nf.cochain.dhnf

Discrete Hodge Normal Form (DHNF)

Contract ID: nf.cochain.dhnf | Family: shape | Index Status: draft | AuthorityEffect: None

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1. Executive Summary & Metadata

Discrete Hodge Normal Form (DHNF) enforces structural normalization and explicit defect routing in the shape regime.

Property	Value
Canonical ID	nf.cochain.dhnf
Short Name	DHNF
Full Name	Discrete Hodge Normal Form
Family	shape (Relational, Algebraic & Cochain Shape Normal Forms (Data at Rest & Structural Decompositions))
Qualification	Extended Form Contract (Normal Form Contract)
Data / Math Posture	at_rest
Index Status	draft
Theory Status	established
Verification Status	absent
Contract Version	v0
Governance Fence	AuthorityEffect: None (certified_implies_authorized: false)

Foundational Treatise Specification

• Levels: Level 17 - 18
• Spec Files: ['18_grammar.md', '17_math.md']
• Anomaly Excluded: Conflating exact gradient updates with topological loop obstructions or boundary curls.
• Preserved Invariant: Discrete 1-cochain admits unique orthogonal direct sum decomposition c = d_0 p + h + d_1^* q.
• Lossless Reconstruction: Summing exact, harmonic, and coexact components reconstructs original cochain identically.

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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 (DHNF Candidate)	Anomaly risk: treating a gradient ; update, a harmonic cycle, and a ; coexact curl as the same kind o...
2. Normalization / Gate	DHNF Verification Gate	a discrete 1-cochain spli... (certified_implies_authorized: false)
3. After / Certified Output	Certified DHNF Output	Invariant: ['exact_part', ; 'harmonic_part', 'coexact_part', ; 'inner_products']

Contract-Declared Worked Cases

- ref: coda_all_universe/mssn_sn_tower/definitions.md
  section: '3.5'

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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.cochain.dhnf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	treating a gradient update, a harmonic cycle, and a coexact curl as the same kind of cochain
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: a discrete 1-cochain splits uniquely into exact, harmonic, and coexact summands for declared compatible rational inner products on C^0, C^1, C^2; summands are unique, while potentials (p, q) are unique up to gauge choice Protected Skeleton: ['exact_part', 'harmonic_part', 'coexact_part', 'inner_products']
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 nonzero harmonic part is a component of a successful decomposition. It is not a failure of the form
• a nonzero harmonic cochain is a residual_harmonic_class providing a canonical cohomology representative; it is not an obstruction in isolation and promotes to an obstruction only relative to a declared protected law via OBNF
• the orthogonal split is stated for a finite cell complex and declared positive definite rational inner products on C^0, C^1, C^2
• unique decomposition applies to the orthogonal summands (d0 p, h, d1* q); potentials p in C^0 and q in C^2 require a declared gauge selector for uniqueness
• this directory does not execute the decomposition
• does not imply authorization

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• cell_complex
• coboundary_d0
• coboundary_d1
• inner_product_c0
• inner_product_c1
• inner_product_c2
• cochain_c

5.2 Required Evidence

None declared.

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	d1 composed with d0 is zero
criterion	Check	c equals d0 p + h + d1-star q in the declared rational arithmetic
criterion	Check	the three summands are pairwise orthogonal
criterion	Check	h is closed and coclosed

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
RecordHarmonicComponent	Preserve
RejectNilpotence	Refuse
RejectReconstruction	Reject

5.5 Declared Failure Modes

• symptom: d1 d0 != 0, diagnosis: cell complex coboundaries do not satisfy Stokes nilpotence, terminal: RejectNilpotence
• symptom: d0 p + h + d1* q does not equal c, diagnosis: decomposition remainder non-zero in rational arithmetic, terminal: RejectReconstruction

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Deep-Dive Level Treatise

AuthorityEffect: None.
Level: Level 19 (Topological Substrate / Discrete Hodge Decomposition & Harmonic Orthogonality)
Evidential Grade: NATIVE IMPLEMENTED
Contract Reference: nf.cochain.dhnf
Mathematical Posture: Discrete Exterior Calculus & Combinatorial Hodge Theory
Data Posture: Substrate

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1. Schematic & Conceptual Architecture

Discrete Hodge Normal Form (DHNF) provides a canonical, unique orthogonal decomposition of discrete cochains and network flows:

• Any $k$-cochain $\omega \in C^k(K, \mathbb{Q})$ decomposes uniquely into three mutually orthogonal components: $$\omega = d\alpha + \delta\beta + \gamma$$ where:
  • $d\alpha$ is the exact component (gradient flow / coboundary),
  • $\delta\beta$ is the coexact component (solenoidal / rotational flow),
  • $\gamma$ is the harmonic component ($\Delta \gamma = 0$, representing global non-bounding topological circulation).
• Evaluates the combinatorial Hodge Laplacian: $$\Delta_k = \delta_{k+1} d_k + d_{k-1} \delta_k = d_k^T d_k + d_{k-1} d_{k-1}^T$$

               Total Cochain Space C^k
  ┌─────────────────────────────────────────────────────────────┐
  │ \omega = d\alpha (Exact) ⊕ \delta\beta (Coexact) ⊕ \gamma (Harmonic)│
  └──────────────────────────────┬──────────────────────────────┘
                                 │
             Orthogonal Hodge Decomposition
                                 ▼
  d\alpha ⟂ \delta\beta ⟂ \gamma   (Mutually Orthogonal under <·,·>)

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2. Mathematical Invariants & Quadrilateral Properties

1. Orthogonal Decomposition Theorem: For any simplicial complex with inner product $\langle u, v \rangle = u^T W v$ (where $W$ is a diagonal positive-definite weight matrix): $$\ker(\Delta_k) \cong H^k(K, \mathbb{Q})$$ Harmonic cochains correspond bijectively to the cohomology classes of the complex.

2. The Tripartite Orthogonality: $$\langle d\alpha, \delta\beta \rangle = 0, \quad \langle d\alpha, \gamma \rangle = 0, \quad \langle \delta\beta, \gamma \rangle = 0$$

3. The Quadrilateral Invariants:

   • Anomaly Excluded: Conflating rotational circulation with potential gradients, undetected topological obstructions in network routing.
   • Protected Skeleton: Harmonic cohomology basis $\gamma$, potential fields $\alpha$, and stream functions $\beta$.
   • Erased Structure: Gauge ambiguity in exact and coexact potential definitions.
   • Reconstruction: Orthogonal sum $d\alpha + \delta\beta + \gamma = \omega$.

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3. Terminal Routes & Verification Logic

Discrete Cochain \omega & Hodge Laplacian \Delta_k
      │
      ├── Exact orthogonal decomposition verified ────► Preserve (DHNFSatisfied)
      │
      ├── Residual mass \Xi > 0 after decomposition ──► Reject (HodgeDecompositionDivergence)
      │
      └── Non-positive inner product matrix ──────────► Reject (NonPositiveGramMatrix)

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4. Test Evidence & Local Execution

Verified in test_physics_fermion_optimization_and_hodge.py:

• verify_dhnf(complex, cochain): Solves the linear system for $\alpha, \beta, \gamma$, verifying that $\langle d\alpha, \delta\beta \rangle = 0$ and $\Delta \gamma = 0$ with zero rational residual.
• Asserts that when $H^1(K) = 0$ (directly connected complex), the harmonic component $\gamma$ vanishes identically.

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5. Epistemic Claim Quadrant

1. What it supports:
   • Canonical orthogonal routing and obstruction decomposition for discrete workflows and network flows.
   • Exact rational computation of discrete Helmholtz-Hodge potentials.
2. What it does not support:
   • Continuous differential forms on Riemannian manifolds without triangulation discretization.
3. The narrowest defensible claim:
   • DHNF provides a unique, coordinate-free orthogonal decomposition of discrete cochains on finite cell complexes.
4. The next test needed:
   • Property tests verifying stability of the harmonic component $\gamma$ under simplicial barycentric subdivision.

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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.cochain.dhnf (DHNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (Accept, RecordHarmonicComponent, RejectNilpotence, RejectReconstruction) 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 treating a gradient update, a harmonic cycle, and a coexact curl as the same kind of cochain is either excluded under a discrete 1-cochain splits uniquely into exact, harmonic, and coexact summands for declared compatible rational inner products on C^0, C^1, C^2; summands are unique, while potentials (p, q) are unique up to gauge choice 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.cochain.dhnf.

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7. Canonical YAML Contract (contracts/nf.cochain.dhnf.yaml)

Click to expand full canonical YAML contract (contracts/nf.cochain.dhnf.yaml)

schema_version: smplx.nf.contract.v0
schema_ref: ./normal_form_contract_schema.yaml
contract_kind: normal_form_contract
identity:
  id: nf.cochain.dhnf
  name: Discrete Hodge Normal Form
  short_name: DHNF
  version: v0
  theory_status: established
  contract_status: draft
  verification_status: absent
  family: shape
  status: draft
classification:
  hierarchy_level: normal_form
  subject_kind:
  - cochain
  - finite_cell_complex
  data_posture: at_rest
  mathematical_posture: formal_algebraic
purpose:
  excludes_anomaly: treating a gradient update, a harmonic cycle, and a coexact curl
    as the same kind of cochain
  preserves_invariant: a discrete 1-cochain splits uniquely into exact, harmonic,
    and coexact summands for declared compatible rational inner products on C^0, C^1, C^2;
    summands are unique, while potentials (p, q) are unique up to gauge choice
  protected_skeleton:
  - exact_part
  - harmonic_part
  - coexact_part
  - inner_products
  erased_structure: []
  non_claims:
  - a nonzero harmonic part is a component of a successful decomposition. It is not
    a failure of the form
  - a nonzero harmonic cochain is a residual_harmonic_class providing a canonical cohomology
    representative; it is not an obstruction in isolation and promotes to an obstruction
    only relative to a declared protected law via OBNF
  - the orthogonal split is stated for a finite cell complex and declared positive
    definite rational inner products on C^0, C^1, C^2
  - unique decomposition applies to the orthogonal summands (d0 p, h, d1* q); potentials
    p in C^0 and q in C^2 require a declared gauge selector for uniqueness
  - this directory does not execute the decomposition
  - does not imply authorization
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  section: '3.5'
  formal_statement: >
    On a finite cell complex with positive definite inner products on C^0, C^1, and C^2
    taking rational values, every discrete 1-cochain c admits a unique orthogonal
    decomposition c = d0 p + h + d1* q into exact, harmonic, and coexact summands, with
    p a 0-cochain, h harmonic, and q a 2-cochain. The three summands are pairwise orthogonal
    and uniquely determined. Canonical potentials p, q require a declared gauge selector.
    The harmonic summand satisfies d1 h = 0 and d0* h = 0. Adding the three summands returns
    c with no remainder in the declared rational arithmetic. Nilpotence d1 d0 = 0 is required
    of the complex.
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_the_hodge_sum_not_a_relational_join
    qualifies_as_extended_form: true
    anomaly_exclusion: conflating an exact gradient, a harmonic cycle, and a coexact
      curl
    lossless_join:
      status: direct_sum
      statement: the exact, harmonic, and coexact summands add back to c
    dependency_preservation:
      status: orthogonality_retained
      statement: each pair of summands has inner product zero, and nilpotence of the
        coboundary is retained
    syntactic_decision:
      status: decomposition_certificate
      statement: >
        The orthogonal split is computed from the declared complex, compatible inner products
        on C^0, C^1, C^2, and the cochain instance. A nonzero harmonic component is recorded
        and the decomposition remains valid.
declarations_required:
- cell_complex
- coboundary_d0
- coboundary_d1
- inner_product_c0
- inner_product_c1
- inner_product_c2
- cochain_c
evidence_required: []
validation_criteria:
- d1 composed with d0 is zero
- c equals d0 p + h + d1-star q in the declared rational arithmetic
- the three summands are pairwise orthogonal
- h is closed and coclosed
terminal_routes:
- Accept
- RecordHarmonicComponent
- RejectNilpotence
- RejectReconstruction
terminal_route_map:
  Accept: Preserve
  RecordHarmonicComponent: Preserve
  RejectNilpotence: Refuse
  RejectReconstruction: Reject
certification:
  certificate_required: false
  certificate_kind: null
  obstruction_class_required: false
  loss_ledger_required: false
  verifier_required: false
  loss_ledger_trigger_id: null
authorization:
  authorized_by: null
  certified_implies_authorized: false
failure_modes:
- symptom: d1 d0 != 0
  diagnosis: cell complex coboundaries do not satisfy Stokes nilpotence
  terminal: RejectNilpotence
- symptom: d0 p + h + d1* q does not equal c
  diagnosis: decomposition remainder non-zero in rational arithmetic
  terminal: RejectReconstruction
examples:
- ref: coda_all_universe/mssn_sn_tower/definitions.md
  section: '3.5'
references:
- coda_all_universe/mssn_sn_tower/definitions.md