Domain-specific naming rules for math/ units: classification, head word decision table, and modifier rules.

Table of contents

Flatfile Agential Resource System Naming Math Unit

What this is

The naming rules passed to skills/name-unit.md when naming a math entity. This document IS the naming_rules argument. To improve math entity naming, edit this file only — the skill picks it up automatically.

Classification

Classify the object as exactly one of:

Class	Test
Object	Can you point at it and examine its parts? It has structure and exists.
Quality	Is it a yes/no condition that holds or fails of something else?
Relation	Does it describe how two or more things stand relative to each other?
Act	Is it defined entirely by what it does to its input?

Tiebreaker: when Object and any other class both apply, classify as Object and encode the secondary character as a modifier (e.g. SaturationNucleus not Saturating).

Head words — Object class

Choose the most specific criterion that applies. More specific beats more general.

Morphism-type objects (endomorphisms and maps)

Nucleus — an idempotent, inflationary endomorphism on a Heyting algebra satisfying $j(a \wedge b) = j(a) \wedge j(b)$ , $a \leq j(a)$ , $j(j(a)) = j(a)$ . USE only for this specific algebraic concept. Do NOT use for arbitrary maps. Examples: SaturationNucleus, TransferNucleus.

Morphism — a structure-preserving map between two objects of the same algebraic type, where the map is defined by what structure it must preserve (not by a formula). USE when defining what it means to be a map from one structured object to another of the same kind. Example: UniverseMorphism.

Functor — a structure-preserving map between categories; preserves composition and identities. USE when both source and target are categories and the map respects categorical structure. Do NOT use Map when the structure being preserved is categorical.

Map — a function defined by an explicit pointwise formula, without a strong structural preservation claim. USE when the map is given by a formula and its preservation properties (if any) are proved separately. Example: FiberNormMap.

Retraction — a map $r: X \to Y$ such that $r \circ i = \text{id}_Y$ for some inclusion $i: Y \hookrightarrow X$ . USE only when the unit defines or characterises a retraction pair.

Sheaf-theoretic objects

Sheaf — a presheaf satisfying the gluing (descent) axiom for a given Grothendieck topology. USE only when the descent condition has been verified or is the subject of the unit. Do NOT use when descent is not yet established.

Presheaf — a contravariant functor $C^{op} \to \mathbf{Set}$ (or into algebraic structures) where the gluing condition has not yet been verified or is not the point. USE during construction, before sheafification. Example: PropositionalPresheaf.

Topos — a category that is a Grothendieck topos (sheaves on a site) or an elementary topos (has a subobject classifier, power objects, all finite limits). USE when the object IS a topos. Do NOT use for Sites or Presheaf categories. Example: AmbientTopos.

Site — a (small) category equipped with a Grothendieck topology (a coverage or sieve system). USE when the object is the input to sheafification — it has covering families but is not yet a topos. Example: HistorySite.

Categorical objects (not sheaf-theoretic)

Category — an abstract category without a distinguished topology or sheaf theory. USE when the object is a category of structured objects. Example: UniverseCategory.

Groupoid — a category in which every morphism is invertible. USE when the object encodes symmetries, equivalences, or orbits as a categorical structure. Example: HyperversalPatternGroupoid.

Doctrine — a 2-categorical fibration, specifically a hyperdoctrine: a functor from a base category into ordered categories or Heyting algebras. USE when the object is a Lawvere-style doctrine or hyperdoctrine. Example: LawvereDoctrine.

Comonad — a comonad on a category (a functor with counit and comultiplication satisfying comonad laws). USE when the unit defines or characterises a comonad. Example: DirectedComonad.

Algebraic objects

Algebra — a set (or object) with finitely many operations satisfying specified equations. USE when the primary structure is algebraic operations: Heyting algebra, nuclear algebra, etc. Example: NuclearHeytingAlgebra.

Fiber — specifically the stalk $H_t$ of the relational presheaf at a history $t \in T$ : the Heyting algebra at that point, carrying $\sigma_t$ and $\Delta_t$ . USE when the object IS this stalk. Do NOT use for arbitrary fibers of other sheaves. Examples: SaturationStableFiber, TransferStableFiber.

Language — a formal logical syntax: a grammar defining terms, formulas, or proof rules. USE when the object is defined by its syntax, not by algebraic operations. Examples: FiberNuclearLanguageAxiom, RelationalHyperverseLanguage.

Monoid — a set with an associative binary operation and identity. USE when the algebraic structure is specifically monoid-level (no inversion, no lattice). Example: HistoryMonoid.

Geometric/topological objects

Space — a set with a topology or metric. USE when the object has classical points and an explicit topology, and is NOT presented as a category of sheaves.

Locale — a pointless topological space (a frame of opens, without assuming classical points). USE when the object is defined by its lattice of opens. Example: HyperversalStableLocale.

Hull — the closure of a set under a specified operation (orbit closure, convex hull, quasicrystal hull). USE when the object is defined as the smallest X containing Y closed under some operation. Example: HyperversalQuasicrystalHull.

Moduli (→ ModuliSpace) — a space parametrising equivalence classes of structured objects. USE when the object classifies structures up to isomorphism. Example: UniverseModuliSpace.

Universe-level objects

Universe — specifically a Relational Universe $R$ satisfying the self-generation axiom $R = U_G(R)$ , or a generalisation of it. Do NOT use for arbitrary large objects. Example: RelationalUniverse.

Hyperverse — specifically $\mathcal{H}$ or a structure at the hyperversal level (a universe of universes). Example: RelationalHyperverse.

Indexed/sequential objects

Tower — a sequential diagram $X_0 \to X_1 \to \cdots$ indexed by $\mathbb{N}$ or an ordinal. USE when the object IS such a diagram, not just a collection of levels. Example: DepthFiltrationTower.

Sequence — an indexed family without a specified colimit or limit structure. USE when the indexing is the point but the directed structure is not. Example: SpectralSequence.

Named structural patterns

Quartet — four objects arranged in the specific nuclear quartet configuration: two adjunctions sharing a middle object, forming a square. USE only for this pattern.

Dyad — two objects in a specific duality relation (construction–observation, limit–colimit). USE only for this two-part duality structure.

Tetrad — four objects in a configuration distinct from the quartet (e.g. a diamond or parallel pair). USE only for this specific four-part pattern.

Head words — Quality class

MUST use an adjective. If no standard adjectival form exists, use the noun + -ness.

Common quality head words in this corpus: Completeness, Soundness, Stability, Aperiodicity, Idempotence, Convergence, Locality, Decidability.

Head words — Relation class

MUST use a flat verb (no -ing, -ed, -s). The name states the relation as a verb.

Common relation head words: Commute, Restrict, Correspond, Separate.

Head words — Act class

MUST use a gerund (the -ing form of the operative verb). The name states what the act does.

Common act head words: Settling, Saturating, Normalizing, Tunneling.

Modifier rules

Modifiers MUST precede the head in order: [scope] [property] [head].

Scope — the mathematical domain the unit lives in: Relational, Fiber, Universe, History, Hull, Hyperverse, Hyperversal, Tower, Depth.
Property — what makes this instance specific: Stable, Transfer, Nuclear, Saturation, Commutation, New, Raw.

When a modifier is derived from a verb: MUST use the attributive noun form. The -ing rule applies ONLY to Act heads. MUST NOT apply to modifiers.

Use the most specific name the math supports. Do NOT omit modifiers for brevity.

Validation

After assembling a candidate name:

Confirm no existing unit has the same id.
Confirm the head word criterion is satisfied by the mathematical content.
If the unit’s heading type prefix is “Theorem —” or “Conjecture —”: the name describes the SUBJECT of the theorem, not the theorem itself. The unit’s name reflects what mathematical object or property the result concerns. (Example: NoetherFiber is a theorem about fiber algebras, named for its subject.)