Prouhet–Thue–Morse constant¶

The real number whose binary expansion is the Thue–Morse sequence.

Version: v1 · 2026-09-08 · History
Domain-specific #: 6267
Origin domain: number theory
Subdomain: number theory

Core Idea¶

The constant tau is the convergent sum of the nth Thue–Morse bit divided by 2^(n+1), equivalently the binary real 0.011010011001… . Automatic-sequence recursion fixes every binary digit, while generating functions and infinite products expose arithmetic representations and transcendence properties. 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 number theory. It is the domain-specific identity determined by the digit in every binary position equals the corresponding Thue–Morse symbol under the declared nonterminating expansion convention.

Scope of Application¶

Prouhet–Thue–Morse constant belongs to number theory and is useful where the analyst can specify the typed number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets, then evaluate the digit in every binary position equals the corresponding Thue–Morse symbol under the declared nonterminating expansion convention. The scope is broad within that domain but bounded by the need for the digit in every binary position equals the corresponding Thue–Morse symbol under the declared nonterminating expansion convention. 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 the digit in every binary position equals the corresponding Thue–Morse symbol under the declared nonterminating expansion convention 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 Prouhet–Thue–Morse constant 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 Prouhet–Thue–Morse constant. Prouhet–Thue–Morse constant 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¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed 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 the digit in every binary position equals the corresponding Thue–Morse symbol under the declared nonterminating expansion convention independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse the typed number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases and comparison targets, Automatic-sequence recursion fixes every binary digit, while generating functions and infinite products expose arithmetic representations and transcendence properties., and type the carrier, state every parameter and convention in the definition, test that the digit in every binary position equals the corresponding Thue–Morse symbol under the declared nonterminating expansion convention, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Prouhet–Thue–Morse constant Domain-specific

Parents (1) — more general patterns this builds on

Prouhet–Thue–Morse constant is a kind of Representation Prime

The proposed strict upward parent is prime:representation.

Hierarchy path (1) — routes to 1 parentless root

Prouhet–Thue–Morse constant → Representation → Abstraction

Neighborhood in Abstraction Space¶

Prouhet–Thue–Morse constant sits in a moderately populated region (40^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Numeration & Arithmetic Representations (15 abstractions)

Nearest neighbors

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