Locally catenative sequence¶
A locally catenative sequence is a word sequence governed by a fixed recurrence in which each sufficiently late word is the concatenation, in fixed order, of specified earlier words.
Core Idea¶
A locally catenative sequence is an infinite sequence of finite words in which every sufficiently late word is obtained by concatenating a fixed finite pattern of earlier words. Formally, there are positive offsets \(i1,\ldots,ik\) such that \(w(n)=w(n-i1)w(n-i2)\cdots w(n-ik)\) for all \(n\) beyond the largest offset. “Local” means that the construction of the next word consults only specified relative positions in the sequence; “catenative” means that the construction joins whole earlier words without interleaving their symbols. The offsets and order remain stable across the recurrence.
Scope of Application¶
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Combinatorics on words. Repeated concatenation supports proofs about factors, prefixes, recurrence, and subword complexity.
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Fibonacci-word constructions. Fixed offsets and factor order produce canonical symbolic analogues of numerical recurrences.
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Morphic limits. Compatible prefix growth can connect the recurrence to an infinite fixed or morphic word.
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Length recurrences. Word lengths inherit numerical relations, but the symbolic sequence retains additional noncommutative information.
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Algorithmic generation. Initial words, offsets, threshold index, and factor order define efficient production and recognition tasks.
Clarity¶
Locally catenative sequence names a word-valued recurrence in which each sufficiently late word is an ordered concatenation of words at fixed earlier offsets. ‘Local’ refers to those relative positions, while ‘catenative’ preserves whole-word order rather than combining symbols arithmetically. This prevents the induced recurrence for word lengths from being mistaken for the full construction, since concatenation is generally noncommutative.
Manages Complexity¶
A locally catenative sequence reduces an infinite family of words to finitely many initial words, fixed offsets, and one ordered concatenation rule. The analyst tracks the recurrence rather than constructing or storing every late word independently. Lengths obey an induced linear recurrence, while prefixes, suffixes, factor occurrences, and growth retain the noncommutative information of word order. Changing offsets or concatenation order creates explicit branches.
Abstract Reasoning¶
Recurrence move. From the fixed offsets and ordered concatenation rule, generate all sufficiently late words and prove properties by induction. Length move. Map concatenation to addition to derive a numerical recurrence for word lengths, while withholding conclusions that depend on symbol order. Factor move. Use prefix, suffix, and recurrence structure to infer repeated factors or morphic behavior. Boundary move. Changing the order of earlier words can leave lengths unchanged while changing the sequence, so length evidence cannot identify the catenative word recurrence. Initialization move.
Knowledge Transfer¶
Within the home domain. Locally catenative sequences transfer across combinatorics on words, morphic sequences, and symbolic dynamics when each term is formed by concatenating a fixed local pattern of earlier terms. Seed words, recurrence indices, concatenation order, lengths, and factor structure retain formal meanings. Beyond the home domain (C — formal sequence class). The definition applies literally to any word sequence satisfying the recurrence, independent of alphabet interpretation. Its boundary is strict: numerical recurrences using addition are not catenative unless symbols or words are concatenated, and visual self-similarity alone does not establish the required local generative rule.
Relationships to Other Abstractions¶
Current abstraction Locally catenative sequence Domain-specific
Parents (1) — more general patterns this builds on
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Locally catenative sequence is a kind of Recursion Prime
Locally catenative sequence is a domain-specific kind of Recursion: A locally catenative sequence is a word sequence governed by a fixed recurrence in which each sufficiently late word is the concatenation, in fixed order, of specified earlier words.
Hierarchy path (1) — routes to 1 parentless root
- Locally catenative sequence → Recursion
Neighborhood in Abstraction Space¶
Locally catenative sequence sits in a crowded region of the domain-specific corpus (39th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Formal Sequences & Language Structure (16 abstractions)
Nearest neighbors
- Square-free word — 0.92
- Unavoidable Pattern — 0.89
- Arithmetic Progression — 0.87
- Well-Formed Formula — 0.86
- Formal Theory — 0.86
Computed from structural-signature embeddings · 2026-10-08