Notation & Definitions

The core terms and symbols, each defined once

The program is built from a small, fixed vocabulary, introduced across several papers. This page collects the core terms in one place, each defined once and as precisely as a reference entry allows, 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 reads as the spine of the construction. Definitions here 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}. No topology, metric, dynamics, or background geometry is assumed; everything else in the program is extracted from this data. (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 it constitutes: 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. The symmetries the world has by virtue of its own comparison structure, 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 decide the world's structure. (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 world distinguishes its own states. (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 what "closed" means in the program. Under intrinsicality and local distinguishability it is a theorem, not an assumption; it is not universal — 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. The closure interpretation is that a self-contained world cannot maintain such an omission without an external selector. This is the engine behind 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. (Chapter 13, §§13.4–13.8.)

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. The geometric results (curvature, the Einstein boundary, the three-sphere) are conditional on this bridge; it is the principal open bridge named on the Objections page. (Chapter 12, §§12.6–12.9; Chapter 13, §§13.12–13.13.)

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. The basis of the four-axis taxonomy. (Foundational Closure and Primitive Structural Input, Definition 4.3.)

Each term above is established rigorously at the cited location. The plain-language account of how they fit together is in The Idea; the result-by-result development in the Overview; the mechanism worked by hand in the example.