
evidence_required:
- simplicial_complex_dimension: integer
- interior_hinges: list
- boundary_hinges: list
- edge_lengths: list
- hinge_volumes: list
- analytical_angle_jacobian: list
- schlafli_contraction_vector: list
- contraction_infinity_norm: float
- boundary_variation_witness: dict
- gauge_symmetry_witness: {present: boolean, zero_modes: list}
schema_ref: ./normal_form_contract_schema.yaml