Reciprocal distribution¶
A bounded positive distribution whose density is proportional to one over the variable, equivalently uniform after logarithmic transformation.
Core Idea¶
For positive bounds a below b, the log-uniform density is one divided by x times the logarithmic span, with cumulative probability linear in log x. Uniform probability across equal logarithmic intervals creates multiplicative rather than additive scale neutrality and makes reciprocation preserve the distribution after swapping inverse bounds. 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 probability distributions. It is the domain-specific identity determined by support is a positive finite interval, normalization uses its logarithmic ratio, and log X is uniform on the corresponding log interval.
Scope of Application¶
Reciprocal distribution belongs to probability distributions and is useful where the analyst can specify the typed probability distributions carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate support is a positive finite interval, normalization uses its logarithmic ratio, and log X is uniform on the corresponding log interval. The scope is broad within that domain but bounded by the need for support is a positive finite interval, normalization uses its logarithmic ratio, and log X is uniform on the corresponding log interval. 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 support is a positive finite interval, normalization uses its logarithmic ratio, and log X is uniform on the corresponding log interval 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 Reciprocal distribution 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 Reciprocal distribution. Reciprocal distribution 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 probability distributions 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 support is a positive finite interval, normalization uses its logarithmic ratio, and log X is uniform on the corresponding log interval independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of probability distributions because they reuse the typed probability distributions carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Uniform probability across equal logarithmic intervals creates multiplicative rather than additive scale neutrality and makes reciprocation preserve the distribution after swapping inverse bounds., and type the carrier, state every parameter and convention in the definition, test that support is a positive finite interval, normalization uses its logarithmic ratio, and log X is uniform on the corresponding log interval, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Reciprocal distribution Domain-specific
Parents (1) — more general patterns this builds on
-
Reciprocal distribution is a kind of Scale Invariance Prime
The proposed strict upward parent is
prime:scale_invariance.
Hierarchy paths (2) — routes to 2 parentless roots
- Reciprocal distribution → Scale Invariance → Invariance
- Reciprocal distribution → Scale Invariance → Symmetry
Neighborhood in Abstraction Space¶
Reciprocal distribution sits in a crowded region of the domain-specific corpus (35th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Probability Measures & Random Variables (36 abstractions)
Nearest neighbors
- Truncated normal distribution — 0.91
- Modified half-normal distribution — 0.91
- Geometric standard deviation — 0.90
- Hyperbolic distribution — 0.90
- Ratio distribution — 0.89
Computed from structural-signature embeddings · 2026-09-08