Complete sequence¶

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

Version: v2 · 2026-09-19 · History
Domain-specific #: 3795
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Additive Number Theory → Mathematics

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.^[1] 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.

The load-bearing residual is not the broad topic of additive number theory. It is the domain-specific identity determined by every positive integer has at least one finite representation using each sequence term at most once. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that every positive integer has at least one finite representation using each sequence term at most once fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test. This gives the entry an operational identity rather than merely a historical label.

A useful analysis keeps three layers separate. The constitutive layer says what must be true: every positive integer has at least one finite representation using each sequence term at most once. The evidential layer asks what observation or proof warrants the claim: 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. The use layer asks what reasoning becomes available once the identity is established: recognizing and comparing instances of Complete sequence, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions. Conflating the layers is the most common source of scope inflation.

How would you explain it like I'm…

Coins That Make Every Amount

Imagine a special set of coins, and you may use each coin only once. If you can make every amount, 1, 2, 3, and on and on, by picking some of those coins, then the list of coins is complete. Coins worth 1, 2, 4, 8 and so on work.

Every Number From Using Each Once

A complete sequence is a list of positive whole numbers where every positive whole number can be made by adding up some of them, using each one at most once. For example, 1, 2, 4, 8, 16 and so on is complete: 11 is 8 + 2 + 1. There is a handy check. If the numbers you already have can make every amount up to some total, then a new number that is no bigger than one more than that total keeps everything covered with no gaps. If a new number is too big, some amount gets skipped.

Sums of Distinct Terms Cover All Integers

A sequence of positive integers is complete if every positive integer can be written as a sum of some of its terms, using each term zero times or once. The conventions matter, such as whether terms can repeat in the list and whether they must be positive, so they should be stated. The standard test works step by step, with terms in increasing order. The first few terms can make every number from 1 up to their total sum. A new term that is at most one more than that sum extends the covered range with no gaps; a bigger term leaves a gap. The powers of two are the classic example of a complete sequence.

In additive number theory, a sequence of positive integers is complete if every positive integer has at least one finite representation as a sum of terms of the sequence, each term used at most once. Conventions about ordering, repetition, and positivity must be fixed explicitly. The key mechanism is the covered interval: if the initial terms can represent every integer from 1 up to their partial sum, then appending a term no larger than one plus that sum extends the covered interval without gaps. This gives a recursive test for completeness when the terms are sorted in nondecreasing order. The condition is existence of a representation, not uniqueness, so completeness should not be confused with unique-representation systems like binary.

Structural Signature¶

Carrier: the typed additive number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets
Inputs or antecedent state: the exact additive number theory carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Complete sequence
Constitutive operation: 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.
Invariant: every positive integer has at least one finite representation using each sequence term at most once
Recognition test: 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
Output or consequence: recognizing and comparing instances of Complete sequence, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions
Failure boundary: the carrier is mistyped, the condition that every positive integer has at least one finite representation using each sequence term at most once fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test

What It Is Not¶

It is not the whole field of additive number theory. The field contains many questions and methods that do not instantiate Complete sequence.
It is not its most familiar example. A canonical instance directly demonstrates that every positive integer has at least one finite representation using each sequence term at most once. exhibits the structure, but the example is evidence for the abstraction rather than its definition.
It is not the neighboring catalog concept Additive basis. An additive basis typically permits a bounded or unrestricted number of selected terms and may allow repetition; a complete sequence uses distinct subset sums.
It is not a claim that every boundary case has one uncontested classification. a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Complete sequence must control the decision
It is not an unrestricted metaphor for any process that seems similar. Outside additive number theory, the vocabulary and validity conditions do not transfer literally.

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.^[2]

Definition and recognition. Determine whether a proposed instance satisfies the constitutive conditions rather than merely sharing terminology.
Construction or evolution. Track how the exact additive number theory carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Complete sequence are converted, constrained, or organized by 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..
Comparison. Compare instances using carrier, parameters, convention, domain, scale, boundary conditions, evidence, exact versus approximate form, and limiting behavior, without treating convenience measures as the definition.
Boundary analysis. Diagnose cases where a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Complete sequence must control the decision and state which convention or theorem controls the decision.
Downstream reasoning. Use the established identity to support recognizing and comparing instances of Complete sequence, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions while preserving the assumptions under which the inference is valid.

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. The disciplined statement is: given the exact additive number theory carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Complete sequence, the structure counts as Complete sequence exactly when every positive integer has at least one finite representation using each sequence term at most once.

This format also separates identity from measurement. Empirical, computational, or documentary proxies support recognition only under declared validity and uncertainty assumptions; formal cases require proof rather than measurement. Measurements can be noisy, implementations can approximate, and proofs can use equivalent characterizations; none of those facts licenses changing the object being measured. When reports disagree, first check scope and convention, then data or proof, and only then interpret the disagreement as substantive.

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.

The compression has a price. A single label can hide canonical, generalized, restricted, approximate, computational, empirical, and historically variant formulations of Complete sequence. Good use therefore carries a small declaration of assumptions alongside the name. The abstraction manages complexity when it reduces the state space of the question while keeping the failure boundary visible; it mismanages complexity when the label substitutes for that boundary analysis.

Abstract Reasoning¶

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.
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. This step prevents the canonical example from becoming the definition.
Derive consequences. From every positive integer has at least one finite representation using each sequence term at most once, infer recognizing and comparing instances of Complete sequence, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions. Record each assumption used so that a later change of setting does not silently preserve an invalid conclusion.
Test adversarial cases. Examine a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Complete sequence must control the decision and an object that resembles Complete sequence in purpose or vocabulary but does not satisfy its invariant is outside the class. A robust identity explains why the first is convention-sensitive and why the second is outside the class.
Compare and refine. Use carrier, parameters, convention, domain, scale, boundary conditions, evidence, exact versus approximate form, and limiting behavior to compare legitimate instances, and refine the model when discrepancies reflect hidden variation rather than failure of the abstraction itself.

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. A theorem, diagnostic, or modeling warning can travel when those roles remain literal. For example, the distinction between constitutive identity and a convenient observable transfers from A canonical instance directly demonstrates that every positive integer has at least one finite representation using each sequence term at most once. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, or implementation..^[n1]

Transfer outside the home domain is weaker. The skeletal pattern—type the carrier, apply the defining mechanism of Complete sequence, preserve its invariant, and derive only consequences licensed by the stated boundary—may suggest an analogy, but the domain-specific mechanisms, admissible evidence, and consequences do not come along automatically. The safe transfer procedure maps each role explicitly, checks the invariant again, and refuses the name when only a superficial resemblance remains.

Examples¶

Canonical¶

A canonical instance directly demonstrates that every positive integer has at least one finite representation using each sequence term at most once. The example exposes the carrier and directly tests that every positive integer has at least one finite representation using each sequence term at most once; changing incidental notation preserves the identity, while removing that condition destroys it. This example is canonical because every role can be inspected: the carrier is the typed additive number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets; the operative rule is 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 invariant is every positive integer has at least one finite representation using each sequence term at most once; and the result supports recognizing and comparing instances of Complete sequence, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions.^[1] Changing incidental notation or scale leaves the structure intact, while removing every positive integer has at least one finite representation using each sequence term at most once destroys the classification.

Mapped back: 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. → every positive integer has at least one finite representation using each sequence term at most once → recognizing and comparing instances of Complete sequence, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions

Applied / In Practice¶

An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, or implementation. The applied case qualifies only because the same invariant and boundary test remain literal under changed parameters or implementation. The applied case is not licensed merely by vocabulary. It qualifies because the same recognition test—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—can be run and because the same failure boundary—the carrier is mistyped, the condition that every positive integer has at least one finite representation using each sequence term at most once fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test—remains meaningful.^[2] The case also shows why practical outputs should report assumptions, resolution, and uncertainty instead of a naked label.

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Axiomatic identity vs. operational recognition. The defining conditions may be exact while empirical or computational recognition is approximate. Neither pole can be removed without changing the analytical task. Diagnostic: Can the reviewer state both the exact condition and the evidence used to infer it?
T2: Local roles vs. global consequence. The mechanism is enacted through local relations, but the abstraction is usually valued for a global classification or prediction. Neither pole can be removed without changing the analytical task. Diagnostic: Does the claimed global result actually follow from the declared local conditions?
T3: Ideal form vs. finite representation. Theory states a clean invariant while data structures, measurements, or proofs expose only finite representations. Neither pole can be removed without changing the analytical task. Diagnostic: Would increasing resolution converge toward the same classification?
T4: Canonical convention vs. legitimate variants. A standard formulation supports communication, while variants may preserve the same core under changed assumptions. Neither pole can be removed without changing the analytical task. Diagnostic: Which role is invariant across variants, and which convention-specific conclusion changes?
T5: Compression vs. hidden assumptions. The name compresses a complex argument but can conceal prerequisites. Neither pole can be removed without changing the analytical task. Diagnostic: Can each downstream inference be traced to an explicit assumption?
T6: Autonomous residual vs. reduction to catalog neighbors. The candidate uses broader structures but adds an identity-bearing residual. Neither pole can be removed without changing the analytical task. Diagnostic: After subtracting the proposed parent and named neighbors, does the constitutive residual still support independent diagnostics?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is type the carrier, apply the defining mechanism of Complete sequence, preserve its invariant, and derive only consequences licensed by the stated boundary. Its identity-bearing terms—Complete sequence, carrier, parameter, invariant, boundary, evidence, model, transformation, and application—derive their meaning from additive number theory and cannot be replaced by generic systems language without losing the tests that distinguish valid from invalid instances.

This mixed character explains why the abstraction is reusable inside the domain yet does not meet the Prime bar. The structure organizes reasoning, but its claims still depend on domain-specific objects, evidence, and intervention semantics.

Structural Core vs. Domain Accent¶

The structural core consists of a carrier, 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., a recognition invariant, and a consequence. That skeleton may resemble patterns elsewhere, especially type the carrier, apply the defining mechanism of Complete sequence, preserve its invariant, and derive only consequences licensed by the stated boundary. The domain accent is not decorative: Complete sequence, carrier, parameter, invariant, boundary, evidence, model, transformation, and application determine what counts as an admissible carrier, a valid transition, and successful evidence.

The abstraction therefore remains domain-specific. A cross-domain reuse that preserves only words such as 'balance,' 'cut,' 'sequence,' 'loss,' or 'simulation' is metaphor. Literal transfer requires the original role structure and diagnostics, which in this case remain anchored in additive number theory.

This entry is a kind of Decomposition.

It remains its own entry because its identity requires that every positive integer have at least one finite representation using each term of the sequence at most once.

Relationships to Other Abstractions¶

Current abstraction Complete sequence Domain-specific

Parents (1) — outgoing typed relations

Complete sequence is a kind of Decomposition Prime

What makes it its own entry: the domain-specific identity determined by every positive integer has at least one finite representation using each sequence term at most once.

Hierarchy path (1) — routes to 1 parentless root

Complete sequence → is a kind of Decomposition

Neighborhood in Abstraction Space¶

Complete sequence sits in a crowded region of the domain-specific corpus (5^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Numeral Systems & Integer Sequences (22 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25

Not to Be Confused With¶

Additive basis. An additive basis typically permits a bounded or unrestricted number of selected terms and may allow repetition; a complete sequence uses distinct subset sums.
One canonical example. An instance demonstrates the structure but does not define the whole abstraction.
Measurement or implementation of Complete sequence. A proxy or realization is evidence for the abstraction, not the abstraction itself.
Generalized Complete sequence. An extension qualifies only when its changed axioms and retained invariant are stated.

Notes¶

[n1] Source cited in the frozen article, 'The main operations of the Fibonacci arithmetic'. ↩

References¶

[1] J. L Brown, 'Note on Complete Sequences of Integers', The American Mathematical Monthly, 1961, doi:10.2307/2311150. registry ↩a ↩b

[2] A. K Srinivasan, 'Practical numbers', Current Science, 1948.not authoritative registry ↩a ↩b