nf.physics.cdt_spectral_dimension

CDT Spectral Dimension & Dynamical Dimensional Reduction Normal Form (CDT-SPEC-NF)

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

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1. Executive Summary & Metadata

CDT Spectral Dimension & Dynamical Dimensional Reduction Normal Form (CDT-SPEC-NF) enforces structural normalization and explicit defect routing in the physics regime.

Property	Value
Canonical ID	nf.physics.cdt_spectral_dimension
Short Name	CDT-SPEC-NF
Full Name	CDT Spectral Dimension & Dynamical Dimensional Reduction 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	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 (CDT-SPEC-NF Candidate)	Anomaly risk: non-stochastic ; transition matrices, non-monotonic ; return probabilities, negative...
2. Normalization / Gate	CDT-SPEC-NF Verification Gate	stochasticity sum_j M_ij ... (certified_implies_authorized: false)
3. After / Certified Output	Certified CDT-SPEC-NF Output	Invariant: ; ['dual_simplex_adjacency_graph', ; 'heat_kernel_diffusion_matrix',...

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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_spectral_dimension
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	non-stochastic transition matrices, non-monotonic return probabilities, negative spectral dimensions, and premature finite-size saturation masking intermediate scaling
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: stochasticity sum_j M_ij = 1, return probability monotonicity P(tau+1) <= P(tau) with P(0) = 1, logarithmic scaling d_s(tau) = -2 d(log P) / d(log tau), and Planck-to-macroscopic dimensional reduction d_s(Planck) ~ 2 to d_s(macro) ~ 4 Protected Skeleton: ['dual_simplex_adjacency_graph', 'heat_kernel_diffusion_matrix', 'return_probability_spectrum', 'logarithmic_spectral_dimension_curve', 'finite_volume_saturation_boundary']
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 claim recovery of infinite-volume asymptotic d_s = 4.00 on finite complexes without thermodynamic extrapolation
• does not claim that discrete random walks replace continuous Laplace-Beltrami spectral analysis on smooth manifolds

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• dual_graph: adjacency of 4-simplices sharing 3-faces
• diffusion_engine: discrete transition matrix M and matrix power trace operator
• return_probability: P(tau) = (1/N) Tr(M^tau)
• spectral_dimension: d_s(tau) = -2 d(log P(tau)) / d(log tau)

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
DynamicalDimensionalReductionVerified	DynamicalDimensionalReductionVerified
MonolithicDimensionFixed	MonolithicDimensionFixed
NonStochasticDiffusionError	NonStochasticDiffusionError
FiniteSizeSaturationReached	FiniteSizeSaturationReached

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_spectral_dimension (CDT-SPEC-NF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (Unsupported, DynamicalDimensionalReductionVerified, MonolithicDimensionFixed, NonStochasticDiffusionError) 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 non-stochastic transition matrices, non-monotonic return probabilities, negative spectral dimensions, and premature finite-size saturation masking intermediate scaling is either excluded under stochasticity sum_j M_ij = 1, return probability monotonicity P(tau+1) <= P(tau) with P(0) = 1, logarithmic scaling d_s(tau) = -2 d(log P) / d(log tau), and Planck-to-macroscopic dimensional reduction d_s(Planck) ~ 2 to d_s(macro) ~ 4 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_spectral_dimension.

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

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

schema_version: smplx.nf.contract.v0
contract_kind: normal_form_contract
identity:
  id: nf.physics.cdt_spectral_dimension
  name: CDT Spectral Dimension & Dynamical Dimensional Reduction Normal Form
  short_name: CDT-SPEC-NF
  version: v0
  theory_status: admitted
  contract_status: draft
  verification_status: absent
  family: physics
  status: draft
classification:
  hierarchy_level: normal_form
  subject_kind:
  - discrete_heat_kernel_diffusion
  - spectral_dimension
  - dynamical_dimensional_reduction
  - cdt_dual_graph_random_walk
  data_posture: in_motion
  mathematical_posture: formal_algebraic
purpose:
  excludes_anomaly: non-stochastic transition matrices, non-monotonic return probabilities,
    negative spectral dimensions, and premature finite-size saturation masking intermediate
    scaling
  preserves_invariant: stochasticity sum_j M_ij = 1, return probability monotonicity
    P(tau+1) <= P(tau) with P(0) = 1, logarithmic scaling d_s(tau) = -2 d(log P) /
    d(log tau), and Planck-to-macroscopic dimensional reduction d_s(Planck) ~ 2 to
    d_s(macro) ~ 4
  protected_skeleton:
  - dual_simplex_adjacency_graph
  - heat_kernel_diffusion_matrix
  - return_probability_spectrum
  - logarithmic_spectral_dimension_curve
  - finite_volume_saturation_boundary
  erased_structure:
  - embedding_coordinates
  - unphysical_acausal_diffusion_leaks
  non_claims:
  - does not claim recovery of infinite-volume asymptotic d_s = 4.00 on finite complexes
    without thermodynamic extrapolation
  - does not claim that discrete random walks replace continuous Laplace-Beltrami
    spectral analysis on smooth manifolds
declarations_required:
- dual_graph: adjacency of 4-simplices sharing 3-faces
- diffusion_engine: discrete transition matrix M and matrix power trace operator
- return_probability: P(tau) = (1/N) Tr(M^tau)
- spectral_dimension: d_s(tau) = -2 d(log P(tau)) / d(log tau)
validation_criteria:
- transition_stochasticity: every row of M sums to 1 within machine precision
- return_probability_decay: P(0) = 1 and P(tau) monotonically decreases with diffusion
    time tau
- planckian_spectral_reduction: at short diffusion times tau, d_s(tau) contracts toward
    2 (1.8 <= d_s <= 2.5)
- macroscopic_spectral_emergence: at intermediate diffusion times, d_s(tau) expands
    toward 4 (3.5 <= d_s <= 4.5)
- finite_size_cutoff_audit: as tau -> infinity, finite complex uniformization forces
    d_s -> 0
terminal_routes:
- Unsupported
- DynamicalDimensionalReductionVerified
- MonolithicDimensionFixed
- NonStochasticDiffusionError
- FiniteSizeSaturationReached
terminal_route_map:
  dimensional_reduction: DynamicalDimensionalReductionVerified
  monolithic: MonolithicDimensionFixed
  stochastic_error: NonStochasticDiffusionError
  saturated: FiniteSizeSaturationReached
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  formal_statement: non-stochastic transition matrices, non-monotonic return probabilities,
    negative spectral dimensions, and premature finite-size saturation masking intermediate
    scaling. stochasticity sum_j M_ij = 1, return probability monotonicity P(tau+1)
    <= P(tau) with P(0) = 1, logarithmic scaling d_s(tau) = -2 d(log P) / d(log tau),
    and Planck-to-macroscopic dimensional reduction d_s(Planck) ~ 2 to d_s(macro)
    ~ 4
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_not_a_relational_project_join
    anomaly_exclusion: non-stochastic transition matrices, non-monotonic return probabilities,
      negative spectral dimensions, and premature finite-size saturation masking intermediate
      scaling
    lossless_join:
      status: declared_reconstruction
      statement: stochasticity sum_j M_ij = 1, return probability monotonicity P(tau+1)
        <= P(tau) with P(0) = 1, logarithmic scaling d_s(tau) = -2 d(log P) / d(log
        tau), and Planck-to-macroscopic dimensional reduction d_s(Planck) ~ 2 to d_s(macro)
        ~ 4
    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
  DynamicalDimensionalReductionVerified: Preserve
  MonolithicDimensionFixed: Preserve
  NonStochasticDiffusionError: Reject
  FiniteSizeSaturationReached: Escalate
authorization:
  authorized_by: null
  certified_implies_authorized: false
certification:
  certificate_required: true
  certificate_kind: nf.physics.cdt_spectral_dimension_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