nf.physics.bianchi_regge

Regge-Bianchi Normal Form (arXiv:1904.01966 Recovery) (RBNF)

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

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1. Executive Summary & Metadata

Regge-Bianchi Normal Form (arXiv:1904.01966 Recovery) (RBNF) enforces structural normalization and explicit defect routing in the physics regime.

Property	Value
Canonical ID	nf.physics.bianchi_regge
Short Name	RBNF
Full Name	Regge-Bianchi Normal Form (arXiv:1904.01966 Recovery)
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	admitted
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 (RBNF Candidate)	Anomaly risk: discrete Riemann ; tensor antisymmetry violations, ; non-vanishing cyclic Jacobi sum...
2. Normalization / Gate	RBNF Verification Gate	exact discrete 1st and 2n... (certified_implies_authorized: false)
3. After / Certified Output	Certified RBNF Output	Invariant: ; ['discrete_riemann_tensor', ; 'ricci_tensor_contraction',...

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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.bianchi_regge
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	discrete Riemann tensor antisymmetry violations, non-vanishing cyclic Jacobi sums, and unclosed bone boundary flux
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: exact discrete 1st and 2nd Bianchi identities and discrete vacuum Einstein field equations Protected Skeleton: ['discrete_riemann_tensor', 'ricci_tensor_contraction', 'ricci_scalar_curvature', 'bone_boundary_closure', 'discrete_einstein_residuals']
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 support that discrete Bianchi identities imply continuum Einstein-Hilbert solutions without continuum limits
• does not eliminate non-local holonomy effects around finite simplicial loops

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• bone_density: rho_p
• deficit_angle: epsilon_p
• bone_orientation_tensor: U_{alpha beta}^p
• hinge_length_or_area: L_p

5.2 Required Evidence

None declared.

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
Unsupported	Unsupported
BianchiExact	BianchiExact
FirstBianchiViolation	FirstBianchiViolation
SecondBianchiViolation	SecondBianchiViolation
EinsteinResidualNonZero	EinsteinResidualNonZero

5.5 Declared Failure Modes

All defects route explicitly through the Terminal Route Map above.

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Deep-Dive Level Treatise

AuthorityEffect: None.
Level: Level 21 (Discrete Spacetime / Discrete Regge Calculus, Lorentzian Foliations & Deficit Angles)
Evidential Grade: NATIVE IMPLEMENTED
Contract Reference: nf.physics.bianchi_regge, nf.physics.lorentzian_cdt
Mathematical Posture: Discrete Differential Geometry, Regge Calculus, Lorentzian Spacetime Foliation, and Causal Dynamical Triangulations
Data Posture: Physical Geometry / Computational Spacetime

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1. Schematic & Conceptual Architecture

Regge-CDT Normal Form enforces geometric, metric, and causal consistency for piecewise-linear discrete spacetimes:

• Discrete Regge Calculus: Replaces continuous Riemannian/Lorentzian metrics with piecewise-flat simplicial geometries defined exclusively by squared edge lengths $s_e = \ell_e^2$. Curvature is concentrated entirely on codimension-2 hinges (bones $h$), measured via deficit angles $\epsilon_h$: $$\epsilon_h = 2\pi - \sum_{\sigma \supset h} \theta_h(\sigma)$$
• Bianchi Identity & Regge Variation: The discrete Einstein-Hilbert-Regge action: $$S_{\text{Regge}} = \sum_h A_h \epsilon_h - \Lambda \sum_\sigma V_\sigma$$ satisfies the discrete contracted Bianchi identity: the variation of deficit angles vanishes identically under vertex displacements when edge lengths vary ($\sum_h A_h \frac{\partial \epsilon_h}{\partial s_e} = 0$).
• Lorentzian Causal Dynamical Triangulation (CDT): Solves the instability of Euclidean dynamical triangulations by enforcing a strict global proper-time foliation $t \in {1, 2, \dots, T}$. Only spacelike and timelike edges are admitted, preventing baby-universe branching and acausal topology change.

       Simplicial Hinge h                 Foliated Sandwich t -> t+1
    ┌───────────────────────┐            ┌───────────────────────────┐
    │ Bone / Hinge: h in S  │            │ Spatial Slice S_t (t=k)   │
    │ Dihedral Angles: theta│            │ Simplex Types: (4,1), (3,2│
    │ Deficit: 2pi - sum(th)│            │ Spatial Slice S_{t+1}     │
    └───────────┬───────────┘            └─────────────┬─────────────┘
                │                                      │
                └───────────────────┬──────────────────┘
                                    ▼
                    Regge-CDT Normal Form
          1. Exact Triangle Inequalities & Positivity
          2. Discrete Bianchi Identity: sum_h A_h d(eps_h) = 0
          3. Strict Lorentzian Causal Foliation (No CTCs)
          4. Wick-Rotated Transfer Matrix Spectral Dimension

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2. Mathematical Invariants & Quadrilateral Properties

1. Quadrilateral Decomposition:

   • Anomaly Excluded: Non-Euclidean/non-Lorentzian dihedral angles, imaginary triangle areas, spatial topology changes within proper-time slices, closed timelike curves (CTCs), and violation of discrete contracted Bianchi identities.
   • Protected Skeleton: Simplicial complex connectivity $\mathcal{K}$, squared edge lengths $s_e \in \mathbb{Q}_{>0}$, spatial foliation index $t \in \mathbb{N}$, and causal simplex classification (e.g. $(4,1)$, $(3,2)$, $(2,3)$, $(1,4)$ in 4D).
   • Erased Structure: Coordinate charts, diffeomorphism gauge redundancies, continuous metric tensor fields $g_{\mu\nu}(x)$, and continuous Christoffel symbols $\Gamma^\alpha_{\beta\gamma}$.
   • Reconstruction: Explicit metric tensor evaluation via Cayley-Menger determinants, deficit angle sums $\epsilon_h$, discrete Ricci scalar curvature, and path-integral transfer matrices over causally admissible triangulations.

2. Cayley-Menger Positivity & Triangle Inequalities: Every $n$-simplex $\sigma$ must possess positive Cayley-Menger volume: $$V_n(\sigma)^2 = \frac{(-1)^{n+1}}{2^n (n!)^2} \det \begin{pmatrix} 0 & 1 & 1 & \dots & 1 \ 1 & 0 & s_{01} & \dots & s_{0n} \ 1 & s_{10} & 0 & \dots & s_{1n} \ \vdots & \vdots & \vdots & \ddots & \vdots \ 1 & s_{n0} & s_{n1} & \dots & 0 \end{pmatrix} > 0$$ All 2-faces must strictly satisfy the triangle inequality $\ell_{ij} + \ell_{jk} > \ell_{ik}$.

3. Causal Foliation Invariant: A triangulation $\mathcal{T}$ is in Lorentzian CDT Normal Form if and only if:

   • Vertices are partitioned into discrete time hyperplanes: $V = \bigsqcup_{t=1}^T V_t$.
   • Edges are either purely spacelike ($e = (u, v)$ with $t(u) = t(v)$) or purely timelike ($t(v) = t(u) + 1$).
   • No spacelike-to-timelike topology hopping or degenerate pinchings exist.

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3. Terminal Routes & Verification Logic

Input Simplicial Configuration (K, s_e, t)
   │
   ├─► Check 1: Metric Positivity & Triangle Inequalities
   │            Verify Cayley-Menger det(M) > 0 and s_e > 0.
   │            FAIL ──► Terminal Reject: DegenerateGeometryResidual
   │
   ├─► Check 2: Dihedral Angle & Deficit Angle Computation
   │            Compute theta_h(sigma) via arccos of edge-length rationales.
   │            Calculate deficit angle eps_h = 2pi - sum(theta_h).
   │            FAIL ──► Terminal Reject: DihedralBranchCutResidual
   │
   ├─► Check 3: Discrete Bianchi Identity Gate
   │            Assert sum_{h \supset e} A_h (d eps_h / d s_e) == 0.
   │            FAIL ──► Terminal Reject: BianchiViolationResidual
   │
   ├─► Check 4: Causal Foliation & Sandwich Structure (CDT)
   │            Verify each simplex is of type (d, 1) or (d-1, 2) across slices.
   │            Confirm zero spatial boundary changes within slices.
   │            FAIL ──► Terminal Reject: AcausalTopologyChangeResidual
   │
   └─► PASS: Canonical Regge-CDT Normal Form Attested

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4. Test Evidence & Local Execution

Verification suite is covered in:

• omega_engine_education/tests/normal_forms/test_tranche17_regge_bianchi_and_higher_coboundaries.py
• omega_engine_education/tests/normal_forms/test_tranche26_unified_retraction_dag_and_regge_synthesis.py

Run targeted verification:

py -3.12 -m pytest omega_engine_education/tests/normal_forms/test_tranche17_regge_bianchi_and_higher_coboundaries.py -v

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5. Epistemic Claim Quadrant

1. What It Supports:

   • Formal verification that discrete simplicial complexes possess well-defined, non-degenerate piecewise-linear geometries with positive volumes.
   • Verification of the discrete contracted Bianchi identity on closed simplicial triangulations.
   • Verification of strict time-foliated causal structures avoiding acausal branching and baby universes in discrete quantum gravity simulations.

2. What It Does Not Support:

   • Does not claim convergence to a specific smooth 4-dimensional Einstein manifold in the continuum limit without Monte Carlo finite-size scaling analyses.
   • Does not certify quantum unitarity for configurations violating reflection positivity.

3. Narrowest Defensible Claim: Under declared discrete simplicial connectivity and squared edge lengths, Regge-CDT Normal Form ensures under declared assumptions that deficit angles are computable, satisfy the discrete Bianchi identity, and respect global Lorentzian foliation constraints.

4. Next Test Needed: Implement dynamic Wick rotation transfer matrix tests certifying spectral dimension $d_s \approx 4$ on macroscopic scales for CDT ensembles.

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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.bianchi_regge (RBNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (Unsupported, BianchiExact, FirstBianchiViolation, SecondBianchiViolation) 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 discrete Riemann tensor antisymmetry violations, non-vanishing cyclic Jacobi sums, and unclosed bone boundary flux is either excluded under exact discrete 1st and 2nd Bianchi identities and discrete vacuum Einstein field equations 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.bianchi_regge.

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

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

schema_version: smplx.nf.contract.v0
contract_kind: normal_form_contract
identity:
  id: nf.physics.bianchi_regge
  name: Regge-Bianchi Normal Form (arXiv:1904.01966 Recovery)
  short_name: RBNF
  version: v0
  theory_status: admitted
  contract_status: draft
  verification_status: absent
  family: physics
  status: draft
classification:
  hierarchy_level: normal_form
  subject_kind:
  - simplicial_riemann_curvature
  - bianchi_identities
  data_posture: at_rest
  mathematical_posture: formal_algebraic
purpose:
  excludes_anomaly: discrete Riemann tensor antisymmetry violations, non-vanishing
    cyclic Jacobi sums, and unclosed bone boundary flux
  preserves_invariant: exact discrete 1st and 2nd Bianchi identities and discrete
    vacuum Einstein field equations
  protected_skeleton:
  - discrete_riemann_tensor
  - ricci_tensor_contraction
  - ricci_scalar_curvature
  - bone_boundary_closure
  - discrete_einstein_residuals
  erased_structure:
  - continuous_christoffel_symbols
  non_claims:
  - does not support that discrete Bianchi identities imply continuum Einstein-Hilbert
    solutions without continuum limits
  - does not eliminate non-local holonomy effects around finite simplicial loops
declarations_required:
- bone_density: rho_p
- deficit_angle: epsilon_p
- bone_orientation_tensor: U_{alpha beta}^p
- hinge_length_or_area: L_p
validation_criteria:
- riemann_antisymmetry: R_{alpha beta mu nu} = - R_{beta alpha mu nu} = - R_{alpha
    beta nu mu}
- first_bianchi_cyclic: R_{[alpha beta mu] nu} = R_{alpha beta mu nu} + R_{beta mu
    alpha nu} + R_{mu alpha beta nu} = 0
- ricci_scalar_relation: R = 2 * rho_p * epsilon_p
- second_bianchi_closure: sum_p epsilon_p * U^p = 0
- discrete_einstein_vacuum: sum_{p \supset e} epsilon_p * (dL_p / dl_e) = 0
terminal_routes:
- Unsupported
- BianchiExact
- FirstBianchiViolation
- SecondBianchiViolation
- EinsteinResidualNonZero
terminal_route_map:
  all_satisfied: BianchiExact
  first_violation: FirstBianchiViolation
  second_violation: SecondBianchiViolation
  einstein_residual: EinsteinResidualNonZero
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  formal_statement: discrete Riemann tensor antisymmetry violations, non-vanishing
    cyclic Jacobi sums, and unclosed bone boundary flux. exact discrete 1st and 2nd
    Bianchi identities and discrete vacuum Einstein field equations
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_not_a_relational_project_join
    anomaly_exclusion: discrete Riemann tensor antisymmetry violations, non-vanishing
      cyclic Jacobi sums, and unclosed bone boundary flux
    lossless_join:
      status: declared_reconstruction
      statement: exact discrete 1st and 2nd Bianchi identities and discrete vacuum
        Einstein field equations
    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:
  Unsupported: Unsupported
  BianchiExact: Preserve
  FirstBianchiViolation: Reject
  SecondBianchiViolation: Reject
  EinsteinResidualNonZero: Reject
authorization:
  authorized_by: null
  certified_implies_authorized: false
certification:
  certificate_required: true
  certificate_kind: nf.physics.bianchi_regge_certificate
  obstruction_class_required: false
  loss_ledger_required: false
  loss_ledger_trigger_id: null
  verifier_required: true
evidence_required: []
schema_ref: ./normal_form_contract_schema.yaml