# 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)](<https://en.wikipedia.org/wiki/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)](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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)](<https://en.wikipedia.org/wiki/Run-length_encoding>) |
| 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)](<https://en.wikipedia.org/wiki/Sequence>) |
| 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](<https://en.wikipedia.org/wiki/Poincar%C3%A9_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)](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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)](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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)](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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)](<https://en.wikipedia.org/wiki/Hyperbolic_group>) |
| 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)](<https://en.wikipedia.org/wiki/Metric_space>) |
| 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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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)](<https://en.wikipedia.org/wiki/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)](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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](<https://en.wikipedia.org/wiki/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)](<https://en.wikipedia.org/wiki/Apache_Flink>) |