Theory and Cochain Substrate

Theory and Cochain Substrate: From Codd Relations to Boundary-Quotient Normal Forms (BQNF & CODA_MUR)

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1. The Four-Pillar Quadrilateral of a Normal Form

As formalized in contracts/_golden_nfs/standard.yaml and contracts/_golden_nfs/normal_form_definitions.yaml, a structural specification qualifies as a True Normal Form (TNF) when it satisfies four pillars:

1. Pillar I — Explicit Anomaly Exclusion: Names and eliminates a specific class of data-modification or state-transition redundancy (e.g., repeating groups in 1NF, partial-key dependence in 2NF, transitive dependence in 3NF, non-superkey determinants in BCNF, independent multivalued Cartesian bloat in 4NF, non-key-implied join dependencies in 5NF).
2. Pillar II — Lossless Reconstruction Theorem: Establishes non-loss recovery under declared operators:
   • Heath's Theorem (1971): Binary projection $R = (XY, XZ)$ is lossless under functional dependencies iff $X \to Y$ or $X \to Z$.
   • Fagin's Theorem (1977): Binary projection $R = (XY, XZ)$ is lossless under multivalued dependencies iff $X \twoheadrightarrow Y \mid Z$.
   • Rissanen's Theorem (1977): $n$-ary decomposition is lossless iff the join dependency $\bowtie[R_1, \dots, R_k]$ holds.
   • Cochain Direct Sum: For discrete 1-cochains (nf.cochain.dhnf, nf.ddc.hodge_canonical), $c = d_0 p + h + d_1^* q$ reconstructs $c$ identically over $\mathbb{Q}$.
3. Pillar III — Dependency Preservation Analysis: Explicitly states whether the union of component dependencies implies the original dependency closure ($(\bigcup_i \pi_{R_i}(\Sigma))^+ = \Sigma^+$), as achieved by Bernstein synthesis for 3NF and EKNF, or whether cross-component joins are required (as can occur in BCNF).
4. Pillar IV — Syntactic Decidability from Constraints: Conformance is decided from the declared schema $(H(R), \Sigma)$ rather than scanning a single accidental table body. Where unrestricted constraints exceed decidability (nf.classical.dknf, nf.logic.qanf), bounded evaluators emit BudgetUnknown or Unsupported rather than silently guessing.

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2. Data in Motion: Boundary-Quotient Normal Form (nf.change.bqnf)

While classical normal forms govern data at rest, distributed caches, materialized views, and streaming pipelines operate on data in motion under Change Data Capture (CDC) events (INSERT, UPDATE, DELETE).

Over an admitted query boundary $\partial Q$, a write event induces a defect 1-cochain $\omega \in C^1$. BQNF evaluates the quotient class in:

$$Q^1 = C^1 / \operatorname{im}(d^0)$$

• Zero Defect ($\omega = 0 \implies \text{Preserve}$): Unrelated column updates outside $\partial Q$ leave the view unchanged.
• Exact Coboundary ($\omega = d^0 \alpha \iff [\omega] = 0 \in Q^1 \implies \text{Repair}$): The view is updated incrementally in $O(1)$ via local potential delta $\alpha$ without scanning the base table.
• Closed Residual ($[\omega] \ne 0, d^1\omega = 0 \implies \text{Recompute} / \text{Invalidate}$): Non-trivial cohomology defect (e.g., destructive deletion of a MAX/MIN witness without an auxiliary priority-queue cochain, or quantile drift exceeding $\epsilon$ tolerance). Routes to bounded partition recomputation.
• Boundary Failure ($d^1\omega \ne 0$ or missing CDC evidence $\implies \text{Invalidate} / \text{ClosureFailure}$): Fails closed when required before/after images are missing.

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3. Data in Representation: CODA_MUR Transport (nf.coda.mur.transport)

When data moves across representation boundaries $F: \operatorname{Rep}{\text{source}} \to \operatorname{Rep}{\text{target}}$ (e.g., relational OLTP table $\to$ document cache or search index), CODA_MUR (nf.coda.mur.transport) enforces the commutation square:

$$D_{\text{target}} \circ F = \bar{F} \circ D_{\text{source}} \oplus \operatorname{Loss}(F)$$

1. Descent Compatibility Witness ($\chi_F$): Verifies that applying source mutation $D_{\text{source}}$ and transporting via $\bar{F}$ agrees with transporting via $F$ and applying target mutation $D_{\text{target}}$.
2. Sealed Loss Ledger (nf.evidence.loss_ledger, nf.ddc.loss_ledger): Every unmapped attribute or coarsened precision is recorded under trigger coda_mur.unmapped_field_drop. Dropping a field without a loss-ledger entry routes immediately to LossLedgerMissing (Refuse).
3. Target Verifier Precedence: A transport certificate never bypasses target-side verification (AuthorityLaundering -> Refuse).

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4. Epistemic Claim Hygiene Quadrant

1. What it supports:
   • Supports formal classification of relational schemas, streaming CDC coboundary repairs, and loss-ledger-backed representation transport in this model.
2. What it does not support:
   • Does not claim that an unverified external database engine preserves constraints without executing the corresponding verifier.
3. The narrowest defensible claim:
   • In this model, separating data at rest (1NF–6NF), data in motion (BQNF), and data in representation (CODA_MUR) prevents silent conflation of storage redundancy, cache staleness, and translation loss.
4. The next test needed:
   • End-to-end property tests composing 3NF decomposition with BQNF CDC maintenance and CODA_MUR projection transport.