---
slug: "Distance"
name: "Distance"
type: "Primitives"
layer: "Physics"
ref: "Distance#dfdf"
sema_id: "sema:Distance#mh:SHA-256:dfdfe0610f91741e27f87042fc396eab765c4e48aa20fdfc0dfa7152d1333893"
tags: ["Physics","Primitives"]
updated: 2026-11-11
mindscape: "bootstrap"
index_url: https://mindscape.world/scapes/bootstrap.json
url: https://mindscape.world/bootstrap/Distance/
related: ["ConceptBlend","DriftWatch"]
---

# Distance

Metric-axiom function d(x,y) on a space — substrate for similarity, proximity, and spatial reasoning

Layer: Physics · Category: Primitives

## Mechanism

A function d(x, y) on a space that assigns a non-negative real number to each pair of points, satisfying the metric axioms: (1) non-negativity, d ≥ 0; (2) identity of indiscernibles, d(x, y) = 0 iff x = y; (3) symmetry, d(x, y) = d(y, x); (4) triangle inequality, d(x, z) ≤ d(x, y) + d(y, z). The substrate of 'how close' in any space — geographic, vector-embedding, cognitive-similarity, network-hop, graph-theoretic. Every metric space has a distance function; agents cannot redefine what metric-ness requires. Pseudo-metrics (which relax identity of indiscernibles) and quasi-metrics (which relax symmetry) are distinct descendants, not specializations of Distance itself.

## Invariants

- Non-negativity: d(x, y) ≥ 0 for all x, y in the space.
- Identity of indiscernibles: d(x, y) = 0 if and only if x = y.
- Symmetry: d(x, y) = d(y, x).
- Triangle inequality: d(x, z) ≤ d(x, y) + d(y, z).

## Failure modes

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## Builds on

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## Used by

- [ConceptBlend](https://mindscape.world/scapes/bootstrap/ConceptBlend.md) (Mind; References)
- [DriftWatch](https://mindscape.world/scapes/bootstrap/DriftWatch.md) (Society; References)

## Identity

- Sema ID: `sema:Distance#mh:SHA-256:dfdfe0610f91741e27f87042fc396eab765c4e48aa20fdfc0dfa7152d1333893`
- Ref: `Distance#dfdf`

## Links

- [Pattern page](https://mindscape.world/bootstrap/Distance/)
- [View in the interactive graph](https://mindscape.world/bootstrap/#Distance)
- [JSON](https://mindscape.world/scapes/bootstrap/Distance.json)
- [Index of all patterns](https://mindscape.world/scapes/bootstrap.json)