Glossary

Terms are defined once. Cross-references point to the anchor where the formal definition appears.

A

Adjunction: A pair of functors $F \dashv G$ with a natural bijection $\text{Hom}(F(A), B) \cong \text{Hom}(A, G(B))$. Captures optimal translation between representations. See A9.

Anchor: A formal definition that later chapters depend on. The 32 anchors (A1–A32) constitute the book's mathematical spine. See Appendix F.

Attested witness: A witness whose verification depends on authority rather than computation. Returns ok_if_trusted(authority) rather than a definitive verdict. See A2c.

C

Category: A collection of objects and morphisms between them, with composition and identity laws. The structural foundation for the book's formal machinery. See Appendix A.

Certification: The process of verifying a proposed claim and producing a typed witness or structured failure. Certification follows proposal in the Third Mode pipeline. See A19b.

Coherence: Agreement on overlaps; the sheaf condition. Local data glues to global data iff restrictions to overlaps agree. See A13.

Coherence budget: Maximum computational cost allocated to coherence checking per operation. Systems must declare their coherence budgets explicitly. See A21.

Commitment set: A set of propositions that the system has asserted. Consistency requires $C \nvdash \bot$. See A1.

Conservative extension: An extension $\Sigma \to \Sigma'$ that preserves all $\Sigma$-truths. Old theorems remain valid; no migration required. See A17b.

Context: A view or scope within which truth is assessed. Contexts form a site with covers and restriction maps. See A12, A12b.

Cover: A family $\{U_i \to U\}$ that jointly covers $U$ in the sense specified by the Grothendieck topology. See A12.

D

Decidable witness: A witness whose verification is total and deterministic, always returning ok or fail. See A2c.

Dossier: See Predicate package.

E

Equivalence: Identity up to maps that are mutually inverse up to natural isomorphism. Written $A \simeq B$. Weaker than isomorphism in that the inverses need only be natural isomorphisms, not strict identities. See A10.

Epistemic status: The truth value of a proposition relative to a view: true, false, or undetermined. Depends on both the view and its logic. See A4.

F

Fibration: A functor $p: E \to B$ satisfying the lifting property. Captures dependent types: the fiber $E_b$ over $b \in B$ represents types valid in context $b$. See A14.

Functor: A structure-preserving map between categories, sending objects to objects and morphisms to morphisms while respecting composition and identity. See Appendix A, A9.

G

Gluing: The sheaf condition's second requirement: compatible local sections amalgamate to a unique global section. If $s_i \in F(U_i)$ agree on overlaps, there exists unique $s \in F(U)$ restricting to each $s_i$. See A13.

Grothendieck topology: A specification of which families of morphisms count as covers for each object. Defines the site structure on a category. See A12b.

H

Hard invariant: A constraint that must hold; violation triggers rejection. Contrast with soft constraint. See A18.

I

Invariant: A property that survives allowed transformations. Under group action $G \curvearrowright X$, function $f$ is invariant iff $f(g \cdot x) = f(x)$ for all $g$. See A7.

Isomorphism: Identity up to structure-preserving bijection. Written $A \cong B$. Requires maps $f: A \to B$ and $g: B \to A$ with $g \circ f = \text{id}_A$ and $f \circ g = \text{id}_B$. See A8.

L

Locality: The sheaf condition's first requirement: global sections are determined by their local restrictions. If $s|_{U_i} = t|_{U_i}$ for all $i$ in a cover, then $s = t$. See A13.

Logic selection: The choice of inference rules for a view. Classical logic admits excluded middle; intuitionistic logic does not. Different views may operate under different logics. See A15.

M

Matching family: A collection of local sections $\{s_i \in F(U_i)\}$ that agree on overlaps: $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$. A matching family glues iff the sheaf condition holds. See A13.

Morphism: An arrow between objects in a category. Morphisms compose associatively and have identity elements. See Appendix A.

N

N-ary event: An event with multiple participants in distinct roles, represented as a first-class object rather than decomposed into binary edges. See A31.

P

Predicate invention: Signature extension $\Sigma \to \Sigma'$ introducing a new predicate with obligations: local definability, overlap agreement, invariant preservation. The Third Mode's central operation. See A17.

Predicate package (dossier): A predicate with its full specification: signature, intension, tests, invariants, provenance, scope. See A24.

Presheaf: A contravariant functor $F: \mathcal{C}^{\text{op}} \to \mathbf{Set}$ assigning data to each context with restriction maps. A sheaf is a presheaf satisfying locality and gluing. See A13.

Probabilistic witness: A witness whose verification terminates with statistical confidence rather than certainty. Returns confidence bounds alongside the verdict. See A2c.

Proposal operator: A function $P$ that takes a query and returns scored candidates. Typically implemented via embedding similarity or retrieval. Produces hypotheses, not certified truths. See A19.

Provenance: The origin and derivation history of a claim. Typed as $\Gamma \vdash \pi : \Pi(p)$, asserting that $\pi$ is evidence for $p$ in context $\Gamma$. See A2.

R

Refusal obligation: The requirement that a system refuse impossible queries with a machine-checkable proof: small unsat core plus derivation chain. See A27.

Restriction map: A morphism $F(U) \to F(V)$ induced by inclusion $V \subseteq U$. Takes global data and restricts it to a smaller context. See A12.

S

Scope: A downward-closed set of contexts in which an equivalence or predicate is valid. All equivalences are scoped. See A30.

Sense boundary: A context-indexed equivalence relation that partitions terms into senses. Different contexts may draw boundaries differently. See A6.

Sheaf: A presheaf satisfying locality and gluing. The formal model for coherent data: local information glues to global information iff it agrees on overlaps. See A13.

Site: A category equipped with a Grothendieck topology. The structure on which presheaves and sheaves are defined. See A12b.

Soft constraint: A constraint whose violation incurs cost but does not trigger rejection. Contrast with hard invariant. See A18.

Signature: A specification $\Sigma = (T, P, I)$ of types, predicates, and constraints. Adding a predicate is a signature morphism $\Sigma \to \Sigma'$. See A3.

T

Third Mode: A system that natively supports predicate invention with conservative extension, witnessed equivalence with scoped transport, view structure with sheaf-condition gluing, and explicit coherence cost accounting. See A32.

Touchstone: One of 10 recurring test cases (T1–T10) that demonstrate failure modes and eventual resolutions. See Appendix E.

Transport: Moving a property along an equivalence. Given $e: A \simeq B$ and $P: A \to \text{Type}$, transport defines $P': B \to \text{Type}$. Substitution requires transport certificates. See A10, A16.

U

Univalence (full): The HoTT axiom $(A \simeq B) \simeq (A = B)$: equivalent types are equal. Provides the strongest substitution principle. See Appendix D.

Univalence (engineering): The practical surrogate A16: witnessed equivalence plus transport certificates licenses substitution. Does not assume full univalence is computationally realizable.

V

Versioning rules: The protocol for evolving schemas: conservative extension (safe) vs. breaking change (requires migration). See A26.

View: A context with its associated local theory $T(U) = (\Sigma, I, L_U)$, including signature, constraints, and logic. See A4, A12.

W

Witness: Evidence for a claim, typed by verification regime (decidable, probabilistic, or attested). Downstream consumers must receive $(p, \pi)$, not bare $p$. See A2b, A2c.

Witnessed sameness: Identity backed by explicit evidence. Three levels: equality (=), isomorphism ($\cong$), equivalence ($\simeq$). All require witnesses; all are scoped. See A10.

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Term Mapping: This Book vs. Standard Literature

The Third Mode uses terms from multiple fields. This table maps our usage to standard terminology and explains departures.

Our Term	Standard Term(s)	Field	Why Different
Anchor	Definition, Axiom	Mathematics	"Anchor" emphasizes load-bearing role in argument structure
Coherence	Sheaf condition, Descent	Algebraic geometry	"Coherence" foregrounds the systems concern (data agreement)
Commitment set	Belief set, Knowledge base	Belief revision, KR	"Commitment" emphasizes assertional force, not mere belief
Context	View, Scope, Module, Theory	Databases, Logic, SE	Standard term; we add logic annotation (CWA/OWA)
Conservative extension	Conservative extension	Logic	Standard term, unchanged
Cover	Cover, Covering family	Sheaf theory	Standard term, unchanged
Decidable witness	Computable certificate	CS theory	"Decidable" aligns with complexity theory usage
Epistemic status	Truth value, Model membership	Logic	Emphasizes observer-relative nature
Gluing	Descent, Patching	Algebraic geometry	"Gluing" is more intuitive for systems audience
Invariant	Invariant	Group theory, SE	Standard term, unchanged
Matching family	Matching family, Compatible family	Sheaf theory	Standard term, unchanged
N-ary event	Reified event, Event object	KR, Databases	Emphasizes arity; aligns with Davidson/Parsons
Obstruction	Cohomology class, Descent obstruction	Algebraic geometry	"Obstruction" emphasizes operational failure mode
Predicate invention	Predicate invention, Concept learning	ILP, ML	Standard term from ILP literature
Predicate package	Concept, Definition, Schema	KR, Databases	"Package" emphasizes bundled obligations
Presheaf	Presheaf, Functor	Category theory	Standard term, unchanged
Probabilistic witness	Soft evidence, Uncertain certificate	Probabilistic reasoning	"Witness" aligns with proof-theoretic usage
Provenance	Provenance, Lineage	Databases	Standard term, unchanged
Restriction map	Restriction, Pullback	Sheaf theory	Standard term, unchanged
Scope	Context, Domain of discourse	Logic	"Scope" emphasizes bounded validity
Scoped equivalence	Contextual identity, Indexed relation	Type theory, KR	Emphasizes scope annotation
Sense boundary	Word sense, Concept	WSD, Ontology	"Boundary" emphasizes partition structure
Sheaf	Sheaf	Algebraic geometry	Standard term, unchanged
Site	Site	Topos theory	Standard term, unchanged
Third Mode	(novel)	—	Novel term for the integrated framework
Touchstone	Test case, Benchmark	SE, ML	"Touchstone" emphasizes diagnostic role
Transport	Transport, Path induction	HoTT	Standard term from HoTT literature
Univalence	Univalence	HoTT	Standard term, unchanged
Witness	Proof term, Certificate	Type theory, CS	"Witness" aligns with existential witness usage

Notes on Field Alignment

Category Theory / Sheaf Theory: We use standard terminology (presheaf, sheaf, cover, site, restriction, gluing) without modification. Readers familiar with algebraic geometry or topos theory will recognize the concepts.

Type Theory / HoTT: We use standard terminology (transport, univalence, fibration) where it applies. We distinguish "engineering univalence" (A16) from "full univalence" (Appendix D) because computational realizability is an open problem.

Databases / Knowledge Representation: We use "context" rather than "view" or "scope" to emphasize the logic-annotated structure. "Predicate package" and "commitment set" are non-standard but map to familiar concepts.

Machine Learning / ILP: "Predicate invention" is standard in ILP. "Proposal operator" is novel but maps to retrieval/ranking in RAG systems.

Software Engineering: "Invariant," "versioning," and "conservative extension" are standard. "Coherence budget" is novel but extends familiar cost modeling.

Deliberate Departures

Some terms are deliberately non-standard:

• Anchor vs. "Definition": Anchors have a specific role in the book's dependency structure. Not all definitions are anchors.
• Touchstone vs. "Test case": Touchstones are recurring diagnostic scenarios, not one-time tests.
• Third Mode vs. any existing term: The integrated framework is novel; no existing term covers it.
• Commitment vs. "Belief": Commitments have assertional force and retraction costs; beliefs do not.