nf.physics.cdt_zolotarev_overlap

4D CDT Rational Zolotarev & Deflated Overlap Dirac Operator Contract (CDT-ZOLOTAREV-NF)

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

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1. Executive Summary & Metadata

4D CDT Rational Zolotarev & Deflated Overlap Dirac Operator Contract (CDT-ZOLOTAREV-NF) enforces structural normalization and explicit defect routing in the physics regime.

Property	Value
Canonical ID	nf.physics.cdt_zolotarev_overlap
Short Name	CDT-ZOLOTAREV-NF
Full Name	4D CDT Rational Zolotarev & Deflated Overlap Dirac Operator Contract
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 (CDT-ZOLOTAREV-NF Candidate)	Anomaly risk: un-isolated fermion ; doublers, sign-approximation ; errors on the absolute spectral...
2. Normalization / Gate	CDT-ZOLOTAREV-NF Verification Gate	verified Zolotarev ration... (certified_implies_authorized: false)
3. After / Certified Output	Certified CDT-ZOLOTAREV-NF Output	Invariant: ; ['simplicial_wilson_hermitian_hamiltonian_Hw', ; 'absolute_spectral_gap_bound',...

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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.cdt_zolotarev_overlap
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	un-isolated fermion doublers, sign-approximation errors on the absolute spectral interval, false certification from trivial GW residual solutions (R = I), unhandled intermediate spectral gaps, multishift solver nonconvergence, and un-deflated spectral collapse
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: verified Zolotarev rational sign approximation on the certified positive spectral interval [lambda_min(
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)

• a small Ginsparg-Wilson residual (1/a)gamma_5(I - R^2) does not alone establish an overlap Dirac operator; the trivial assignment R = I yields zero GW residual without approximating sign(H_w). The approximation must be certified against the spectrum of H_w^2
• signed minimum eigenvalue lambda_min(H_w) is not the spectral gap around zero; the gap must be certified as the positive lower bound min |lambda| = sqrt(lambda_min(H_w^2)) on the undeflated subspace
• the work formula O(N_iter * nnz(D_w)) bounds the cost of applying the operator to a single vector, not constructing or inverting the full dense overlap matrix
• intermediate spectral gaps between stiffening and gapped thresholds represent unresolved ranges requiring higher rational degree or deflation, not mathematical obstructions
• does not claim uniform convergence on gapless triangulations without verified low-mode deflation

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• hermitian_wilson_operator: H_w = gamma_5 D_w on simplicial complex
• absolute_spectral_interval: certified lower bound lambda_min_abs > 0 and upper bound lambda_max_abs on undeflated subspace
• zolotarev_rational_coefficients: degree-m optimal minimax rational approximation R_m(x) to sign(x) on [lambda_min_abs, lambda_max_abs] with error delta_Z
• multishift_krylov_solver: simultaneous solution of (H_w^2 + d_l I) x_l = b across all rational poles d_l > 0
• low_mode_deflation: projection Pi_low onto eigenspace with |lambda| < lambda_min_abs
• single_action_cost_bound: per-vector application cost bound C_action = O(m * N_iter * nnz(H_w) + N_defl * N)

5.2 Required Evidence

• spectral_interval_bounds: {'lambda_min_abs': 'float', 'lambda_max_abs': 'float'}
• rational_degree_m: integer
• zolotarev_minimax_error_bound: float
• multishift_cg_residuals: list
• deflated_eigenvalues: list
• deflated_eigenvectors_orthogonality: float
• operator_vector_action_cost: {'krylov_iterations': 'integer', 'nnz_count': 'integer', 'flops_per_action': 'integer'}
• ginsparg_wilson_algebraic_residual: float

5.3 Validation Criteria

Criterion ID	Name	Condition
``	****
``	****
``	****
``	****
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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
ZolotarevOverlapCertified	Preserve
ZolotarevSpectralStiffeningObstruction	Refuse
IntermediateSpectralRange	Quarantine
ZolotarevToleranceBreach	Reject
SolverConvergenceFailure	Escalate
LanczosDivergenceDefect	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.cdt_zolotarev_overlap (CDT-ZOLOTAREV-NF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (ZolotarevOverlapCertified, ZolotarevSpectralStiffeningObstruction, IntermediateSpectralRange, ZolotarevToleranceBreach) 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 un-isolated fermion doublers, sign-approximation errors on the absolute spectral interval, false certification from trivial GW residual solutions (R = I), unhandled intermediate spectral gaps, multishift solver nonconvergence, and un-deflated spectral collapse is either excluded under verified Zolotarev rational sign approximation on the certified positive spectral interval [lambda_min(|H_w|), lambda_max(|H_w|)], multishift Krylov convergence, Ginsparg-Wilson residual bound ||{D, gamma_5} - a D gamma_5 D|| <= tau_GW, and scoped sparse matrix-vector action cost 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.cdt_zolotarev_overlap.

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

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

schema_version: smplx.nf.contract.v0
contract_kind: normal_form_contract
identity:
  id: nf.physics.cdt_zolotarev_overlap
  name: 4D CDT Rational Zolotarev & Deflated Overlap Dirac Operator Contract
  short_name: CDT-ZOLOTAREV-NF
  version: v0
  theory_status: established
  contract_status: draft
  verification_status: absent
  family: physics
  status: draft
classification:
  hierarchy_level: normal_form
  subject_kind:
  - zolotarev_rational_minimax
  - lanczos_krylov_subspace
  - multishift_conjugate_gradient
  - low_mode_deflation
  - causal_dynamical_triangulations
  data_posture: in_motion
  mathematical_posture: operational
purpose:
  excludes_anomaly: un-isolated fermion doublers, sign-approximation errors on the
    absolute spectral interval, false certification from trivial GW residual solutions
    (R = I), unhandled intermediate spectral gaps, multishift solver nonconvergence,
    and un-deflated spectral collapse
  preserves_invariant: verified Zolotarev rational sign approximation on the certified
    positive spectral interval [lambda_min(|H_w|), lambda_max(|H_w|)], multishift Krylov
    convergence, Ginsparg-Wilson residual bound ||{D, gamma_5} - a D gamma_5 D|| <= tau_GW,
    and scoped sparse matrix-vector action cost
  protected_skeleton:
  - simplicial_wilson_hermitian_hamiltonian_Hw
  - absolute_spectral_gap_bound
  - zolotarev_minimax_rational_coefficients
  - multishift_conjugate_gradient_residuals
  - low_mode_deflation_subspace
  - ginsparg_wilson_residual
  - single_action_flop_accounting
  erased_structure:
  - dense_spectral_diagonalization
  - redundant_single_shift_linear_inversions
  non_claims:
  - a small Ginsparg-Wilson residual (1/a)*gamma_5*(I - R^2) does not alone establish
    an overlap Dirac operator; the trivial assignment R = I yields zero GW residual
    without approximating sign(H_w). The approximation must be certified against the
    spectrum of H_w^2
  - signed minimum eigenvalue lambda_min(H_w) is not the spectral gap around zero;
    the gap must be certified as the positive lower bound min |lambda| = sqrt(lambda_min(H_w^2))
    on the undeflated subspace
  - the work formula O(N_iter * nnz(D_w)) bounds the cost of applying the operator
    to a single vector, not constructing or inverting the full dense overlap matrix
  - intermediate spectral gaps between stiffening and gapped thresholds represent
    unresolved ranges requiring higher rational degree or deflation, not mathematical
    obstructions
  - does not claim uniform convergence on gapless triangulations without verified
    low-mode deflation
declarations_required:
- hermitian_wilson_operator: H_w = gamma_5 D_w on simplicial complex
- absolute_spectral_interval: certified lower bound lambda_min_abs > 0 and upper bound lambda_max_abs on undeflated subspace
- zolotarev_rational_coefficients: degree-m optimal minimax rational approximation R_m(x) to sign(x) on [lambda_min_abs, lambda_max_abs] with error delta_Z
- multishift_krylov_solver: simultaneous solution of (H_w^2 + d_l I) x_l = b across all rational poles d_l > 0
- low_mode_deflation: projection Pi_low onto eigenspace with |lambda| < lambda_min_abs
- single_action_cost_bound: per-vector application cost bound C_action = O(m * N_iter * nnz(H_w) + N_defl * N)
validation_criteria:
- absolute_spectral_gap_certified: "verified lower bound lambda_min_abs = sqrt(lambda_min(H_w^2|_{im(I - Pi_low)})) > 0"
- rational_minimax_error_bound: "analytical Zolotarev error satisfies sup_{x in [lambda_min_abs, lambda_max_abs]} |R_m(x) - sign(x)| <= delta_Z <= 1e-8"
- multishift_solve_convergence: "all shifted linear systems achieve relative residual ||(H_w^2 + d_l I)x_l - b|| / ||b|| <= tau_CG"
- ginsparg_wilson_algebraic_consistency: "||D_rat gamma_5 + gamma_5 D_rat - a D_rat gamma_5 D_rat|| <= tau_GW as a necessary algebraic consistency check"
- gapped_complex_routing: certified gap lambda_min_abs >= 0.05 routes to ZolotarevOverlapCertified
- witnessed_stiffening_obstruction: certified closure lambda_min_abs < 1e-4 with exhibited low-mode witness routes to ZolotarevSpectralStiffeningObstruction
- intermediate_spectral_range: gap between 1e-4 and 0.05 routes to IntermediateSpectralRange
terminal_routes:
- ZolotarevOverlapCertified
- ZolotarevSpectralStiffeningObstruction
- IntermediateSpectralRange
- ZolotarevToleranceBreach
- SolverConvergenceFailure
- LanczosDivergenceDefect
- Unsupported
terminal_route_map:
  ZolotarevOverlapCertified: Preserve
  ZolotarevSpectralStiffeningObstruction: Refuse
  IntermediateSpectralRange: Quarantine
  ZolotarevToleranceBreach: Reject
  SolverConvergenceFailure: Escalate
  LanczosDivergenceDefect: Reject
  Unsupported: Unsupported
extensions:
  condition_to_local_route_map:
    gapped_subcubic_certified: ZolotarevOverlapCertified
    gapless_stiffening_witnessed: ZolotarevSpectralStiffeningObstruction
    intermediate_gap_unresolved: IntermediateSpectralRange
    tolerance_breach: ZolotarevToleranceBreach
    solver_failed_to_converge: SolverConvergenceFailure
    lanczos_defect: LanczosDivergenceDefect
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  formal_statement: Overlap Dirac operator contract certifying degree-m Zolotarev rational
    sign approximation over certified absolute spectral intervals [lambda_min(|H_w|),
    lambda_max(|H_w|)], multishift Krylov pole convergence, explicit low-mode deflation,
    and scoped per-vector operational complexity.
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_not_a_relational_project_join
    anomaly_exclusion: un-isolated fermion doublers, broken GW relation, false sign
      approximation certificates, and un-deflated spectral collapse
    lossless_join:
      status: declared_reconstruction
      statement: certified rational Zolotarev overlap action with bounded minimax
        error and low-mode deflation
    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:
  ZolotarevOverlapCertified: Preserve
  ZolotarevSpectralStiffeningObstruction: Refuse
  IntermediateSpectralRange: Quarantine
  ZolotarevToleranceBreach: Reject
  SolverConvergenceFailure: Escalate
  LanczosDivergenceDefect: Reject
  Unsupported: Unsupported
authorization:
  authorized_by: null
  certified_implies_authorized: false
certification:
  certificate_required: true
  certificate_kind: nf.physics.cdt_zolotarev_overlap_certificate
  obstruction_class_required: false
  loss_ledger_required: false
  loss_ledger_trigger_id: null
  verifier_required: true
evidence_required:
- spectral_interval_bounds: {lambda_min_abs: float, lambda_max_abs: float}
- rational_degree_m: integer
- zolotarev_minimax_error_bound: float
- multishift_cg_residuals: list
- deflated_eigenvalues: list
- deflated_eigenvectors_orthogonality: float
- operator_vector_action_cost: {krylov_iterations: integer, nnz_count: integer, flops_per_action: integer}
- ginsparg_wilson_algebraic_residual: float
schema_ref: ./normal_form_contract_schema.yaml