Add RDF/TrustGraph projection for typed claim formulas

[kunyuan]

Context

Gaia now has an executable predicate-logic surface through claim(formula=...): typed PredicateSymbol / FunctionSymbol declarations, UserPredicate applications, ClaimAtom, equality, causal markers, and finite-domain quantifier lowering. This gives Gaia a stronger internal semantics than a plain subject-predicate-object graph because formulas carry arity, domain/type constraints, quantifier scope, claim references, and lowering behavior into operators / factor graphs.

At the same time, systems such as TrustGraph show that RDF-style graphs are useful as an interoperability, indexing, retrieval, and provenance layer. The two designs are compatible if RDF is treated as a projection/interchange representation rather than the canonical Gaia formula kernel.

Proposal

Add a Gaia RDF / TrustGraph bridge that can project typed Gaia claim formulas into RDF named graphs while preserving enough structure to round-trip or audit the original formula.

Suggested mapping:

• Domain -> RDF class / SKOS concept
• PredicateSymbol -> RDF property or class, with arity/domain metadata
• FunctionSymbol -> RDF property/function node with result range metadata
• UserPredicate atom -> RDF triple where lossless, or an explicit atom node when arguments/provenance require it
• Equals / measured values -> value binding or measurement node
• Forall / Exists / Implies / compound formulas -> reified formula nodes or named graph structures, not flattened into a single RDF edge
• ClaimAtom -> RDF link to the source Gaia claim IRI
• Formula provenance -> PROV-O / named graph metadata with source claim, source span, package, and compiler/lowering metadata

Non-goals

• Do not replace Gaia's canonical typed formula AST with RDF triples.
• Do not flatten quantified or compound formulas into simple SPO edges.
• Do not let LLM-extracted RDF predicates enter Gaia formulas without typed symbol validation.

Why this matters

This would let Gaia benefit from TrustGraph-like graph retrieval, ontology selection, cross-document linking, and provenance/explainability while preserving Gaia's own semantics for formula validation, lowering, and probabilistic inference.

Acceptance criteria

• Define a minimal RDF projection schema for Gaia domains, symbols, atomic formulas, claim references, and provenance.
• Support direct triple projection for simple atoms where no logical structure is lost.
• Support reified/named-graph projection for quantifiers and compound formulas.
• Add round-trip or audit tests showing that projection does not silently erase quantifier scope, arity/domain information, or claim identity.
• Document that RDF is an external graph/interchange layer, while Gaia typed formula AST remains the canonical source of formula semantics.