Complete sequence

A sequence of natural numbers whose distinct finite subset sums represent every positive integer.

Version

v1 · 2026-09-08 · History

Domain-specific #

3795

Origin domain

additive number theory

Subdomain

additive number theory

Core Idea

A sequence is complete when each positive integer equals a sum of zero-or-one copies of its terms, with order, repetition, and positivity conventions explicitly fixed. Early terms create a covered interval; a new term no larger than one plus the previous covered sum extends that interval without gaps, supporting recursive completeness tests. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application

Complete sequence belongs to additive number theory and is useful where the analyst can specify the typed additive number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets, then evaluate every positive integer has at least one finite representation using each sequence term at most once. The scope is broad within that domain but bounded by the need for every positive integer has at least one finite representation using each sequence term at most once. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making every positive integer has at least one finite representation using each sequence term at most once the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Complete sequence can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Complete sequence. Complete sequence compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed additive number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express every positive integer has at least one finite representation using each sequence term at most once independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of additive number theory because they reuse the typed additive number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets, Early terms create a covered interval; a new term no larger than one plus the previous covered sum extends that interval without gaps, supporting recursive completeness tests., and type the carrier, state every parameter and convention in the definition, test that every positive integer has at least one finite representation using each sequence term at most once, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Complete sequence Domain-specific

Parents (1) — more general patterns this builds on

• Complete sequence is a kind of Decomposition Prime

  The proposed strict upward parent is prime:decomposition.

Hierarchy path (1) — routes to 1 parentless root

• Complete sequence → Decomposition

Neighborhood in Abstraction Space

Complete sequence sits in a crowded region of the domain-specific corpus (8^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Number-Theoretic Sequences & Classes (37 abstractions)

Nearest neighbors

• Highly composite number — 0.93
• Prime triplet — 0.93
• Multiplicative partition — 0.93
• Arithmetic function — 0.92
• Arithmetic number — 0.92

Computed from structural-signature embeddings · 2026-09-08