---
slug: density
name: Density
one_liner: "An intensive per-unit-extent quantity: an extensive amount divided by the extent it occupies (mass per volume, population per area, probability per length), turning a size-dependent count into a size-independent local concentration that can vary over space as a field and be integrated back to recover the total."
type: prime
schema_version: 2
status: density_v2_draft
categories: [structural, physical, mathematical]
origin_domain: physics
alternate_origin_domains: [statistics, demography, ecology, chemistry, information_theory, computer_science]
aliases: [intensive_quantity, per_unit_extent_concentration, amount_per_extent, local_concentration, mass_density, probability_density, spectral_density]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-04 spatialization/scaling, flavors 4a point→field and 4c micro→macro) — a genus convergence across many inputs: the intensive, per-unit-extent form obtained by normalizing an extensive amount by the extent it occupies (mass/volume, population/area, probability/length, items/bucket). Discovered in the op-04 full-corpus sweep 2026-07-03. MODERATE-WEAK hold (verified absent from corpus; established scientific name across physics/statistics/demography). STATUS: HOLD pending owner keep/cut — this is the WEAKEST and most fold-prone of the op-04 holds."
- "DOMINANT KEEP/CUT CONSIDERATION: density is very likely a CHILD or FACET of `field` (a separate, STRONGER op-04 stub) — a density is typically a scalar FIELD, i.e. density-at-every-point. Recommend the owner decide keep-as-child-of-field vs fold-into-field, and evaluate `density` and `field` TOGETHER rather than in isolation."
- "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` FIRST (parent/genus candidate), then `gradient`, `scaling_and_scale_dependence`, `aggregation`, and probability; if kept, draw a subsumption edge field→density (density = an intensive scalar field) + a composition/derivation edge from `aggregation`; 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 — an extensive quantity (one that scales with system size) divided by the extent it occupies (a length, area, volume, or more abstract measure) to yield an intensive quantity (independent of total size), which may vary from point to point and integrates back to recover the extensive total — is a genuinely substrate-neutral form. It recurs across structurally-unlike domains: mass density in physics, probability density in statistics, population density in demography, species/biomass density in ecology, concentration in chemistry, spectral and energy density in signal processing and thermodynamics, and load/occupancy density in computing. These share only the abstract normalize-by-extent move, so domain_breadth scores 5. Structural abstraction is 4 because the core is a pure ratio-of-measures statement requiring no physical substance, but stays just short of 5 because the concept leans on a notion of "extent"/measure that carries a mild geometric residue. Transfer evidence is strong (independently named and load-bearing in physics, statistics, demography, and information theory), though the concept sits close enough to `field` that some of that transfer evidence may belong to the field genus rather than to density as an independent prime.'
similarity_to_nearest_existing_prime:
  slug: field
  similarity: 0.6
structural_framed:
  label: structural
  aggregate: 0.33
  criteria: {vocab_travels: 0.75, evaluative_weight: 0.15, institutional_origin: 0.1, human_practice_bound: 0.15, import_vs_recognize: 0.35}
  rationale: 'Reads clearly structural. The core claim — an amount normalized by the extent it occupies to give a size-independent local concentration — is recognized rather than imported, and travels cleanly into statistics (probability density), demography (population density), and information theory (spectral density) without carrying any physics vocabulary; "mass per volume" is one instance, not the concept. There is almost no evaluative or institutional residue: normalize-by-extent is a mathematical operation with no human-practice binding. The only reason aggregate is not lower is the mild geometric framing carried by "extent"/"measure," which keeps import_vs_recognize and vocab_travels off the floor. Framing is comparable to its parent-candidate `field`.'
phase_b_passes_completed: []
---
# Density

## Core Idea

Density is an *intensive, per-unit-extent* quantity: an amount divided by the extent it occupies. Its anatomy has three moving parts. Start with an *extensive* quantity — one that scales with the size of the system: total mass, total population, total probability, a count of items. Divide it by the *extent* it is spread across — a length, an area, a volume, or a more abstract measure. The result is an *intensive* quantity — a local concentration that is independent of the total size of the system: mass per volume, population per area, probability per unit length, items per bucket. The load-bearing move is *normalization by extent*, which converts a size-dependent count into a size-independent local rate. Two systems of very different total size can share the same density; a single system can have the same density whether you weigh a gram of it or a ton.

Once normalized, the density need not be uniform: it can *vary from point to point*, giving a density field — density-at-every-location — and this field can be *integrated back over the extent* to recover the original extensive total. This closes the loop between the intensive local quantity and the extensive global one: differentiate the total (per unit extent) to get the density, integrate the density (over the extent) to get the total back. That reciprocity — an extensive total that is the integral of an intensive density, and an intensive density that is the local derivative of the total — is the structural signature the corpus does not currently name at the level of a bare count or amount. A raw count tells you *how much*; a density tells you *how much per unit of where*.

Read structurally, the same shape recurs across genuinely unlike domains. In *physics* it is mass density (mass per volume) and its kin — charge, current, energy, and momentum densities. In *statistics* it is the probability density function: total probability one, spread continuously over an outcome space, with the density integrating back to probabilities of regions. In *demography and ecology* it is population, species, or biomass density (individuals per area). In *chemistry* it is concentration (amount of substance per volume). In *signal processing and thermodynamics* it is spectral density and energy density. In *computing* it is load or occupancy (items per slot). Across all of these the abstract engine is identical: an extensive amount normalized by the extent it occupies, yielding an intensive field that integrates back to the total — the biological, statistical, and physical cases are species of one genus.

## Notes

**Provenance.** Produced by operator 04 (spatialization / scaling), flavors 4a (point → field) and 4c (micro → macro), as a *genus convergence* across many inputs during the full-corpus op-04 sweep on 2026-07-03: hundreds of "amount of X per unit extent" inputs (mass per volume, population per area, probability per length, items per bucket, energy/current/spectral density) resolve to the same intensive, per-unit-extent structure. Established scientific term across physics (mass density), statistics (probability density function), demography and ecology (population density), chemistry (concentration), and information theory (spectral density) — substrate-neutral, not a proposer coinage. Verified absent from the corpus. Manifest: MODERATE-WEAK hold — the weakest and most fold-prone of the op-04 holds.

**For owner evaluation (keep/cut).** The case FOR: density is a genuinely foundational, ubiquitous *intensive-quantity* concept — per-unit-extent normalization — that the corpus does not name, and it recurs load-bearingly across physics, statistics, demography, ecology, chemistry, and information theory. It fills a real gap: the corpus has `gradient` and diffusion acting *on* densities/fields but no prime for the intensive quantity itself. The RISKS Kurt should weigh, in order of severity: (1) **density is most likely a CHILD or FACET of `field`** (a separate, stronger op-04 stub) — a density is typically a *scalar field*, i.e. density-at-every-point, so the dominant question is placement, not existence; recommend deciding keep-as-child-of-`field` vs fold-into-`field`, and evaluating the two stubs *together*. (2) **Partial overlap with `aggregation`** — a density is an aggregate amount divided by extent, so it is close to "aggregation normalized by extent," and also brushes the general intensive/extensive distinction. (3) **Operation-not-prime risk** — density may be a *mathematical operation* (normalize an extensive amount by extent) rather than a standalone abstraction, in which case it belongs as a facet of `field` or a derivation note rather than its own node. Sharp keep/cut question: **is density a distinct prime, a child of `field`, or a facet folded into `field`?** The proposer's honest recommendation is to resolve this by evaluating `density` and `field` side by side, and to treat fold-into-`field` (or keep-as-explicit-child-of-`field`) as the most probable outcome.

**At incorporation.** Re-run similarity against `field` **first** (the parent/genus candidate — a density is an intensive scalar field), then `gradient` (acts on a density; sibling not parent), `scaling_and_scale_dependence` (intensive/extensive scaling behavior), `aggregation` (density = aggregate amount ÷ extent), and probability (probability density as a leading instance). If kept as an independent node: draw a subsumption edge `field` → `density` (density is an intensive scalar field), a derivation/composition edge from `aggregation` (aggregate amount ÷ extent), and sibling edges to `gradient` and `scaling_and_scale_dependence`. If folded: record density as an explicit facet/example under `field` with the intensive/extensive integrate-back reciprocity preserved as a body note. Either way, write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
