nf.physics.lorentzian_cdt

4D Lorentzian Regge Causal Triangulation Normal Form (LCNF)

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

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1. Executive Summary & Metadata

4D Lorentzian Regge Causal Triangulation Normal Form (LCNF) enforces structural normalization and explicit defect routing in the physics regime.

Property	Value
Canonical ID	nf.physics.lorentzian_cdt
Short Name	LCNF
Full Name	4D Lorentzian Regge Causal Triangulation 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 (LCNF Candidate)	Anomaly risk: closed timelike ; curves (CTCs), spatial slice ; foliation branching, unverified...
2. Normalization / Gate	LCNF Verification Gate	global proper time foliat... (certified_implies_authorized: false)
3. After / Certified Output	Certified LCNF Output	Invariant: ; ['edge_signatures_and_squared_lengths', ; 'simplex_causal_types',...

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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.lorentzian_cdt
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	closed timelike curves (CTCs), spatial slice foliation branching, unverified spatial Cauchy topology changes, and metric signature incompatibilities under Lorentzian Pachner surgery
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: global proper time foliation t -> t+1, strictly verified causal Pachner move legality preserving initial carrier topology, compatible Lorentzian edge length assignments (l_s^2 > 0, l_t^2 < 0), and invariant spatial Euler characteristic Protected Skeleton: ['edge_signatures_and_squared_lengths', 'simplex_causal_types', 'cauchy_slice_foliation', 'certified_initial_slice_topology', 'causal_pachner_move_witnesses', 'face_gluing_compatibility']
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)

• Euler characteristic conservation chi(Sigma_t) is an invariant consistency check, not a complete topology certificate; topologies with identical Euler characteristics (e.g., S^3 and T^3 with chi=0) are distinguished by checked move legality and initial carrier homology
• monotonic time labels certify directed graph acyclicity; they do not alone establish that simplices realize compatible nondegenerate Lorentzian metrics without explicit metric data and face-gluing compatibility
• does not support that discrete CDT converges to continuous 4D de Sitter spacetime without quantum Monte Carlo infinite volume limits
• does not eliminate Wick rotation path integral measure ambiguities

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• edge_signature_map: edge_id -> {spacelike (l_s^2 > 0), timelike (l_t^2 < 0)}
• simplex_type_map: simplex_id -> {(4,1), (3,2), (2,3), (1,4)}
• initial_carrier_topology: declared manifold class and verified homology of Sigma_0
• foliation_slice_registry: slice_id -> spatial 3-triangulation
• lorentzian_metric_parameters: ratio alpha = -l_t^2 / l_s^2 satisfying Regge Lorentzian nondegeneracy (alpha > 7/12 in 4D)
• causal_pachner_moves: sequence of local bistellar replacements with verified replacement complexes and boundary matches

5.2 Required Evidence

• initial_carrier_topology: {'manifold_type': 'string', 'homology_ranks': 'list'}
• slice_foliation_indices: list
• edge_lengths_squared: dict
• causal_pachner_moves_log: list
• move_legality_witnesses: list
• euler_characteristic_sequence: list
• simplex_lorentzian_gram_signs: list
• face_gluing_compatibility_verified: boolean
• time_directed_acyclicity_witness: boolean

5.3 Validation Criteria

Criterion ID	Name	Condition
``	****
``	****
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``	****
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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
CausalFoliationPreserved	Preserve
ClosedTimelikeCurveDetected	Refuse
SpatialTopologyBreach	Reject
MetricCompatibilityDefect	Reject
IllegalPachnerMove	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.lorentzian_cdt (LCNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (CausalFoliationPreserved, ClosedTimelikeCurveDetected, SpatialTopologyBreach, MetricCompatibilityDefect) 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 closed timelike curves (CTCs), spatial slice foliation branching, unverified spatial Cauchy topology changes, and metric signature incompatibilities under Lorentzian Pachner surgery is either excluded under global proper time foliation t -> t+1, strictly verified causal Pachner move legality preserving initial carrier topology, compatible Lorentzian edge length assignments (l_s^2 > 0, l_t^2 < 0), and invariant spatial Euler characteristic 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.lorentzian_cdt.

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

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

schema_version: smplx.nf.contract.v0
contract_kind: normal_form_contract
identity:
  id: nf.physics.lorentzian_cdt
  name: 4D Lorentzian Regge Causal Triangulation Normal Form
  short_name: LCNF
  version: v0
  theory_status: established
  contract_status: draft
  verification_status: absent
  family: physics
  status: draft
classification:
  hierarchy_level: normal_form
  subject_kind:
  - lorentzian_simplicial_geometry
  - causal_dynamical_triangulations
  data_posture: in_motion
  mathematical_posture: formal_algebraic
purpose:
  excludes_anomaly: closed timelike curves (CTCs), spatial slice foliation branching,
    unverified spatial Cauchy topology changes, and metric signature incompatibilities
    under Lorentzian Pachner surgery
  preserves_invariant: global proper time foliation t -> t+1, strictly verified causal
    Pachner move legality preserving initial carrier topology, compatible Lorentzian
    edge length assignments (l_s^2 > 0, l_t^2 < 0), and invariant spatial Euler characteristic
  protected_skeleton:
  - edge_signatures_and_squared_lengths
  - simplex_causal_types
  - cauchy_slice_foliation
  - certified_initial_slice_topology
  - causal_pachner_move_witnesses
  - face_gluing_compatibility
  erased_structure:
  - continuous_spacetime_diffeomorphisms
  - acausal_spacelike_topology_changes
  non_claims:
  - Euler characteristic conservation chi(Sigma_t) is an invariant consistency check,
    not a complete topology certificate; topologies with identical Euler characteristics
    (e.g., S^3 and T^3 with chi=0) are distinguished by checked move legality and initial
    carrier homology
  - monotonic time labels certify directed graph acyclicity; they do not alone establish
    that simplices realize compatible nondegenerate Lorentzian metrics without explicit
    metric data and face-gluing compatibility
  - does not support that discrete CDT converges to continuous 4D de Sitter spacetime
    without quantum Monte Carlo infinite volume limits
  - does not eliminate Wick rotation path integral measure ambiguities
declarations_required:
- edge_signature_map: edge_id -> {spacelike (l_s^2 > 0), timelike (l_t^2 < 0)}
- simplex_type_map: simplex_id -> {(4,1), (3,2), (2,3), (1,4)}
- initial_carrier_topology: declared manifold class and verified homology of Sigma_0
- foliation_slice_registry: slice_id -> spatial 3-triangulation
- lorentzian_metric_parameters: ratio alpha = -l_t^2 / l_s^2 satisfying Regge Lorentzian nondegeneracy (alpha > 7/12 in 4D)
- causal_pachner_moves: sequence of local bistellar replacements with verified replacement complexes and boundary matches
validation_criteria:
- proper_foliation: all spacelike edges connect vertices at the same time slice t; all timelike edges connect t to t+1
- no_ctc: causal vertex ordering is strictly irreflexive and transitive (t_u < t_v for all timelike edges (u,v))
- lorentzian_metric_admissibility: ratio alpha = -l_t^2 / l_s^2 strictly satisfies simplex metric nondegeneracy on all simplex causal types
- face_gluing_compatibility: shared faces between adjacent simplices possess identical induced piecewise-flat Lorentzian geometry
- move_legality_topology_preservation: every spatial transition is composed of verified local causal Pachner moves that preserve PL 3-manifold homeomorphism to Sigma_0
- euler_characteristic_conservation: spatial Euler characteristic chi(Sigma_t) remains constant across all slices as a necessary consistency check
terminal_routes:
- CausalFoliationPreserved
- ClosedTimelikeCurveDetected
- SpatialTopologyBreach
- MetricCompatibilityDefect
- IllegalPachnerMove
- Unsupported
terminal_route_map:
  CausalFoliationPreserved: Preserve
  ClosedTimelikeCurveDetected: Refuse
  SpatialTopologyBreach: Reject
  MetricCompatibilityDefect: Reject
  IllegalPachnerMove: Refuse
  Unsupported: Unsupported
extensions:
  condition_to_local_route_map:
    all_valid: CausalFoliationPreserved
    ctc_detected: ClosedTimelikeCurveDetected
    move_topology_breach: SpatialTopologyBreach
    lorentzian_metric_incompatibility: MetricCompatibilityDefect
    unverified_pachner_surgery: IllegalPachnerMove
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  formal_statement: Lorentzian causal dynamical triangulation normal form preserving
    global proper time foliation, acyclic causal ordering, compatible Lorentzian simplex
    metrics, and certified spatial 3-manifold topology under verified causal Pachner surgeries.
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_not_a_relational_project_join
    anomaly_exclusion: closed timelike curves (CTCs), spatial slice foliation branching,
      spatial topology changes, and Lorentzian metric degeneracy
    lossless_join:
      status: declared_reconstruction
      statement: global proper time foliation t -> t+1 and invariant spatial 3-manifold
        topology certified through legal move sequence from Sigma_0
    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:
  CausalFoliationPreserved: Preserve
  ClosedTimelikeCurveDetected: Refuse
  SpatialTopologyBreach: Reject
  MetricCompatibilityDefect: Reject
  IllegalPachnerMove: Refuse
  Unsupported: Unsupported
authorization:
  authorized_by: null
  certified_implies_authorized: false
certification:
  certificate_required: true
  certificate_kind: nf.physics.lorentzian_cdt_certificate
  obstruction_class_required: false
  loss_ledger_required: false
  loss_ledger_trigger_id: null
  verifier_required: true
evidence_required:
- initial_carrier_topology: {manifold_type: string, homology_ranks: list}
- slice_foliation_indices: list
- edge_lengths_squared: dict
- causal_pachner_moves_log: list
- move_legality_witnesses: list
- euler_characteristic_sequence: list
- simplex_lorentzian_gram_signs: list
- face_gluing_compatibility_verified: boolean
- time_directed_acyclicity_witness: boolean
schema_ref: ./normal_form_contract_schema.yaml