Dottie number¶
The unique real fixed point of cosine, satisfying cos x equals x when the angle is measured in radians.
Core Idea¶
Radian measure is constitutive, the real fixed point differs from infinitely many complex solutions and decimal approximations are not the identity. Cosine maps a suitable real interval into itself as a contraction, so iteration converges to the unique crossing of cosine and the identity line. 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 mathematical constants. It is the domain-specific identity fixed by the cosine function in radians, fixed-point equation, real domain and interval, monotonic uniqueness proof, contraction or convergence condition, iterative sequence, numerical value and error, transcendence status and distinction from complex roots are explicit.
Scope of Application¶
Dottie number belongs to mathematical constants and is useful where the analyst can specify the typed mathematical constants carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the cosine function in radians, fixed-point equation, real domain and interval, monotonic uniqueness proof, contraction or convergence condition, iterative sequence, numerical value and error, transcendence status and distinction from complex roots are explicit. The scope is broad within that domain but bounded by the need for the cosine function in radians, fixed-point equation, real domain and interval, monotonic uniqueness proof, contraction or convergence condition, iterative sequence, numerical value and error, transcendence status and distinction from complex roots are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the cosine function in radians, fixed-point equation, real domain and interval, monotonic uniqueness proof, contraction or convergence condition, iterative sequence, numerical value and error, transcendence status and distinction from complex roots are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
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 Dottie number. Dottie number 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 mathematical constants carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the cosine function in radians, fixed-point equation, real domain and interval, monotonic uniqueness proof, contraction or convergence condition, iterative sequence, numerical value and error, transcendence status and distinction from complex roots are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of mathematical constants because they reuse the typed mathematical constants carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Cosine maps a suitable real interval into itself as a contraction, so iteration converges to the unique crossing of cosine and the identity line., and type the carrier, state every parameter and convention in the definition, test that the cosine function in radians, fixed-point equation, real domain and interval, monotonic uniqueness proof, contraction or convergence condition, iterative sequence, numerical value and error, transcendence status and distinction from complex roots are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Dottie number Domain-specific
Parents (1) — more general patterns this builds on
-
Dottie number is a kind of Fixed Point Prime
The proposed strict upward parent is
prime:fixed_point.
Hierarchy path (1) — routes to 1 parentless root
- Dottie number → Fixed Point
Neighborhood in Abstraction Space¶
Dottie number sits in a crowded region of the domain-specific corpus (30th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Mathematical Types, Functions & Infinity (33 abstractions)
Nearest neighbors
- Geometric progression — 0.91
- Differentiation of trigonometric functions — 0.91
- Proportionality (mathematics) — 0.90
- Real-valued function — 0.90
- Pi — 0.90
Computed from structural-signature embeddings · 2026-09-08