Single-crossing condition¶

An ordered-comparison property under which the difference between two functions changes sign at most once and in a fixed direction.

Version: v1 · 2026-09-08 · History
Domain-specific #: 6749
Origin domain: monotone comparative statics
Subdomain: monotone comparative statics

Core Idea¶

Single crossing orders types and choices so once one alternative becomes weakly preferred it remains preferred for all higher types, supporting monotone selection and comparative-statics results without full differentiability. A difference or preference comparison is evaluated along an ordered variable, and its sign pattern is constrained from reversing after the designated crossing. 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¶

Single-crossing condition belongs to monotone comparative statics and is useful where the analyst can specify the typed monotone comparative statics carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the ordered domain and alternatives, compared functions or preference relation, weak or strict formulation, crossing direction, treatment of equality intervals and implications for monotone choice or distribution order are explicit. The scope is broad within that domain but bounded by the need for the ordered domain and alternatives, compared functions or preference relation, weak or strict formulation, crossing direction, treatment of equality intervals and implications for monotone choice or distribution order are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the ordered domain and alternatives, compared functions or preference relation, weak or strict formulation, crossing direction, treatment of equality intervals and implications for monotone choice or distribution order 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 Single-crossing condition. Single-crossing condition 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 monotone comparative statics carrier, including its objects, 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 ordered domain and alternatives, compared functions or preference relation, weak or strict formulation, crossing direction, treatment of equality intervals and implications for monotone choice or distribution order are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of monotone comparative statics because they reuse the typed monotone comparative statics carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, A difference or preference comparison is evaluated along an ordered variable, and its sign pattern is constrained from reversing after the designated crossing., and type the carrier, state every parameter and convention in the definition, test that the ordered domain and alternatives, compared functions or preference relation, weak or strict formulation, crossing direction, treatment of equality intervals and implications for monotone choice or distribution order are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Single-crossing condition Domain-specific

Parents (1) — more general patterns this builds on

Single-crossing condition is a kind of Constraint Prime

The proposed strict upward parent is prime:constraint.

Hierarchy path (1) — routes to 1 parentless root

Single-crossing condition → Constraint

Neighborhood in Abstraction Space¶

Single-crossing condition sits in a moderately populated region (51^st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Mathematical Types, Functions & Infinity (33 abstractions)

Nearest neighbors

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