---
slug: spatial_random_field
name: Spatial Random Field
one_liner: "A random quantity indexed by position — a random value at every point of a space whose values at nearby points are spatially correlated over a characteristic correlation length — the spatial analog of a stochastic process (randomness indexed by time)."
type: prime
schema_version: 2
status: density_v2_draft
categories: [structural, mathematical, computational]
origin_domain: probability_and_statistics
alternate_origin_domains: [geostatistics, machine_learning, cosmology, image_analysis, ecology]
aliases: [random_field, gaussian_random_field, markov_random_field, spatial_gaussian_process, stochastic_field]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-04 spatialization/scaling, flavor 4a point→field) — the SPATIALIZED prime obtained by indexing a single random value BY POSITION: a random quantity defined at every point of a space, with values at nearby points correlated (spatial autocorrelation over a characteristic correlation length), where a single random variable has no spatial structure at all. Discovered in the op-04 sweep 2026-07-03. MODERATE hold; CONVERGENCE ×2 (two independent inputs). Verified absent from corpus; established scientific name — random field / Gaussian random field / Markov random field (spatial statistics, geostatistics/kriging); Gaussian process over space (ML). STATUS: HOLD pending owner keep/cut."
- "COMPOSITION QUESTION (flag prominently) — a spatial random field is arguably the COMPOSITION `field` (op-04 stub) × randomness (a *stochastic* field), NOT necessarily a new atomic prime. The owner may prefer to record it as a composition of `field` and probability/`stochastic_process` rather than mint it. Recommend evaluating it TOGETHER with the `field` stub. Secondary overlap: `stochastic_process` (the TEMPORAL analog — randomness indexed by time; spatial_random_field is the spatial analog)."
- "Created 2026-07-03; STUB ONLY (frontmatter + Core Idea + Notes). SI/SF/similarity are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT written."
- "Phase-C action: re-similarity-check vs `field` (op-04 stub — plausible parent), `stochastic_process` (temporal analog), `distributional_assumption`/`probability` (the input), and `markov_process` (whose spatial analog is a Markov random field); resolve the composition-vs-atomic question; if kept, likely draw a subsumption edge `field` → `spatial_random_field` plus a composition/sibling edge to `stochastic_process`; write a v1 draft and sync the one_liner across v1/v2 per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 4
  domain_breadth: 5
  structural_abstraction: 4
  transfer_evidence: 4
  reasoning: 'The bare structure — a random quantity indexed by position, such that values at nearby points are correlated over a characteristic correlation length while values far apart decorrelate — is a genuinely substrate-neutral mathematical form. It recurs wherever randomness carries spatial structure: ore grades / soil properties / rainfall over a region in geostatistics (kriging), pixel-label fields in image analysis (Markov random fields), primordial density fluctuation fields in cosmology, spatial covariate fields in Gaussian-process ML models, and species-distribution fields in ecology. These are structurally-unlike siblings sharing only the abstract form (a random value per location + a spatial covariance function + a correlation length + stationarity/isotropy properties), so domain_breadth scores 5. Structural abstraction is 4 because the core is a pure probability-theoretic statement (a random function on an index set that is a space) requiring no particular substrate; it stays short of 5 because the useful cases lean on smoothness/stationarity assumptions that are modeling choices rather than pure structure. Transfer evidence is strong — the construct is independently and canonically named across statistics, geostatistics, ML, cosmology, and image analysis.'
similarity_to_nearest_existing_prime:
  slug: field
  similarity: 0.60
structural_framed:
  label: structural
  aggregate: 0.35
  criteria: {vocab_travels: 0.7, evaluative_weight: 0.1, institutional_origin: 0.1, human_practice_bound: 0.2, import_vs_recognize: 0.5}
  rationale: 'Reads structural. The core claim — a random value at every point of a space, with a spatial covariance function and a correlation length governing how quickly nearby values decorrelate — is recognized rather than imported, and travels cleanly from geostatistics into image analysis, cosmology, and Gaussian-process ML without carrying any one domain''s vocabulary. There is essentially no evaluative or institutional residue; the light framing residue is that "field," "correlation length," and "stationarity/isotropy" are technical modeling constructs (keeping import_vs_recognize off the floor), but the underlying object — a random function on a spatial index set — is substrate-neutral. Comparable framing to its likely parent `field` and to `stochastic_process`.'
phase_b_passes_completed: []
---
# Spatial Random Field

## Core Idea

A spatial random field is a random quantity *indexed by position*: a random value assigned to every point of a space, such that values at nearby points are *correlated* and values far apart are (typically) not. Base-level probability and `distributional_assumption` describe a single random variable — one draw, one value, no notion of *where* — and the feature absent at that single random value is precisely spatial structure: a lone random variable has no location, no neighbors, and no correlation length. Spatialize it by flavor 4a (point → field) — attach a random value to every location and let the joint law of those values depend on their spatial arrangement — and a new object appears, whose load-bearing anatomy is a *random value per location*, a *spatial covariance (or correlation) function*, a *correlation length* (the distance over which values stay similar before they decorrelate), and *stationarity / isotropy* properties (whether the statistics look the same everywhere and in every direction).

The load-bearing distinction the base object does not name is *spatial autocorrelation*: not merely that many random variables exist, but that their joint distribution ties nearby ones together, so knowing the value here informs the value a short distance away. This is what separates a spatial random field from an unstructured collection of independent random variables scattered over a space — the correlation function, and the correlation length that summarizes it, are the whole point. The field is the spatial analog of a *stochastic process* (a random quantity indexed by *time*): where a stochastic process supplies randomness-over-time with temporal autocorrelation, a spatial random field supplies randomness-over-space with spatial autocorrelation; the two are siblings under the more general "random function on an index set."

Read structurally, the same form appears wherever randomness is spatially organized, and the siblings are genuinely unlike. In *geostatistics*, it is the field of ore grades, soil properties, or rainfall over a region, and kriging is best-linear-unbiased prediction on such a field. In *image analysis*, a Markov random field models pixel labels whose values are conditionally dependent on their neighbors. In *cosmology*, the primordial density fluctuations are a Gaussian random field whose power spectrum encodes the correlation structure. In *machine learning*, a Gaussian process over a spatial domain is exactly a spatial random field used for interpolation and uncertainty quantification. In *ecology*, a species-distribution field is a spatial random field over a landscape. Across these, the shared abstract object is a random value per location plus a spatial covariance function with a characteristic correlation length; the specific quantity (grade, label, density, response) is one instantiation of the genus.

## Notes

**Provenance.** Produced by operator 04 (spatialization / scaling), flavor 4a (point → field), from *probability and randomness*: take a single random value and index it by position, yielding a random value at every point of a space with spatial autocorrelation over a characteristic correlation length. Surfaced in the op-04 sweep on 2026-07-03 with CONVERGENCE ×2 (two independent inputs arrived at the same target). Established scientific construct across multiple literatures: *random field / Gaussian random field / Markov random field* in spatial statistics; *kriging / geostatistics* (Matheron) for prediction on such fields; *Gaussian process over space* in machine learning; *Gaussian random field* in cosmology (density-fluctuation power spectra); *Markov random field* in image analysis. It is the spatial analog of a stochastic process (randomness indexed by time). Verified absent from the corpus. Manifest: MODERATE hold, convergence ×2.

**For owner evaluation (keep/cut).** The case FOR: spatial_random_field is a *canonical, named scientific construct* (random field / Gaussian process / kriging — not a proposer coinage), verified absent, arrived at by two independent inputs, with a clean spatialized anatomy — spatially-correlated randomness with a correlation length — and genuinely unlike siblings spanning geostatistics, ML, cosmology, image analysis, and ecology. The RISKS Kurt should weigh: (1) **COMPOSITION vs ATOMIC (flag prominently)** — a spatial random field is arguably just the composition `field` (op-04 stub) × randomness, i.e. a *stochastic field*; the owner may prefer to record it as a composition of `field` and probability/`stochastic_process` rather than mint a new atomic prime. This is the sharp keep/cut question, and it is best resolved by **evaluating this stub jointly with the `field` stub.** (2) **Overlap with `stochastic_process`** — the temporal analog (randomness indexed by time); the two are siblings under "random function on an index set," and the discriminating claim is that the index here is a *space* (spatial autocorrelation, isotropy) rather than a time line. If both are kept they should be sibling nodes, not merged. (3) **Placement** — whether it sits as a child of `field` (subsumption: it *is* a field, specifically a random one) or as a distinct node; the proposer leans toward a subsumption edge `field` → `spatial_random_field`.

**At incorporation.** Re-run similarity against `field` (op-04 stub — plausible parent), `stochastic_process` (temporal analog / sibling), `distributional_assumption` and `probability` (the input), and `markov_process` (whose spatial analog is the Markov random field — screen for the conditional-independence / neighbor-dependence relationship). Resolve the composition-vs-atomic question against the `field` stub first. If kept: likely draw a subsumption edge `field` → `spatial_random_field` plus a composition/sibling edge to `stochastic_process`, and a missing-edge note flagging the `markov_process` → Markov-random-field spatial-analog relationship. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
