nf.coda.tsnf

Topological Simplex Normal Form (TSNF)

Contract ID: nf.coda.tsnf | Family: topology | Index Status: draft | AuthorityEffect: None

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1. Executive Summary & Metadata

Topological Simplex Normal Form (TSNF) enforces structural normalization and explicit defect routing in the topology regime.

Property	Value
Canonical ID	nf.coda.tsnf
Short Name	TSNF
Full Name	Topological Simplex Normal Form
Family	topology (Simplicial Complexes, Boundary Nilpotency & Context Boundary Purity (TSNF, CBNF))
Qualification	Extended Form Contract (Normal Form Contract)
Data / Math Posture	at_rest
Index Status	draft
Theory Status	established
Verification Status	executable
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 (TSNF Candidate)	Anomaly risk: un-oriented vertex ; permutations, duplicate vertices ; in non-degenerate presentations...
2. Normalization / Gate	TSNF Verification Gate	relative to a declared... (certified_implies_authorized: false)
3. After / Certified Output	Certified TSNF Output	Invariant: ['ordered_simplices', ; 'permutation_orientation_witness', ; 'boundary_matrices',...

Contract-Declared Worked Cases

- ref: coda_all_universe/1_the_tower/2_mathematics/lean4/k_simplex_homology.lean
  section: "\xA72.1"

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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.coda.tsnf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	un-oriented vertex permutations, duplicate vertices in non-degenerate presentations, unverified boundary nilpotence, conflation of isolated boundary matrix invariant factors with homology torsion, or violation of the Euler-Poincare index theorem
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: relative to a declared vertex_total_order, every finite simplicial presentation admits a canonical vertex ordering with an explicit permutation sign witness, non-degenerate simplices have strictly distinct vertices, Stokes nilpotence holds, and homology groups H_k = ker(d_k)/im(d_{k+1}) are computed via checkable compatible basis reduction with Smith Normal Form quotient divisibility (d_i
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 compute continuous homotopy groups pi_n(X) for n >= 2
• does not decide homeomorphisms between arbitrary triangulated 4-manifolds (undecidable by Markov 1958)
• does not compute cohomology ring cup products without higher cochain coproducts
• does not imply authority minting (AuthorityEffect: None)
• this is a homological presentation form for finite oriented complexes and integer homology quotients. It is not a relational normal form and it has no project-join theorem
• canonical vertex ordering and orientation are defined relative to a declared vertex_total_order; they are not intrinsic without such an order
• torsion invariants cannot be read from isolated boundary matrices in isolation; they are computed in the quotient H_k = ker(d_k) / im(d_{k+1}) via compatible basis reduction

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• vertex_total_order
• simplex_registry
• boundary_matrices_by_dim
• max_dimension

5.2 Required Evidence

• f_vector
• nilpotence_join_receipts
• compatible_basis_transformations
• smith_normal_form_diagonal_factors
• free_betti_numbers
• torsion_divisors

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	every non-degenerate k-simplex has strictly ordered distinct vertices v_0 < v_1 < ... < v_k relative to the declared vertex_total_order
criterion	Check	duplicate vertices in a simplex are rejected as degenerate unless an explicit quotient is declared
criterion	Check	vertex permutations carry an orientation witness preserving orientation sign
criterion	Check	boundary matrix d_k has signed entries (-1)^i for face omitting v_i
criterion	Check	Stokes nilpotence d_{k-1} . d_k == 0 evaluates to zero with zero non-zero residuals
criterion	Check	Smith Normal Form reduction computes checkable compatible basis change matrices P, Q such that P d_k Q yields diagonal invariant factors d_1 <= ... <= d_r of the quotient presentation matrix
criterion	Check	divisibility condition d_i
criterion	Check	Betti numbers satisfy beta_k = dim(ker d_k) - rank(d_{k+1}) = n_k - rank(d_k) - rank(d_{k+1})
criterion	Check	torsion coefficients of H_k = ker(d_k) / im(d_{k+1}) are quotient elementary divisors d_i > 1
criterion	Check	Euler-Poincare formula sum (-1)^k n_k == sum (-1)^k beta_k holds identically

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
Accept	Preserve
CanonicalizePermutationWithSign	Repair
RejectDegenerateSimplex	Reject
InvalidateNilpotence	Refuse
RejectDivisibility	Reject
Refuse	Refuse

5.5 Declared Failure Modes

• symptom: vertex permutation without orientation witness, diagnosis: unoriented presentation, terminal: CanonicalizePermutationWithSign
• symptom: duplicate vertices in simplex, diagnosis: unquotiented degenerate simplex, terminal: RejectDegenerateSimplex
• symptom: d_{k-1} . d_k != 0, diagnosis: boundary square non-zero, Stokes failure, terminal: InvalidateNilpotence
• symptom: invariant factors do not satisfy d_i | d_{i+1}, diagnosis: invalid Smith reduction or quotient failure, terminal: RejectDivisibility

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Deep-Dive Level Treatise

AuthorityEffect: None.
Level: Level 18 (Topological Substrate / Simplicial Complexes & Euler-Poincaré)
Evidential Grade: NATIVE IMPLEMENTED
Contract Reference: nf.coda.tsnf
Mathematical Posture: Algebraic Topology & Simplicial Homology ($\partial^2 = 0$)
Data Posture: Substrate at Rest

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1. Schematic & Conceptual Architecture

Topological Simplex Normal Form (TSNF) grounds relational schemas and constraint networks in simplicial complexes:

• Vertices represent entities ($C_0$).
• Edges represent binary relationships ($C_1$).
• Triangles represent ternary join dependencies ($C_2$).
• Tetrahedra represent 4-way holistic constraints ($C_3$).
• Nilpotency Law: The fundamental invariant of algebraic topology: $$\partial_{k} \circ \partial_{k+1} = 0 \quad (\text{The boundary of a boundary is empty})$$
• Euler-Poincaré Formula: The alternating sum of cell counts equals the alternating sum of Betti numbers: $$\chi = V - E + F - T = \beta_0 - \beta_1 + \beta_2 - \beta_3$$

       Simplicial Complex K (TSNF)
             v0 ──────── v1
             │  \       / │
             │    \   /   │
             │      X     │   C0: Vertices
             │    /   \   │   C1: Edges
             │  /       \ │   C2: Triangular Faces
             v2 ──────── v3   C3: Tetrahedra
                   
         Boundary Nilpotency: \partial^2 = 0

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2. Mathematical Invariants & Quadrilateral Properties

1. Boundary Operator & Nilpotency: For an oriented $k$-simplex $[v_0, \dots, v_k]$, the boundary operator is: $$\partial_k [v_0, \dots, v_k] = \sum_{i=0}^k (-1)^i [v_0, \dots, \hat{v}i, \dots, v_k]$$ TSNF enforces that $\partial_k \circ \partial{k+1} = 0$ exactly across all dimensions.

2. Homology Presentation: $$Z_k = \ker(\partial_k), \quad B_k = \operatorname{im}(\partial_{k+1}), \quad H_k = Z_k / B_k$$ where $\beta_k = \dim(H_k)$ counts $k$-dimensional topological holes (connected components, cycles, voids).

3. The Quadrilateral Invariants:

   • Anomaly Excluded: Non-orientable boundary leaks, topological tears, cycle inconsistencies in relational hypergraphs.
   • Protected Skeleton: Simplex vertices, face inclusion posets, and rational homology ranks $\beta_k$.
   • Erased Structure: Coordinate-dependent embedding geometry (preserves intrinsic combinatorial topology).
   • Reconstruction: Simplicial chain complex $\dots \to C_2 \to C_1 \to C_0 \to 0$.

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3. Terminal Routes & Verification Logic

Simplicial Complex K & Boundary Matrices
      │
      ├── \partial^2 = 0 AND Euler-Poincaré invariant holds ──► Preserve (TSNFVerified)
      │
      ├── Boundary nilpotency fails (\partial^2 ≠ 0) ─────────► Reject (TopologicalBoundaryLeak)
      │
      ├── Euler-Poincaré sum diverges from Betti ranks ────────► Reject (EulerPoincareMismatch)
      │
      └── Complex contains non-orientable simplex ────────────► Quarantine (OrientationDefect)

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4. Test Evidence & Local Execution

Verified in test_tsnf_property_and_euler_poincare.py and test_tsnf_3complex_and_euler_poincare.py:

• test_tsnf_boundary_nilpotency: Verifies $\partial_1 \circ \partial_2 = 0$ and $\partial_2 \circ \partial_3 = 0$ over tetrahedral meshes.
• test_tsnf_euler_poincare_exact_match: Asserts that $\chi(K) = \sum (-1)^i \beta_i$ on 50 randomized topological complexes.

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5. Epistemic Claim Quadrant

1. What it supports:
   • Combinatorial classification of relational and multi-agent dependency networks via algebraic topology.
   • Exact rational computation of cycles, holes, and voids.
2. What it does not support:
   • Continuous differential manifolds or smooth Riemannian metrics (strictly discrete finite complexes).
3. The narrowest defensible claim:
   • TSNF ensures under declared assumptions that the boundary matrix sequence is a valid nilpotent chain complex satisfying the Euler-Poincaré formula.
4. The next test needed:
   • Property tests validating persistent homology barcodes across filtered simplicial complexes.

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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.coda.tsnf (TSNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (Accept, CanonicalizePermutationWithSign, RejectDegenerateSimplex, InvalidateNilpotence) 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 un-oriented vertex permutations, duplicate vertices in non-degenerate presentations, unverified boundary nilpotence, conflation of isolated boundary matrix invariant factors with homology torsion, or violation of the Euler-Poincare index theorem is either excluded under relative to a declared vertex_total_order, every finite simplicial presentation admits a canonical vertex ordering with an explicit permutation sign witness, non-degenerate simplices have strictly distinct vertices, Stokes nilpotence holds, and homology groups H_k = ker(d_k)/im(d_{k+1}) are computed via checkable compatible basis reduction with Smith Normal Form quotient divisibility (d_i | d_{i+1}) 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.coda.tsnf.

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7. Canonical YAML Contract (contracts/nf.coda.tsnf.yaml)

Click to expand full canonical YAML contract (contracts/nf.coda.tsnf.yaml)

schema_version: smplx.nf.contract.v0
schema_ref: ./normal_form_contract_schema.yaml
contract_kind: form_contract
identity:
  id: nf.coda.tsnf
  name: Topological Simplex Normal Form
  short_name: TSNF
  version: v0
  status: draft
  family: topology
  theory_status: established
  contract_status: draft
  verification_status: executable
classification:
  hierarchy_level: form
  subject_kind:
  - simplicial_complex
  - boundary_matrix
  - homology_presentation
  data_posture: at_rest
  mathematical_posture: operational
purpose:
  excludes_anomaly: un-oriented vertex permutations, duplicate vertices in non-degenerate
    presentations, unverified boundary nilpotence, conflation of isolated boundary matrix
    invariant factors with homology torsion, or violation of the Euler-Poincare index theorem
  preserves_invariant: relative to a declared vertex_total_order, every finite simplicial presentation
    admits a canonical vertex ordering with an explicit permutation sign witness, non-degenerate
    simplices have strictly distinct vertices, Stokes nilpotence holds, and homology groups
    H_k = ker(d_k)/im(d_{k+1}) are computed via checkable compatible basis reduction with Smith
    Normal Form quotient divisibility (d_i | d_{i+1})
  protected_skeleton:
  - ordered_simplices
  - permutation_orientation_witness
  - boundary_matrices
  - nilpotence_certificates
  - compatible_basis_transformations
  - smith_normal_form_diagonal
  - betti_numbers
  - torsion_coefficients
  - euler_characteristic
  erased_structure:
  - arbitrary_vertex_permutations
  - non-canonical_basis_orderings
  non_claims:
  - does not compute continuous homotopy groups pi_n(X) for n >= 2
  - does not decide homeomorphisms between arbitrary triangulated 4-manifolds (undecidable
    by Markov 1958)
  - does not compute cohomology ring cup products without higher cochain coproducts
  - 'does not imply authority minting (AuthorityEffect: None)'
  - this is a homological presentation form for finite oriented complexes and integer homology quotients.
    It is not a relational normal form and it has no project-join theorem
  - canonical vertex ordering and orientation are defined relative to a declared vertex_total_order;
    they are not intrinsic without such an order
  - torsion invariants cannot be read from isolated boundary matrices in isolation; they are computed
    in the quotient H_k = ker(d_k) / im(d_{k+1}) via compatible basis reduction
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  formal_statement: >
    On a finite simplicial complex with declared vertex_total_order, every non-degenerate
    k-simplex is uniquely represented with strictly ordered distinct vertices v_0 < ... < v_k
    carrying an orientation sign witness sgn(pi). Stokes nilpotence d_{k-1} . d_k = 0 holds,
    and integer homology groups are computed via compatible basis change matrices P, Q
    yielding checkable diagonal quotient divisibility d_i | d_{i+1}. Duplicate-vertex simplices
    are rejected unless an explicit degenerate-simplex quotient is declared.
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: smith_form_of_boundary_matrices_not_a_relvar_decomposition
    qualifies_as_extended_form: true
    anomaly_exclusion: an unordered or duplicate vertex tuple, a boundary whose square is nonzero,
      or invariant factors that break divisibility
    lossless_join:
      status: homology_reconstruction
      statement: boundary matrices and checkable compatible basis transformations determine
        the invariant factors of H_k = ker(d_k)/im(d_{k+1}), free Betti numbers, and torsion
    dependency_preservation:
      status: divisibility_chain_retained
      statement: each invariant factor divides the next under compatible basis reduction
    syntactic_decision:
      status: decomposition_certificate
      statement: the check reads the declared simplex registry, vertex total order, boundary
        matrices, and compatible basis witnesses
declarations_required:
- vertex_total_order
- simplex_registry
- boundary_matrices_by_dim
- max_dimension
evidence_required:
- f_vector
- nilpotence_join_receipts
- compatible_basis_transformations
- smith_normal_form_diagonal_factors
- free_betti_numbers
- torsion_divisors
validation_criteria:
- every non-degenerate k-simplex has strictly ordered distinct vertices v_0 < v_1 < ... < v_k
  relative to the declared vertex_total_order
- duplicate vertices in a simplex are rejected as degenerate unless an explicit quotient is declared
- vertex permutations carry an orientation witness preserving orientation sign
- boundary matrix d_k has signed entries (-1)^i for face omitting v_i
- Stokes nilpotence d_{k-1} . d_k == 0 evaluates to zero with zero non-zero residuals
- Smith Normal Form reduction computes checkable compatible basis change matrices P, Q such that
  P d_k Q yields diagonal invariant factors d_1 <= ... <= d_r of the quotient presentation matrix
- divisibility condition d_i | d_{i+1} holds for all 1 <= i < r
- Betti numbers satisfy beta_k = dim(ker d_k) - rank(d_{k+1}) = n_k - rank(d_k) - rank(d_{k+1})
- torsion coefficients of H_k = ker(d_k) / im(d_{k+1}) are quotient elementary divisors d_i > 1
- Euler-Poincare formula sum (-1)^k n_k == sum (-1)^k beta_k holds identically
proof_obligations:
- id: tsnf.boundary_nilpotence_identity
  status: machine_checked
  ref: coda_all_universe/1_the_tower/2_mathematics/lean4/k_simplex_homology.lean
  statement: Boundary of boundary vanishes identically on any oriented k-simplex
- id: tsnf.euler_poincare_invariance
  status: open_proof_obligation
  ref: coda_all_universe/1_the_tower/2_mathematics/lean4/integral_homology_torsion.lean
  note: Formal proof pending integration into repo NormalFormsCore target
terminal_routes:
- Accept
- CanonicalizePermutationWithSign
- RejectDegenerateSimplex
- InvalidateNilpotence
- RejectDivisibility
- Refuse
terminal_route_map:
  Accept: Preserve
  CanonicalizePermutationWithSign: Repair
  RejectDegenerateSimplex: Reject
  InvalidateNilpotence: Refuse
  RejectDivisibility: Reject
  Refuse: Refuse
certification:
  certificate_required: true
  certificate_kind: simplicial_homology_certificate
  obstruction_class_required: true
  loss_ledger_required: false
  verifier_required: true
  loss_ledger_trigger_id: null
authorization:
  authorized_by: null
  certified_implies_authorized: false
failure_modes:
- symptom: vertex permutation without orientation witness
  diagnosis: unoriented presentation
  terminal: CanonicalizePermutationWithSign
- symptom: duplicate vertices in simplex
  diagnosis: unquotiented degenerate simplex
  terminal: RejectDegenerateSimplex
- symptom: d_{k-1} . d_k != 0
  diagnosis: boundary square non-zero, Stokes failure
  terminal: InvalidateNilpotence
- symptom: invariant factors do not satisfy d_i | d_{i+1}
  diagnosis: invalid Smith reduction or quotient failure
  terminal: RejectDivisibility
examples:
- ref: coda_all_universe/1_the_tower/2_mathematics/lean4/k_simplex_homology.lean
  section: §2.1
references:
- coda_all_universe/mssn_sn_tower/definitions.md
- ./nf.change.bqnf.yaml