The Coherence Requirement

Marco Polo describes a bridge, stone by stone. "But which is the stone that supports the bridge?" Kublai Khan asks. "The bridge is not supported by one stone or another," Marco answers, "but by the line of the arch that they form." Kublai Khan remains silent, reflecting. Then he adds: "Why do you speak of the stones? It is only the arch that matters to me." Polo answers: "Without stones there is no arch."

Local predicate invention creates an obligation at every relevant overlap. A5 states that obligation before the later machinery supplies a full construction: new predicates must admit reconciliation and preserve the declared invariants of the views they enter.

Coherence Gap

Generating a candidate meaning and enforcing constraints over represented data are distinct operations. The problem in this chapter arises when a locally proposed predicate is used across partially overlapping contexts without an explicit account of agreement, scope, and invariant preservation.

The preceding chapters motivate four obligations:

• Commitment (A1): State which claims have been accepted and which consequence relation governs their consistency.
• Provenance (A2): Preserve the evidence and identifying links needed to inspect a claim and its derivation.
• Vocabulary (A3): State how a proposed distinction acquires an interpretation and enters the operative signature.
• Epistemic status (A4): Preserve the declared inference regime and distinguish a derived negative from silence under its consistency assumption. Opposed evidence requires a separate comparison.

Existing systems can represent and implement parts of this discipline, and hybrid systems can combine them. The unresolved question is whether a particular implementation enforces the combined obligations on every relevant path, including disagreement and change. Training a model or declaring a schema does not, by itself, answer that question.

A5 names the contract pursued here: contextual predicate admission under invariants. It is one proposed organization of the problem, not an impossibility result for other architectures.

A successful local check leaves the cost of a further comparison open. The next use determines which agreements matter.

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Views and Overlaps

Coherence fails at boundaries; so we name the boundary objects.

A view is a locally coherent theory-plus-data slice: a signature (types and predicates), constraints (invariants that must hold), a provenance regime (what counts as evidence), and the subset of entities or records the view purports to cover. In the fashion catalog, Supplier A's feed is a view. Supplier C's feed is a different view. The merged catalog is a candidate view: it is the result of attempting to glue views, and the question is whether that gluing is justified under an explicit coherence discipline.

Two views overlap when they make claims about common referents—the same entities, the same events, or entities identified as the same under a stated equivalence. Supplier A and Supplier C overlap on every SKU they both sell. The CI team's definition of "production-ready" and the Release team's definition overlap on every build artifact that passes through both pipelines.

Overlap is where coherence is tested. Shared referents make certain disagreements possible to compare. A matching identifier still needs its namespace and interpretation; an overlap of records does not itself establish opposed claims. And when they disagree, the system must have a response—agreement, reconciliation, or explicit fork. Silent collision is the failure mode we are diagnosing.

From here on, the question is not whether a claim is fluent or grounded, but whether it is composable.

Imagine a city covered by overlapping maps. Each map is accurate within its boundaries, but the boundaries overlap. For the maps to form an atlas, they must agree where they overlap. A street drawn vertically on one sheet and horizontally on another may agree after rotation. The atlas needs a declared translation before the drawings can be compared. An actual disagreement under that translation prevents those particular drawings from fitting together unchanged.

This is the structure we need: local views with local truths, overlaps where truths are tested, and a gluing discipline that produces global coherence from local pieces—or else surfaces the obstruction.

In the catalog, a view might be Supplier A's item feed; the overlap is the set of SKUs that both A and C sell; agreement means that when both define a predicate like sustainable, they either coincide on those SKUs or else produce a witnessed reconciliation artifact—a scope restriction, a fork, a preference ordering—that downstream queries can respect.

The proposed agreement artifact records how one view's meanings may meet another's and what evidence supports the permitted use. Existing storage systems can hold that artifact. The engineering obligation is to define its checks and ensure that downstream operations respect their outcomes.

A sheaf supplies one precise version of this local-to-global relation¹. Its restriction maps and matching conditions must be established for the chosen data. Scoped forks and institutional compromises require other contracts; calling them reconciliation does not turn them into exact gluing.

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The Capability Table

The table traces requirements rather than ranking technology families. Its comparison boundary is explicit: a generator proposes outputs; a retriever selects source material; a configured database or logic service evaluates represented definitions and constraints. A deployed system may combine all three and add validation, provenance, and governance. A capability means the operation can be represented or implemented. An enforced guarantee requires a specified checker, execution boundary, and coverage of every relevant update path. A proposed contract is neither of those by declaration alone.

Requirement	Existing support and its limit	Obligation in the proposed discipline	Touchstones
Candidate predicates from unstructured input	Generators and learning systems can propose definitions; retrieval can supply examples. Candidate quality does not establish admissibility.	Retain the proposed definition, evidence, and intended scope for evaluation.	T6, T8
Declared invariants	Configured database and logic services enforce supported checks; the guarantee depends on the constraint language and execution conditions.	Name the invariants and preserve them on the relevant admission and update paths.	T1, T4
Dynamic predicate admission	Views, typed functions, schema evolution, and predicate registries can add represented vocabulary. These mechanisms do not alone settle its meaning or safety across contexts.	Check local grounding, overlap obligations, authority, and the chosen preservation contract.	T6, T9
Context-dependent equivalence	Context-indexed mapping relations can represent scoped equivalence. Merely storing a mapping does not justify a substitution.	Carry evidence, permitted scope, and the conditions for transport.	T2, T7
Compositional generalization	Formal query and programming languages compose specified operations. Learned generalization to unseen combinations is a separate empirical question.	Declare the composition rules and check the resulting claim under them.	T3
Explicit epistemic status	SQL provides NULL-related three-valued evaluation; richer statuses and absence policies can be represented explicitly.	Preserve each view's declared status policy and the meaning of any translation.	T5
N-ary event unity	Ordinary relational event tables and graph relation instances can preserve event grouping. Mutation and constraint checks must retain that grouping.	Bind roles, context, and evidence to an identifiable event and define its permitted updates.	T10
Typed provenance	Typed records and provenance vocabularies can represent sources, agents, and derivations. Representation alone does not establish authenticity or truth.	Bind claims to the required evidence and retain it through composition.	T2, T5

For concrete examples of this existing support, see PostgreSQL's constraints and typed functions, the W3C n-ary relation pattern, and PROV-O. SHACL also specifies graph validation and structured reports. These sources establish particular mechanisms, not automatic conformance to A5 or A17.

Coverage of separate rows does not prove that a combination satisfies the whole contract; gaps in individual components do not prove that no combination can. That requires an implementation-specific account of composition, enforcement, and failure. A5 supplies a proposed contract for that account. The later construction remains subject to the same evidentiary standard.

Two toy models put the table's rows in motion.

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Two Toy Models

Toy Model A: Two Definitions of "Puffy"

Shopper A defines "puffy" as a volume-to-body ratio: a dress is puffy if it occupies visual space disproportionate to the wearer's silhouette. This is a geometric criterion.

Shopper B defines "puffy" as fabric behavior: a dress is puffy if the material holds air or structure rather than draping. This is a material criterion.

On a catalog of 100 dresses:

• 20 are puffy by both definitions
• 15 are puffy by A's criterion only (large volume, but drapy fabric)
• 10 are puffy by B's criterion only (structured but slim silhouette)
• 55 are not puffy by either

If the system asks "is this dress puffy?" and both definitions have contributed, the answer depends on which definition. Without parametrization or reconciliation, the predicate is incoherent—not because people disagree about aesthetics, but because unreconciled definitions propagate into systems that assume a single predicate.

Consider: the catalog's fulfillment logic uses puffy to estimate shipping volume. puffy → shipping_volume_estimate → warehouse_allocation. If fulfillment substitutes the fabric criterion for a rule calibrated to the volume criterion, its estimate can change without the evidence that justified the original rule. Whether the box is wrong then depends on that rule and the garment’s packing requirements.

What coherence requires:

• If A's and B's definitions are different contexts, overlap agreement demands a decision: either the definitions carve the same boundary (prove it), or they don't (fork the predicate with explicit scope).
• If the catalog asserts an invariant—"puffy ⊆ non-fitted"—then both definitions must respect it.
• If A and B disagree on the 25 dresses in the symmetric difference, the system must produce a reconciliation artifact with provenance and justification—not silently prefer one.

Toy Model B: "Production-Ready" in Build Systems

The CI team declares a build "production-ready" if all tests pass and coverage exceeds 80%.

The Release team declares a build "production-ready" if it has passed a security audit, met performance benchmarks, and received formal release approval.

A build can be:

• CI-ready but not release-ready (tests pass, no audit yet)
• Release-ready but CI-failing (manual override, approved by the Release team)
• Both ready
• Neither ready

Downstream automation triggers on "production-ready" without knowing which definition. A CI-ready build gets deployed to production without security audit. The failure is not conceptual ambiguity. It is a system with two overlapping views that lacked a reconciliation discipline.

What coherence requires: "production-ready" is fibered over team or scope. Where CI and Release contexts overlap—the set of builds that both teams touch—the system either proves the definitions coincide, or forks them, or establishes a conjunction ("deploy requires both"). The invariant "deployed builds must be both CI-ready and Release-ready" is a constraint that any coherent gluing must satisfy.

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The Coherence Requirement

We can now state what a Third Mode system must satisfy.

By "well-defined" we mean: the predicate comes with a signature, a domain of applicability, and a method of evaluation (measurement, proof rule, or oracle) sufficient to make its outputs checkable.

The rule compresses: On overlaps, agree with proof, fork with scope, or reconcile with witness. Silent collapse is never permitted.

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Gluing on Overlaps

The coherence problem is geometric: two views overlap, and the overlap must agree.

The views use different absence conventions. The recipient can preserve both reported statuses and decide which inference its use requires. If it seeks one exact section reproducing specified local values, it must establish the A13 hypotheses and agreement on the relevant overlaps. A failure of that agreement excludes that exact gluing; it does not exclude independent investigation or a justified policy for proceeding.

A5’s broader reconciliation obligation also permits an explicit fork or scope restriction. An unfinished check leaves the proposed reliance unfinished. A13 supplies no complete decision procedure for finding every possible reconciliation.

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The Fashion Catalog with Vocabulary Conflict

The fashion catalog now carries four layers of diagnostic:

1. Commitment (Ch 1): String retrieval matched "flowy" without tracking that the evidence was a negation.
2. Provenance (Ch 2): Supplier A reported “70% silk / 30% polyester”; Supplier B reported “100% polyester”; Supplier C supplied the separate weave claim “satin.” The synthesis lost those distinctions.
3. Vocabulary (Ch 3): User asked for "puffy dresses." The catalog lacks an agreed definition and admission contract for that predicate.
4. Epistemic status (Ch 4): Supplier A's NULL means "not sustainable." Supplier C's NULL means "unknown." The merge conflated them.

Add now a fifth layer: vocabulary conflict.

The Marketing team has invented "statement piece"—items with bold visual presence suitable for editorial features. The Merchandising team has invented "bold item"—items with bright colors or unusual patterns that stand out in grid view.

Are these the same predicate? The overlap is significant but not total:

• Some items are "statement piece" in Marketing but not "bold" in Merchandising (architecturally dramatic, muted color)
• Some items are "bold" in Merchandising but not "statement piece" in Marketing (neon print, conventional cut)

This is Toy Model A with different clothes: two locally valid predicates with a nontrivial symmetric difference.

A recommendation using “bold” can legitimately select items that Marketing would not feature; an editorial using “statement piece” can omit some of Merchandising’s standouts. The unsupported step is to treat either result as satisfying the other predicate merely because their extensions overlap.

A5 requires: either these are distinct predicates (fork with scope), or they are the same predicate (prove overlap agreement), or they are related predicates with explicit transport (is "statement_piece" a subset of "bold"? a superset? neither?). The system must produce a witness for whichever answer it gives.

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A5 marks the point where diagnosis becomes a construction problem. Local vocabulary must be allowed to change, yet a change that reaches an overlap owes an account of agreement and invariant preservation. The remaining parts supply candidate machinery for that obligation, beginning with the relations under which two representations may be treated as the same.

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