---
slug: field
name: Field
one_liner: "A quantity defined at every point of a space, so that it varies from place to place — the spatial substrate whose anatomy (values-over-a-domain, gradients, boundaries, spatial correlation, and scalar/vector/tensor type) underlies gradient, diffusion, potential, and flow, which are operations ON a field rather than the field itself."
type: prime
schema_version: 2
status: density_v2_draft
categories: [structural, mathematical, physical]
origin_domain: physics
alternate_origin_domains: [mathematics, geography, image_processing, ecology, economics]
aliases: [scalar_field, vector_field, tensor_field, spatial_field, quantity_over_a_domain, distributed_quantity, value_at_every_point]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-04 spatialization, flavor 4a / point→field) — the GENUS convergence of the op-04 sweep and its STRONGEST find: hundreds of inputs individually spatialized to 'a field of P' (a pointwise value P promoted to P defined over a space). The individual 'field of P' instances are trivial coinages; their convergence is what flags the missing genus — the field itself, the spatial substrate that these coinages all instantiate. Discovered 2026-07-03. STRONG hold (verified absent from corpus; established scientific name — classical field theory; a function over a domain in mathematics). STATUS: HOLD pending owner keep/cut."
- "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."
- "Screens at incorporation: `gradient` (present — the spatial derivative OF a field; field is the more basic object / genus), `diffusion` (a field-evolution process), `receptive_field` (a domain-specific instance/metaphor), `non_locality` (a property some fields have), and `density` (a per-unit-extent scalar field — plausibly a CHILD of field; flag this). Confirm field is the genus/substrate and not a restatement of gradient."
- "Phase-C action: re-similarity-check vs `gradient` (nearest; a field-operator) + `density`/`diffusion`/`receptive_field`/`non_locality`; recommend field as the GENUS with child/related edges gradient→field and density→field, and diffusion/flow noted as field-processes; decide whether scalar/vector/tensor fields are one head or a small family (recommend one head); write a v1 draft and sync the one_liner across v1/v2 per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 5
  domain_breadth: 5
  structural_abstraction: 5
  transfer_evidence: 4
  reasoning: 'The bare structure — a quantity assigned to every point of a space, so that its value varies with position and the object of study is the whole distribution rather than any single value — is about as substrate-neutral as a structural primitive can be. Nothing physical is required: the space can be geometric (physical fields), a function''s domain (mathematics), a geographic surface (GIS elevation/population-density fields), a pixel raster (image processing), a resource landscape (ecology), or a price surface (economics). domain_breadth scores 5 — the siblings are structurally unlike, sharing only the abstract form "value at every point of a domain." structural_abstraction scores 5 because the core makes no appeal to any particular substance: it is purely the promotion of a pointwise value to a value-over-a-space, with named anatomy (gradients as the field''s spatial derivative, boundaries where the domain ends, spatial correlation between nearby points, and field type — scalar/vector/tensor). transfer_evidence is 4 (strong and canonical across physics and mathematics; the everyday/GIS/economics "surface" uses are widespread but less formally named as "fields"). The main tension the owner must weigh is not neutrality but generality — a maximally-general substrate risks being TOO foundational to earn a distinct node; the proposer''s view is that this is exactly what makes it a keystone the spatial layer currently lacks.'
similarity_to_nearest_existing_prime:
  slug: gradient
  similarity: 0.59
structural_framed:
  label: structural
  aggregate: 0.27
  criteria: {vocab_travels: 0.85, evaluative_weight: 0.1, institutional_origin: 0.1, human_practice_bound: 0.1, import_vs_recognize: 0.25}
  rationale: 'Reads strongly structural — one of the most structural op-04 finds. The core claim — a quantity distributed over every point of a space — carries no evaluative, institutional, or human-practice residue; it is recognized as a bare form rather than imported from a home domain, and travels cleanly into mathematics, GIS, image processing, and economics without dragging physics vocabulary with it. The only framing residue is that the canonical NAME ("field," "scalar/vector/tensor field") originates in physics, giving import_vs_recognize a small nonzero weight; the underlying structure (value-at-every-point-of-a-domain) is fully substrate-neutral. Aggregate sits at the low/structural end (~0.22–0.32 band), comparable to the corpus''s most foundational spatial primitives.'
phase_b_passes_completed: []
---
# Field

## Core Idea

A field is a quantity defined at *every* point of a space (or, more abstractly, a domain), so that the quantity varies from place to place. Where a plain value is a single number attached to nothing in particular — a temperature, a pressure, a concentration — a field is that same kind of quantity *spread across a space*: a temperature at every location in a room, a pressure at every point in a fluid, a concentration at every point of a medium. The object of study is no longer the value but the whole *distribution* of values over the domain. This is the promotion op-04 performs in flavor 4a: take a pointwise value P and ask what P looks like when it is defined not at a point but over an entire space — the answer is *a field of P*.

The load-bearing anatomy is a small vocabulary the pointwise value simply does not have. A field has *spatial gradients* — how fast its value changes as you move through the space, the field's spatial derivative. It has *boundaries* or edges — where the domain ends, or where the field's behavior changes character. It has *spatial correlation* — nearby points tend to hold related values, so the field is not arbitrary noise but a structured surface. And it has a *type*: a *scalar field* assigns one number to each point (temperature, elevation, density), a *vector field* assigns a direction-and-magnitude to each point (a flow, a force, a wind map), and a *tensor field* assigns a still richer object. None of these — gradient, boundary, spatial correlation, field-type — is even expressible for a value that lives at a single point; they are exactly the features that appear once the value is distributed over a space and absent before it is.

This matters because the field is the *substrate on which a whole family of already-present primes acts*, yet the substrate itself has no node. A `gradient` is the spatial derivative *of* a field — it presupposes a field to differentiate. `diffusion` is a *process that evolves* a field over time. A potential or a flow *is* a field. The corpus holds these operators-on-fields but not the field they operate on, which is why hundreds of op-04 inputs converged on "a field of P" without any of them landing on the underlying genus. Read structurally, the same shape recurs across unlike domains: a function over its domain in mathematics, an elevation or population-density *surface* in geography and GIS, a pixel raster in image processing, a resource landscape in ecology, a price *surface* in economics. In every case the shared abstract form is one thing — a quantity defined at every point of a space — and the domain supplies only what the points and the values happen to be.

## Notes

**Provenance.** Produced by operator 04 (spatialization / scaling), flavor 4a (point → field), as the promotion of a pointwise value P to *P defined over an entire space*. This was the **strongest find of the op-04 sweep, and structurally distinctive: it is a GENUS convergence rather than a single application.** Hundreds of inputs across the corpus individually spatialized to "a field of P" — a field of preferences, a field of risk, a field of activation, a field of prices, and so on. Each individual "field of P" is a trivial coinage with no independent standing; what carries the signal is that they all *converge on the same underlying object* — the field itself, the spatial substrate every one of those coinages instantiates. That convergence is the strongest possible flag for a missing genus. Surfaced 2026-07-03. Established scientific construct: classical field theory in physics (scalar / vector / tensor fields, the electromagnetic field, the temperature field) and "the field of a function over its domain" in mathematics; also the everyday distributed-quantity sense ("a field of X"). Verified absent from the corpus. Manifest: STRONG hold.

**For owner evaluation (keep/cut).** The case FOR is unusually clean: this is arguably the single clearest gap in the corpus's *spatial layer*. A foundational, maximally-general structural primitive — a quantity distributed over a space — that dozens of existing primes (`gradient`, `diffusion`, potential, flow) *presuppose* but that has no node of its own, and the genus-convergence across hundreds of inputs is the strongest signal the sweep can produce. The RISKS Kurt should weigh: (1) **generality** — is a quantity-over-a-space *too* general or foundational to be a useful prime, or is that exactly what makes it a keystone? The proposer's view is the latter: foundational substrates that many primes silently presuppose are precisely the nodes worth drawing. (2) **relationship to `gradient`, `density`, and `receptive_field`** — `gradient` is the spatial derivative *of* a field (an operator, so field is the genus/substrate, not a synonym); `density` is a per-unit-extent scalar field and is plausibly a *child* of field; `receptive_field` is a domain-specific instance/metaphor. Recommend **field as the genus, with `gradient` and `density` as related/child nodes** rather than mergers. (3) **head count** — whether scalar / vector / tensor fields are one prime or a small family; recommend **one head** (field), with the scalar/vector/tensor distinction handled as field-*type* inside the single prime. Sharp keep/cut question: *is "a quantity defined at every point of a space" a distinct prime, or is it just the presupposed backdrop of `gradient` and `diffusion` and therefore not worth its own node?* — the proposer's answer is that a substrate presupposed by many primes but named by none is exactly the kind of gap the densification pass exists to close.

**At incorporation.** Re-run similarity against `gradient` (nearest existing prime — the spatial derivative OF a field; confirm field is the more basic object and not a restatement), `density` (candidate child — a per-unit-extent scalar field), `diffusion` (a field-evolution process), `receptive_field` (domain-specific instance / metaphor), and `non_locality` (a property some fields have) to confirm no collapse. If kept: draw child / related edges `gradient` → `field` and `density` → `field`, and add missing-edge notes flagging `diffusion` and flow as *field-processes* (processes that evolve a field) rather than kinds of field. Decide the head count explicitly — recommend a single `field` head carrying the scalar/vector/tensor distinction as field-type — and record the decision. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
