Non-atomic terms (NATs)

• Covers: how a function-application term reifies to an opaque constant before the index sees it, or stays structural when it does not.
• Not here: minting a deterministic constant for a rule's existential head → skolem.md; comparing two structural measure terms → quantity.md.
• Assumes: sentex, context, taxonomy, predicate metadata → glossary.md.

A NAT is a function-application term (F arg…) that denotes an entity — (FruitFn AppleTree), (CapitalOf France), (QuantityFn 5 Meter). Pure has no first-class function terms in a stored sentence; every stored token is atomic. NATs are supported by reification, and functions split by declaration into two kinds.

• (reifiableFunction F) — object-denoting. A ground (F a…) is a reified NAT: it reifies to an opaque nat/-namespaced constant K before it reaches the index, so the reified NAT autoindexes exactly like a hand-minted symbol.
• (unreifiableFunction F) — evaluated / interpreted. The application stays a structural NAT — (QuantityFn 5 Meter) keeps its magnitude and unit readable for a downstream prover; it is never minted.

QuantityFn must not be reifiable — reifying it would collapse 5 and Meter into an opaque atom and lose the very structure a unit prover needs.

The strategy: reify before the index

The invasive way to support function terms is to structurally index nested compounds — change the trie key and sentex/index-terms so (color (FruitFn AppleTree) Red) is findable by the nested FruitFn occurrence. That reaches further into the engine than anything else here touches. Reification sidesteps all of it. A reifiable NAT is replaced by an atomic constant on the write path, so:

• the trie key is unchanged — (color K Red) keys [color K Red], K one token;
• sentex/index-terms is unchanged — K is an ordinary symbol subterm;
• the reified NAT autoindexes, is retrieved, and is retracted exactly like any symbol.

There is no structural indexing of unreified or open NATs: the reifiable path needs none, because a reified NAT is a symbol by the time the index sees it.

The data model — all ordinary stored facts

Nothing here is a KV side table. Every mapping is a normal sentex in UniverseContext, so it rides put-sentex, the functor / argument roots, and the inverted term index like any other fact.

Fact	Meaning
(reifiableFunction F) / (unreifiableFunction F)	F's kind — a predicate-metadata mark (vaelii.impl.taxonomy :reifiable / :unreifiable prop), belief-following like transitive/symmetric
(termOfUnit K E)	the constant↔expression map: constant K denotes NAT expression E. The reverse (K → E) index; E → K is the inverted term index
(rewriteOf T E) (compound E)	NAT E reifies to the existing real term T instead of a fresh constant
(resultIsa F T) / (resultGenl F T)	F's output types — materialized on mint as (T K) / (genl K T)
(functionCorrespondingPredicate F P N)	F and P state one relationship: (F a…) = V exactly when P holds of a… with V at argument N

termOfUnit and rewriteOf are quoting predicates (nat/nat-quoting-predicates = #{termOfUnit rewriteOf}): their expression argument is a literal NAT payload that must not itself be reified or type-checked as a term.

(rewriteOf T E) overloads the equality/deprecation rewriteOf, discriminated by shape: a symbol second argument is term equality (the equality partition — see equality.md); a compound second argument is a NAT reify-to-term declaration. The equality integrate arm and wff/equality-problems both skip the compound shape, so it is stored as an inert quoting fact, never entering the closure.

The reifiable gate

A function's kind is metadata cached in the taxonomy, so the per-sentence gate — "does the KB declare any reifiableFunction?" — is a free in-memory set read (nat/any-reifiable-functions? ⇒ seq (tax/props tax :reifiable)), belief-following and needing no mtime cache. A KB that declares no reifiableFunction pays one set read per assert / query and nothing else: the whole subsystem is a gated no-op.

Write path

assert runs maybe-reify-nats first — before expand-consequent, WFF, and the constraint checks:

• every ground, reifiable NAT subterm (F a…) is replaced by its constant K (an existing one via dedup, else a fresh mint) — inner args reified first, then the outer NAT;
• the walk descends into nested non-NAT literals (rule bodies, conjuncts) but leaves the head predicate and every quoting-predicate argument opaque;
• a vector is descended element by element, and every element of it. A vector in a sentence is a list of forms rather than a literal — an exceptWhen's conjuncts, a thereExists's binders — so it has no head predicate to hold opaque and no element to skip. That is what puts an exception's query in the same spelling as the fact it is about, and the cost of stopping at one is not a missing answer: an exception that cannot be answered does not hold (exceptions.md), so the rule fires unguarded and nothing says so;
• unreifiable NATs are left structural — never minted.

Reify-before-WFF is load-bearing. checks/args-problem and disjoint-problem read only top-level args and skip compounds, so the raw compound (FruitFn AppleTree) in an argument slot is neither type-checked nor indexed. The minted constant K carries the materialized resultIsa types, so it is an atomic term the checks can see — and the term index never posts the raw NAT's constituents, which would pollute it (sentex/index-terms descends into ground compounds).

mint-nat! allocates a fresh opaque K, asserts (termOfUnit K E), materializes the result types ((T K) per resultIsa, (genl K T) per resultGenl), and returns K, all at :monotonic strength — a reified NAT's identity and result types are structural, not defeasible. assert stores synchronously, so a second occurrence of E in the same sentence dedups against the first; a (rewriteOf T E) declaration short-circuits the mint to the real term T.

Read path

Before matching, a query goal's ground reifiable NATs are reified to their existing constant (dedup, never mint). An unknown (never-minted) NAT resolves to the reserved nat/no-match sentinel, which can never be a real constant — so an unknown-NAT query returns empty without minting.

It is wired at every read entry, and the enumeration is the whole of the guarantee: sentexes-matching, ask, prove, query, the anytime ask-within / prove-within, and levels 6 and 7 of the lookup stack (levels.md), which are the two that claim to be the engine's own dispatch. One shared step (core/prepare-goal-for-read) does it for all of them, because the failure mode when an entry omits it is not a wrong answer but an empty one — the compound is matched against a store that holds a symbol, and nothing comes back, which is indistinguishable from a KB that was never told.

The corresponding predicate

An ontology that reifies MotherFn usually also has motherOf, and the two are one claim written twice. (functionCorrespondingPredicate F P N) says so: F maps a₁ … a_M to V exactly when (P a₁ … a_{N-1} V a_N … a_M) holds. N is 1-based over P's arguments; omit it and the value takes the last position, which is what (functionCorrespondingPredicate MotherFn motherOf) means and the shape nearly every correspondence has. A value in any other position needs the explicit form — (functionCorrespondingPredicate StreetCornerFn streetCornerOf 1), so (StreetCornerFn Xing North) is the Lot of (streetCornerOf Lot Xing North).

The example is the shipped one: resources/kb/upper/LifeContext.txt states motherOf and fatherOf with MotherFn and FatherFn beside them, so a KB that loads the starter can name somebody by their role before it knows their name.

It is read in both directions, and that is the point: a KB told only one of the two spellings can otherwise reason with only that one.

• value → term. A believed (motherOf Muffet Mary) reifies (MotherFn Muffet) to Mary. The expression names the object the KB already has a name for rather than minting a second one beside it, so the correspondence is a computed rewriteOf target — the same seam, looked up through the predicate instead of declared per expression. It is consulted before the dedup probe, because a real term outranks a placeholder.
• term → value. When no value is known, the mint proceeds and the constant is projected back onto the predicate: (motherOf Muffet K), asserted with the rest of K's bookkeeping and after the result types, since the projected literal is argIsa-checked and K's types are what it is checked against. So the placeholder answers motherOf's questions instead of being a term nothing says anything about. The alternative — minting the constant and leaving the predicate unstated — makes the placeholder invisible to every question the predicate answers.

The two meet at the merge, and that is what makes the order stop mattering. Three things can arrive in any order — the application, the fact naming its value, the declaration — and each of the three landing last has an arm:

Last to arrive	What happens
the application	reifies to the believed value; nothing is minted
the fact	(rewriteOf V K) retires the placeholder, and the equality migration folds every use of K — its termOfUnit map included — onto V
the declaration	each constant already minted for F is equated with its value, or projected when it has none

rewriteOf rather than equals because the two sides are not interchangeable: V is a name somebody wrote and K is a stand-in for not knowing it, so the class needs a term that wins the election rather than whichever one sorts first. A minted constant is :opaque to wff's same-role check — the naming invariants are conventions over names a person chose, and what a reified NAT denotes is settled by its materialized result types — so merging one into an individual is the intended move, not the import bug that check catches.

Several believed declarations for one function decide nothing. Two correspondences are two different claims about what (F a…) denotes, and choosing between them would have to key on a handle, which belief may never do — so neither is read and the application mints as if none were declared. The same for a value: correspondence-value answers only when exactly one is believed.

The declaration is read through the index, not a taxonomy cache. It is consulted once per NAT, where the reify is already probing for a rewriteOf target and a dedup, and a KB declaring none pays one O(1) functor count per assert (nat/any-corresponding-predicates?). Nothing to integrate means nothing for recover to rebuild — a correspondence works across a restart because the declaration is a stored fact and always has been. It bites only for a reifiableFunction: an undeclared function's application is a raw compound the reify pass never visits.

Display / export

nat/expand-expression reverses the map, recursively rebuilding a reified constant back to its functional expression — (color (FruitFn AppleTree) Red), never a raw nat/ symbol. Non-reified NAT content is returned unchanged (same identity); only reified NAT-bearing forms are rebuilt.

A constant is never what a reader sees. It is an implementation of term identity — a nat/-namespaced gensym is not a name anybody wrote — so a display layer resolves it, and vaelii.core carries the pair that lets one: reified-term?, a pure test on the reserved namespace that costs nothing on a KB that has minted none, and term-expression, one hop of the map. One hop rather than the whole expansion, because a caller rendering each term individually — linking it, colouring it — recurses and keeps every level addressable; expand-expression is the flat answer for a caller that wants the form and not the parts. The browser does the former, showing the expression with bold parens and the opening one linking to the constant's own page (web.md, "A reified term is never shown as its constant").

Rename and remove — keeping the 1:1 invariant

Rename is an equality assert. (rewriteOf MalusTree AppleTree) retires AppleTree; the equality migration (equality.md) restates every sentex naming it, including (termOfUnit K (FruitFn AppleTree)), whose rewritten twin is (termOfUnit K (FruitFn MalusTree)). K stays stable — it is not the term renamed — so nested NATs referencing K need no cascade. A rename can collapse two NATs onto one expression; merge-colliding-nats! (run after an equality assert, gated) restores the 1:1 invariant by merging each colliding group's constants into its lexicographically-smallest survivor via a further equality, which migrates their uses. The retired spelling stays a usable question: a goal naming the old term goal-rewrites to the new expression and still resolves to K.

Remove is retraction. When the last live use of a reified NAT goes, its termOfUnit map and materialized types would dangle a raw nat/ symbol — so remove-orphaned-nats! (run after retract! and edit!, gated, suppressed while already removing orphans) collects every constant whose only remaining believed sentexes are its own bookkeeping. A correspondence projection is bookkeeping for this: like a result type it states what the constant is, and counting it as a use would make every placeholder immortal.

Bookkeeping is decided by authorship, not by shape. K's own sentexes are its termOfUnit map plus exactly what mint-nat! wrote about it, and nat/minted-for re-derives that set the way the mint derived it: (T K) for each believed (resultIsa F T), (genl K T) for each (resultGenl F T), F being the function of the expression K maps to, and the correspondence projection. Everything else naming K is somebody's assertion whatever its arity — (prime K) is a claim about K exactly as (noted Author K) is, and a sweep reading arity alone would retract the claim along with the constant. The set is computed behind a delay, and the termOfUnit clause is what makes that pay: a constant a teardown collects has its map and nothing else left, so the common answer is reached without reading the declarations at all. The one constant this keeps alive that a shape test would have collected is one whose resultIsa declaration was retracted after the mint: the materialized (T K) is then a believed sentence no rule supports and nobody has withdrawn, and holding K for it is the direction to err in.

The sweep asks about the region the teardown removed, not about the KB. A constant becomes an orphan only when something that referenced it goes, so the candidates are the constants the departing sentexes named (constants-named-by), and each is settled by one inverted-term-index read — orphan?, since a constant's uses, its map and its materialized types are all sentexes naming it (indexing.md). What a retraction costs is the size of what it removed, and a KB that has minted a hundred thousand NATs the retraction is not about adds nothing to it.

The removals reach the sweep through integrate/*removed-sink*, the record the removal choke point fills while a teardown runs, because they arrive from three places: the dependency-directed sweep; the settle that follows it, where an exception that starts holding blocks a justification and what it solely supported is deleted (exceptions.md); and the orphan sweep's own retractions. The third is how the region grows with the fixpoint — removing an orphan removes the termOfUnit sentex holding its expression, so a reified NAT nested in that expression is a candidate on the next round. A cascade is therefore found by the rule that found the first orphan, and the loop ends on the round that removes nothing.

preview's rollback is the one caller that asks the whole KB instead (orphaned-constants): the batch it undoes runs with the settle sweep off (preview.md) and reaches the orphan sweep at no point, so the claim it owes — the KB is as it was found — is about all of it rather than about one teardown's region.

Where it lives

• vaelii.impl.nat — detectors (reified-nat-symbol?, reifiable-function?, reifiable-ground-nat?), lookups (nat-expression, dedup-constant, rewrite-target, result-isa-types / result-genl-types, correspondence-of / correspondence-value / corresponding-literal), expand-expression, the shared reify walk in both modes, mint-nat!, and the rename / remove / correspondence maintenance — the orphan questions among it (orphan? per constant, orphaned-among over a region, orphaned-constants over the KB, constants-named-by / reified-nats-in for the candidates). Reads the store, taxonomy and belief directly, and reaches assertion through wiring/assert-sentence.
• vaelii.core — the reify call sites: the write-path reify at the head of assert, the read-path reify at the query entries, the post-assert maintenance hooks (merge-colliding-nats!, reconcile-correspondence!), and remove-orphaned-nats! on the retract! and edit! sweeps.
• vaelii.impl.integrate — *removed-sink*, the removal choke point's record of what a teardown took away, which is the region the orphan sweep runs over.
• vaelii.impl.special — the two function-kind prop marks, the correspondence's wff-only entry, and the equality-arm skip for a compound-arg rewriteOf.
• vaelii.impl.wff — function-decl-problems, correspondence-problems, the :opaque role a minted constant carries through the same-role check, and the equality-problems waiver for a compound-arg rewriteOf.
• resources/kb/CoreContext.txt — the seven NAT predicates, declared and documented.

What reification does not cover

• An unreified or open NAT is not structurally indexed. Reification is what makes a NAT findable, so a NAT that cannot be reified is reachable only through the coarser keys (indexing.md).
• An existential rule head is not skolemized here (skolem.md).

The measure-evaluating quantity prover — measure comparison over a dimensionOf / conversionFactor table — reads the structural measures this gate preserves. It lives in quantity.md.