glossary.md

Glossary

This glossary mirrors the in-article glossary for The Geometry of Endless Discovery. Definitions are local to that essay. Each Wikipedia link points to the closest standard article; terms specific to the essay link to the nearest general concept.

Term	Definition	Wikipedia
Alphabet	The state set $K$ from which a traversal takes its values. It may be finite, countably infinite, or a metric space with continuously many points.	Alphabet (formal languages)
State	One possible value $x\in K$. A state is a local configuration, not a complete history or a boundary point.	State (computer science)
Traversal	An infinite ordered sequence $L=(x_0,x_1,\dots)\in K^\omega$. Repetitions are allowed unless an admissible class explicitly forbids them.	Sequence
Prefix	A finite initial history $I_N(L)=(x_0,\dots,x_N)$. It narrows possible completions but generally does not determine one.	Substring
Dwell	A consecutive run in which the traversal remains at the same state. A finite dwell ends; an infinite dwell is a halt.	Run-length encoding (run)
Halting	Eventual constancy: after some index $N$, every later state is the same. Halting is stronger than merely revisiting a state often.	Sequence (eventual constancy)
Recurrence	Return of a state, region, or neighborhood under a specified notion of observation. Exact-point recurrence and regional recurrence are different claims.	Poincaré recurrence theorem
Exact recurrent support	The set $R_{\mathrm{pt}}(L)$ of exact states occurring infinitely many times. It can be empty even when the traversal accumulates somewhere.	Support (mathematics)
$\omega$-limit set	The points approached by subsequences arbitrarily far along the traversal. It retains where the path accumulates but forgets the order of first discovery.	Limit set
Nonatomic measure	A measure assigning zero mass to every individual point. Continuous sampling may therefore avoid exact repeats even while returning infinitely often to positive-measure regions.	Atom (measure theory)
Metric	A distance function satisfying positivity, symmetry, and the triangle inequality. Different metrics here measure different things: state distance, prefix agreement, or discovery agreement.	Metric space
$\varepsilon$-separated set	A collection whose distinct points remain at least $\varepsilon$ apart. It models how many observations can still be distinguished at resolution $\varepsilon$.	Covering number
Packing number	The largest size of an $\varepsilon$-separated subset, up to the convention used. It measures distinguishable capacity at a fixed spatial scale.	Covering number
Prefix tree	The rooted tree $T_K=K^{<\omega}$ whose vertices are finite histories and whose edges append one state. It makes temporal order part of the geometry.	Trie
Ray	An infinite geodesic path beginning at the root. In a prefix tree, a ray chooses exactly one state at every successive depth.	End (graph theory)
Cylinder	The set of all infinite traversals sharing a specified finite prefix—or, later, all novelty profiles sharing specified resolution coordinates. Cylinders are the basic observable neighborhoods.	Cylinder set
Ultrametric	A metric satisfying $d(x,z)\le\max{d(x,y),d(y,z)}$. Agreement until the first differing level naturally produces this stronger triangle law.	Ultrametric space
Gromov-hyperbolic	A geodesic space whose triangles are uniformly thin. Trees are the exact case: every geodesic triangle is a tripod, so a tree is $0$-hyperbolic.	Hyperbolic group
Boundary	The space of asymptotic ray directions, with rays identified when they remain a bounded distance apart. It is a completion object, not a final vertex reached at infinite time.	Hyperbolic group (Gromov boundary)
Proper space	A metric space in which every closed bounded set is compact. Locally finite unit-edge trees are proper; infinitely branching discrete trees are not.	Metric space (proper)
Compact	A topological finiteness condition: every open cover has a finite subcover, equivalently in metric spaces every sequence has a convergent subsequence. Proper hyperbolic boundaries are compact.	Compact space
Metrizable	Having a topology generated by some metric. A compact metrizable novelty space can be handled with sequences, distances, and Cauchy completion.	Metrizable space
Quotient space	A space formed by treating equivalent points as one point. A closed equivalence relation is crucial when the starting space is compact metric and the quotient is expected to remain Hausdorff.	Quotient space (topology)
Ordered novelty word	The finite word $D_j(L)$ listing partition cells in the order they are first encountered at resolution $j$. Dwell and revisits disappear; first-discovery order remains.	String (computer science)
Partition tower	A sequence of finite partitions $\mathcal P_0,\mathcal P_1,\dots$ whose cells shrink and refine one another. Its depth represents improving distinguishability rather than passing time.	Partition of a set
Bonding map	The coarse-graining map $q_{j+1,j}$: send fine cells to parents, then remove repeated parents after their first appearance. It enforces consistency across resolutions.	Inverse limit
Inverse limit	The space $N_P(K)$ of all coordinate sequences compatible under every bonding map. It stores a complete coherent discovery profile across all resolutions.	Inverse limit
Discovery topology	The topology in which two traversals are close when their ordered novelty words agree to fine resolution. It differs from the prefix topology, which measures agreement through time.	Topological space
Metric completion	The complete metric space obtained by adding limits of all Cauchy sequences after zero-distance points are identified. Here it adds coherent ideal novelty profiles.	Complete metric space
RBDS	A Resource-Bounded Distinction System: an observer model in which limited resources determine which objective states are operationally distinguishable. Increasing budget refines the quotient.	Observability
Sufficient memory	A compressed history representation retaining everything required for a specified novelty decision. One-step sufficiency and deterministic online updating require different equivalence relations in general.	Sufficient statistic
Fenwick tree	A binary indexed tree for dynamic prefix sums. Here it indexes an ordered occupancy vector, adding spatial rank and cumulative-count queries without replacing the separate discovery-order log.	Fenwick tree
Keyed streaming state	Updateable state scoped to a stream key. It is an implementation substrate for online memory, not automatically the minimal future-equivalence quotient.	Apache Flink (keyed state)