nf.mssn.bqnf

MSS-N Boundary-Quotient Normal Form (MSSN-BQNF)

Contract ID: nf.mssn.bqnf | Family: change | Index Status: draft | AuthorityEffect: None

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1. Executive Summary & Metadata

MSS-N Boundary-Quotient Normal Form (MSSN-BQNF) enforces structural normalization and explicit defect routing in the change regime.

Property	Value
Canonical ID	nf.mssn.bqnf
Short Name	MSSN-BQNF
Full Name	MSS-N Boundary-Quotient Normal Form
Family	change (Data in Motion & Boundary-Quotient Incremental Maintenance Contracts (BQNF & Cochain Repair))
Qualification	Extended Form Contract (Normal Form Contract)
Data / Math Posture	in_motion
Index Status	draft
Theory Status	proposed
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 (MSSN-BQNF Candidate)	Anomaly risk: uncertified local ; repair / unverified coboundary ; lift during MSS-N level...
2. Normalization / Gate	MSSN-BQNF Verification Gate	incremental repair of MSS... (certified_implies_authorized: false)
3. After / Certified Output	Certified MSSN-BQNF Output	Invariant: ; ['boundary_fingerprint', ; 'c0_repair_potentials',...

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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.mssn.bqnf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	uncertified local repair / unverified coboundary lift during MSS-N level transitions and cycle repairs
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: incremental repair of MSS-N complexes and level transitions preserves declared boundary containment, boundary nilpotence (d1 . d0 == 0), and integer homology exactness (omega = d2_ext c) Protected Skeleton: ['boundary_fingerprint', 'c0_repair_potentials', 'c1_defects', 'd1_incidence_matrix', 'd2_ext_repair_image', 'quotient_homology_class']
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 authorize execution permissions or terminal semantic decisions
• does not ensure under declared assumptions generic simplicial complex homology without pinned finite incidence
• syntactic acceptance of qed leaves receipt UNVERIFIED; independent exact rational solver replay is required before establishing truth
• does not permit unverified boundary enlargement; repair candidates must lie within declared candidate 2-cells

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• boundary_fingerprint: {'vertices': 4, 'edges': 5, 'faces': 1}
• incidence_d1
• incidence_d2
• repair_operator_d2_ext
• quotient_space: Q1 = Z1 / im(d2_ext)
• terminal_routes

5.2 Required Evidence

• candidate_wire_text
• parsed_cycle_b
• computed_d1_b
• obstruction_vector_z
• preimage_candidate_c
• computed_d2_ext_c
• exactness_receipt_status

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	boundary complex incidence matrices d_1, d_2 are declared and pinned before intake
criterion	Check	boundary nilpotence d1 . d2 == 0 holds identically over integers
``	****
criterion	Check	defect z maps to computable obstruction class in Q1 = Z1 / im(d2_ext)
criterion	Check	repair allowed only when obstruction_class == '[0]', equivalently z = d2_ext(c)
criterion	Check	unverified qed literal does not authorize state change without independent solver replay
criterion	Check	nonzero unknown or non-closed obstruction terminalizes loudly

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
Preserve	Preserve
Repair	Repair
Invalidate	Invalidate
Recompute	Recompute
Serialize	Serialize
Lift	Lift
Unsupported	Unsupported
ConservativeInvalidate	ConservativeInvalidate
Escalate	Escalate
ClosureFailure	ClosureFailure
Residual	Escalate

5.5 Declared Failure Modes

• symptom: candidate cycle b has d1 * b != 0, diagnosis: cycle closure failure, terminal: ClosureFailure
• symptom: closed cycle z cannot be expressed as d2_ext(c), diagnosis: non-trivial homology class; unrepairable locally, terminal: Escalate
• symptom: qed present but solver replay unexecuted or failing, diagnosis: unverified claim promotion attempt, terminal: Invalidate
• symptom: candidate adds edge outside pinned 5-edge complex, diagnosis: undeclared boundary enlargement, terminal: Unsupported

---

Deep-Dive Level Treatise

AuthorityEffect: None.
Level: Level 10 (Data in Motion / Incremental Repair & Boundary Quotients)
Evidential Grade: NATIVE IMPLEMENTED
Contract Reference: nf.mssn.bqnf
Mathematical Posture: Discrete Boundary-Quotient Calculus & CDC Streams
Data Posture: In Motion

---

1. Schematic & Conceptual Architecture

Boundary-Quotient Normal Form (BQNF) shifts normalization from static schemas (data at rest) to runtime state transitions (data in motion):

• Evaluates whether an incoming Change Data Capture (CDC) write event $e \in \Delta$ on base tables can be incrementally repaired across dependent views, caches, and materialized observers.
• Uses boundary fingerprints ($\partial Q$) to determine whether a query boundary is breached.
• Emits explicit typed terminal routes rather than crashing or silently serving stale data.

       Write Event e on Base Relvars
                     │
                     ▼
       Boundary Test: e ∩ ∂Q ≠ ∅ ?
                    / \
                  NO   YES
                  /     \
                 /       ▼
                /   Evaluate Residual: Xi = 0 ?
               /         / \
              /        YES  NO
             /         /     \
            ▼         ▼       ▼
        Preserve    Repair   Quarantine / Invalidate

Engine-Relative Realization: $\text{NF} = (\text{invariant}, \text{substrate}, \text{attempt})$

Industrial relational engines do not possess native category-theoretic quotient types. On real engines, BQNF is realized as a certified attempt:

Engine Substrate	CDC / Trigger Mechanism	Certificate Store	Attempt Realization
SQL Server / Azure SQL	AFTER INSERT, UPDATE, DELETE triggers or SQL Server CDC (cdc.fn_cdc_get_net_changes_*)	dbo.RepairCertificate (columns: ChangeId, BoundaryBreached, ResidualNorm, WitnessHash, CertifiedAt)	Procedural evaluation via T-SQL; routes un-repairable events to quarantine status flags
PostgreSQL	Row-level triggers (FOR EACH ROW) or logical decoding replication slots (pg_logical_slot_get_changes)	public.repair_certificate with UUID primary keys and JSONB witness payload	PL/pgSQL event hooks; external daemon verifies witness contracts and unblocks quarantined views

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2. Mathematical Invariants & Quadrilateral Properties

1. Formal Definition: Let $Q$ be an observer query over base relvars $R$. The boundary $\partial Q$ is the set of tuples whose modification can alter the membership or aggregation of $Q(R)$. A system transition $(S, e) \to S'$ is in BQNF with respect to observer $Q$ if and only if:

   • When $e \cap \partial Q = \emptyset$, $Q(S') = Q(S)$ and the transition routes to Preserve.
   • When $e \cap \partial Q \neq \emptyset$, the change satisfies the repair contract ($d\alpha = \Delta$) with residual $\Xi = 0$, routing to Repair.
   • When the change cannot be locally repaired ($\Xi > 0$), it routes to Quarantine or Invalidate with a signed RepairCertificate.

2. The Four Canonical Terminal Routes:

   • Preserve: Event does not touch the boundary; cached state remains strictly valid without recomputation.
   • Repair: Event hits the boundary, but compensating incremental deltas restore exact consistency.
   • Quarantine: Event causes a non-trivial boundary breach; view is fenced from serving production traffic until background recomputation finishes.
   • Invalidate: Boundary severely corrupted; cache is purged and must be rebuilt from ground relvars.

3. The Quadrilateral Invariants:

   • Anomaly Excluded: Silent stale reads, unbounded CDC drift, phantom aggregate corruption, and uncertified cache repairs.
   • Protected Skeleton: Base relvar identities, boundary predicates $\partial Q$, and verified repair certificates.
   • Erased Structure: Intermediate transient CDC states that collapse under delta folding.
   • Reconstruction: Exact incremental view maintenance $Q(S') = Q(S) \oplus \Delta Q(e)$.

---

3. Terminal Routes & Verification Logic

CDC Event e & Materialized Observer Q
      │
      ├── e outside boundary (e ∩ ∂Q = ∅) ──────────► Preserve (ZeroCostInvariance)
      │
      ├── Boundary hit, local repair exact (Ξ = 0) ──► Repair (IncrementalDeltaApplied)
      │
      ├── Boundary hit, local repair diverges ───────► Quarantine (BoundedDiscrepancy)
      │
      └── Unreconciled schema migration ─────────────► Invalidate (CachePurged)

---

4. Test Evidence & Local Execution

Verified in test_bqnf_cdc_property.py and test_chase_algebra_and_bqnf_streaming.py:

• test_bqnf_orders_paid_revenue_incremental_repair: Simulates 50 e-commerce order status transitions; asserts that non-boundary updates route to Preserve and boundary hits route to Repair with verified delta payloads.
• test_bqnf_aggregate_boundary_hit_with_zero_residual: supports that when an aggregate hits the boundary with $\Xi = 0$, the state updates without full table scans.

---

5. Epistemic Claim Quadrant

1. What it supports:
   • checked by this oracle incremental cache maintenance with zero stale read anomalies.
   • Deterministic routing to named terminal states under all CDC event types.
2. What it does not support:
   • Arbitrary ad-hoc SQL rewrites without declared boundary definitions.
3. The narrowest defensible claim:
   • BQNF ensures under declared assumptions that an observer is updated locally if and only if the event does not breach the verified boundary quotient.
4. The next test needed:
   • Property tests under randomized concurrent CDC event reordering to verify abelian commutation of delta repairs.

---

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.mssn.bqnf (MSSN-BQNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (Preserve, Repair, Invalidate, Recompute) 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 uncertified local repair / unverified coboundary lift during MSS-N level transitions and cycle repairs is either excluded under incremental repair of MSS-N complexes and level transitions preserves declared boundary containment, boundary nilpotence (d1 . d0 == 0), and integer homology exactness (omega = d2_ext c) 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.mssn.bqnf.

---

7. Canonical YAML Contract (contracts/nf.mssn.bqnf.yaml)

Click to expand full canonical YAML contract (contracts/nf.mssn.bqnf.yaml)

schema_version: smplx.nf.contract.v0
schema_ref: ./normal_form_contract_schema.yaml
contract_kind: normal_form_contract
identity:
  id: nf.mssn.bqnf
  name: MSS-N Boundary-Quotient Normal Form
  short_name: MSSN-BQNF
  version: v0
  theory_status: proposed
  contract_status: draft
  verification_status: executable
  family: change
  status: draft
classification:
  hierarchy_level: normal_form
  subject_kind:
  - mssn_boundary_packet
  - k0_cycle_repair
  - glyph_path_transition
  - homology_repair_candidate
  data_posture: in_motion
  mathematical_posture: operational
purpose:
  excludes_anomaly: uncertified local repair / unverified coboundary lift during MSS-N
    level transitions and cycle repairs
  preserves_invariant: incremental repair of MSS-N complexes and level transitions
    preserves declared boundary containment, boundary nilpotence (d1 . d0 == 0), and
    integer homology exactness (omega = d2_ext c)
  protected_skeleton:
  - boundary_fingerprint
  - c0_repair_potentials
  - c1_defects
  - d1_incidence_matrix
  - d2_ext_repair_image
  - quotient_homology_class
  erased_structure:
  - raw_comment_lines
  - cosmetic_vertex_relabeling
  non_claims:
  - does not authorize execution permissions or terminal semantic decisions
  - does not ensure under declared assumptions generic simplicial complex homology without pinned finite incidence
  - syntactic acceptance of qed<exact> leaves receipt UNVERIFIED; independent exact
    rational solver replay is required before establishing truth
  - does not permit unverified boundary enlargement; repair candidates must lie within
    declared candidate 2-cells
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  formal_statement: An MSS-N boundary transition or cycle-repair candidate is in Boundary-Quotient
    Normal Form (MSSN-BQNF) when each state modification or cycle candidate is classified
    against a declared boundary complex before any carrier state is patched. In K0
    cycle repair (cycle_repair.ebnf and mssn_transcoder.py), the 1-cycle z is closed
    when d1 z = 0 (z in Z1 = ker d1). The defect z is exact when z = d2_ext c for
    an external 2-cell preimage c, equivalently when its class in Q1 = Z1 / im(d2_ext)
    is zero ([z] = 0). Exact defects repair. A zero defect preserves state. A closed
    non-exact defect (d1 z = 0 but [z] != 0) cannot be repaired locally and escalates.
    A defect with d1 z != 0 is an invalid cycle. Stamping qed<exact> submits the exactness
    assertion; parsing leaves the receipt UNVERIFIED until independent exact solver
    replay certifies that [z] = 0.
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_exact_homological_repair_not_relational_project_join
    anomaly_exclusion: A local complex patch that was not shown to be an exact boundary,
      a silent under-invalidation in a boundary composite, or an unverified qed<exact>
      claim.
    lossless_join:
      status: recomputation_agreement
      statement: The repaired simplicial complex agrees with a full exact computation
        of H_1(K) on the declared complex. That agreement is the reconstruction theorem
        for this form.
    dependency_preservation:
      status: boundary_declared_before_runtime
      statement: The 4 vertices, 5 edges, 1 face, and incidence matrices d_1, d_2
        are declared and pinned before runtime; a candidate write does not enlarge
        that boundary.
    syntactic_decision:
      status: declaration_plus_event_decidable
      statement: The route is computed from the declared boundary matrices and the
        cycle candidate vector. It is syntactically decidable via integer matrix multiplication.
declarations_required:
- boundary_fingerprint:
    vertices: 4
    edges: 5
    faces: 1
- incidence_d1
- incidence_d2
- repair_operator_d2_ext
- quotient_space: Q1 = Z1 / im(d2_ext)
- terminal_routes
evidence_required:
- candidate_wire_text
- parsed_cycle_b
- computed_d1_b
- obstruction_vector_z
- preimage_candidate_c
- computed_d2_ext_c
- exactness_receipt_status
validation_criteria:
- boundary complex incidence matrices d_1, d_2 are declared and pinned before intake
- boundary nilpotence d1 . d2 == 0 holds identically over integers
- candidate chain b is closed: d1 * b == 0
- defect z maps to computable obstruction class in Q1 = Z1 / im(d2_ext)
- repair allowed only when obstruction_class == '[0]', equivalently z = d2_ext(c)
- unverified qed<exact> literal does not authorize state change without independent
  solver replay
- nonzero unknown or non-closed obstruction terminalizes loudly
proof_obligations:
- id: mssn.bqnf.nilpotence
  status: theorem_shaped
  ref: omega_engine_atlas/coda_atlas/F004_papers/001_finite_obstruction_calculus/000_pre_genesis_to_mssn/paper.md
  section: 13.3 (the explicitly declared triangle face)
- id: mssn.bqnf.exactness_solvability
  status: theorem_shaped
  ref: omega_engine_atlas/coda_atlas/F004_papers/001_finite_obstruction_calculus/000_pre_genesis_to_mssn/paper.md
  section: 13 (finite examples; general exact-solver obligations remain separate)
- id: mssn.bqnf.receipt_unverified_firewall
  status: theorem_shaped
  ref: ow/omega_engine_pfp_v0.2/grammar/README.md
  section: Source and wire boundaries
terminal_routes:
- Preserve
- Repair
- Invalidate
- Recompute
- Serialize
- Lift
- Unsupported
- ConservativeInvalidate
- Escalate
- ClosureFailure
- Residual
terminal_route_map:
  Preserve: Preserve
  Repair: Repair
  Invalidate: Invalidate
  Recompute: Recompute
  Serialize: Serialize
  Lift: Lift
  Unsupported: Unsupported
  ConservativeInvalidate: ConservativeInvalidate
  Escalate: Escalate
  ClosureFailure: ClosureFailure
  Residual: Escalate
certification:
  certificate_required: true
  certificate_kind: exact_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: candidate cycle b has d1 * b != 0
  diagnosis: cycle closure failure
  terminal: ClosureFailure
- symptom: closed cycle z cannot be expressed as d2_ext(c)
  diagnosis: non-trivial homology class; unrepairable locally
  terminal: Escalate
- symptom: qed<exact> present but solver replay unexecuted or failing
  diagnosis: unverified claim promotion attempt
  terminal: Invalidate
- symptom: candidate adds edge outside pinned 5-edge complex
  diagnosis: undeclared boundary enlargement
  terminal: Unsupported
references:
- ../../../../omega_engine_atlas/coda_atlas/F004_papers/001_finite_obstruction_calculus/000_pre_genesis_to_mssn/paper.md
- ../../../ow/omega_engine_pfp_v0.2/grammar/cycle_repair.ebnf
- ../../../ow/omega_engine_pfp_v0.2/grammar/README.md
- ./nf.change.bqnf.yaml
- ./normal_form_contract_schema.yaml
obstruction_substrate:
  policy_ref: ../substrate/default_finite_obstruction_policy.yaml
  finite_carrier:
    description: pinned K0 simplicial complex with 4 vertices, 5 edges, 1 face
    max_rank: 5
  cochain_complex:
    C0: 2_cell_preimages_c
    C1: 1_cycles_z
    C2: 0_boundary_checks_d1_z
    d0: d2_ext maps external 2-cells to 1-boundaries
    d1: d1 maps 1-simplices to 0-boundaries
    quotient: Q1 = Z1 / im(d2_ext)
  obstruction_class:
    notation: '[z]'
    exact: '[z] == 0'
    residual_closed: d1(z) == 0 and [z] != 0
    open: d1(z) != 0
  carrier_realization:
    ref: geom.directed_graph_transition.v0
    path: ../geometry_modules/geom.directed_graph_transition.v0.yaml
  geometry_module: null
  gauge_metric_selector: null