Notation & Definitions

Core terms, symbols, and status conventions

The closure program uses a small, fixed vocabulary introduced across several papers. The reference list collects the core terms and status conventions in one place, with each entry defined as precisely as a reference entry allows and with the location in the monograph (Closed Systems from Comparison Completeness) where it is established rigorously. The terms are ordered by dependency — each builds on the ones above it — rather than alphabetically, so the list also records the dependency structure of the construction. The definitions below are reference summaries; the binding statements are in the monograph at the cited locations.

Comparison world (U, C)

A set of states U together with a family C of comparison predicates c : U × U → {0, 1}. Topology, metric, dynamics, and background geometry are not primitive data; everything else in the program is extracted from the comparison structure. (Definition 0.1.1; recalled in Definition 2.2.1.)

Intrinsicality — the closure premise

The requirement that every predicate be invariant under the automorphism group of the comparison world it helps constitute: c(gu, gv) = c(u, v) for all c ∈ C and all g ∈ Aut(U, C). A comparison is a relation among states, not a stipulated labeling of them; this is the primitive-level expression of the intrinsicity standing principle. It is the program's single foundational premise. Under the additional theorem hypotheses — especially local distinguishability — rectangular completeness follows as a theorem; the finite-to-global compactness boundary is discharged separately through the refinement-tower and finite-witness results. (Definition 0.1.1; Standing Principle 1 / SP1; recalled in Definition 2.2.1.)

Automorphism group G = Aut(U, C)

The group of bijections of U preserving every comparison predicate. These are the symmetries determined by the comparison structure itself, with no external reference. (Definition 2.2.3.)

Left and right profiles L(u), R(u)

For a state u, its left profile L(u)(c, w) = c(u, w) records how u behaves as the first argument across all comparisons; its right profile R(u)(c, w) = c(w, u) records how it behaves as the second. A state's complete comparison behavior. (Definition 2.3.1.)

Intrinsic congruences α, β

Two states are α-equivalent when their left profiles coincide (u α v iff L(u) = L(v)), and β-equivalent when their right profiles coincide. These are the finest distinctions the comparison data itself can draw. (Definition 2.3.2.)

Profile (α-class, β-class)

The pair consisting of a state's left class and right class; its position in the product of the two quotients. The admissible features a state carries, recovered from comparison alone. (Via Definitions 2.3.1–2.3.2.)

Canonical factor map Θ : U → X_A × X_B

The map sending each state to its profile pair, Θ(u) = ([u]_α, [u]_β), where X_A = U/α and X_B = U/β are the profile quotients. Its injectivity and surjectivity determine the structure of the comparison world. (Definition 2.4.1.)

Local distinguishability α ∩ β = Δ_U

The condition that no two distinct states share both their left and right profiles; equivalently, that Θ is injective. The comparison data distinguish states up to equality. (Lemma 2.4.2.)

Rectangular completeness (closure) Θ bijective

The condition that for every left profile A ∈ X_A and right profile B ∈ X_B there is a unique state u with [u]_α = A and [u]_β = B — equivalently, that the canonical factor map is a bijection and the world is the full product of its two factors. This is the comparison-stage meaning of "closed" in the program. Under intrinsicality and local distinguishability it is a theorem; constrained worlds (a rigid rod, a heat-bath coupling, symmetric worlds) fail it and are thereby open. (Definition 2.5.1; equivalence with Θ bijective, Theorem 2.6.1; derivation under intrinsicality and local distinguishability, Theorem 2.6.4; the "closed := rectangularly complete" convention, Definition 4.2.1.)

Conservative completion — the completeness mechanism

The dichotomy for an unrealized profile pair: either the formal one-point extension realizing it is internally excluded, or it is conservative — restriction back to U preserves every finite-coordinate comparison algebra supported in U and creates no new profile distinction among the original states. A conservative omission is intrinsically undetectable. On the closure interpretation, such an omission cannot be maintained without an external selector. This is the mechanism used to prove rectangular completeness. (Lemma 2.6.2.)

Quotient semantics — closed-system descent

The principle that admissible reports in a closed system are exactly those that descend to the orbit space of the symmetry group — equivalently, those factoring uniquely through π : X → Phys = X/G. Distinctions not preserved by G are not physically accessible. (Chapter 3, §3.2.)

Subsystem-attribution obstruction — the no-go result

The result that, for a motion within a single diagonal orbit, no admissible report can decide whether it is attributed to subsystem A or to subsystem B. Subsystem attribution is not an invariant of a closed system. (Theorem 3.5.1; Corollary 3.5.2.)

Two loci of enrichment — object locus, morphism locus

The theorem that all freedom in extending the quotient structure enters at exactly two places: the object locus (representative selection on objects) and the morphism locus (transport and holonomy). There is no third independent enrichment datum. (Theorem 5.1.1; Corollary 5.6.1; Theorem 6.6.1; Corollary 6.6.2.)

Transport obstruction — route dependence

The failure of comparison data to transport consistently between points along different routes through the comparison groupoid. Recorded around closed loops, this obstruction is what becomes curvature under smooth realization. (Chapters 6, 12.)

Augmentation filtration / quadratic carrier F², F³, F²/F³

The filtration of the transport (triangle) obstruction by degree. The first visible graded carrier of the obstruction is F²/F³, the first stable nonvanishing layer. Under faithful smooth realization, this degree-2 carrier is identified with realized curvature; later chapters build on the same carrier downstream. The faithfulness of that identification is decomposed by the ghost-free regime below. (Chapter 13, §§13.4–13.8.)

Realization smooth, Hilbert, phase, observer-side, or numerical reading

A reading of intrinsic comparison, quotient, or transport structure into a later mathematical setting. A realized object is not primitive merely because it appears in a theorem; its status is fixed by the interface data and construction that introduce it. (Notation and Conventions; Standing Terminology; Chapters 12–21.)

Faithful smooth realization — the manifold-stage bridge

The condition under which the already-stabilized degree-2 carrier is realized without collapse on a smooth manifold, identifying nonzero stabilized square classes with realized curvature; formally, injectivity of the second-jet comparison map on the stabilized quadratic carrier. The condition decomposes exactly into condition (T) and axiom (D) and is discharged on a canonical realization built from the tower; the explicit interface inputs are the intrinsic detectability axiom (D), together with condition (T) outside the Magnus regime. (Chapter 12, §§12.6–12.9; Chapter 13, §§13.12–13.16.)

Ghost-free regime (T)+(D)

The exact decomposition of the faithfulness condition. Condition (T): the stabilized quadratic carrier is torsion-free — necessary because torsion classes are invisible to every real-linear reading, and a theorem in the Magnus regime, where stage carriers embed in the exterior square of the degree-1 lattice. Axiom (D): uniform defect detectability — the realized transport completion has no small subgroups, so a nonzero failure of closure cannot be infinitesimally small; independent of the tower axioms (profinite counter-model) and equivalent to selecting the Lie branch of the structure trichotomy (Lie / totally disconnected / mixed, by Gleason–Yamabe and van Dantzig). Under (T)+(D), the canonical Malcev realization of the two-step transport quotient has exact square-loop holonomy and an injective second-jet comparison map: faithfulness holds by construction. (Chapter 13, §§13.14–13.16.)

Boundary datum — named interface input

An explicitly stated condition, calibration, interface choice, or realization input that is not part of the primitive comparison data. Boundary data are not hidden primitives; they mark the point at which a realized or quantitative reading adds information beyond the intrinsic theorem stack. (Notation and Conventions; Standing Terminology; CSM Appendix A.)

Conditional theorem — theorem with named hypotheses

A theorem whose assertion includes its stated interface, realization, boundary, or calibration hypotheses. The conditional status is part of the claim's logical content: the theorem proves what follows from the named data, and it does not assert the same conclusion with those data omitted. (Notation and Conventions; Logical Status; CSM Appendix A.)

Closed-world admissibility — the four-axis criterion

The standard against which primitive structural input is assessed: a quantity is admissible if it is invariant under internal symmetries, independent of imposed subsystem cuts, and independent of non-dynamical representational choices. It underlies the four-axis taxonomy: externalization, artificial factorization, premature globalization, and reification of repair. (Foundational Closure and Primitive Structural Input, Definition 4.3.)

Each term above is established rigorously at the cited location. Internal sources: the finitary two-subsystem form is in CCW; the full construction and chapter references are in CSM; the four-axis admissibility vocabulary is in FE; and the faithful-smooth-realization interface is tracked on Logical Status. The first-principles account is in Foundations; the result-by-result development is in Overview; the finite model is in the worked example.