---
slug: mean_field
name: Mean Field
one_liner: "An aggregation in which each unit's detailed interactions with all the specific others are replaced by its interaction with a single population AVERAGE (the mean field), turning an N-body problem into a self-consistent one-body problem — the average each unit sees must equal the average the units collectively produce."
type: prime
schema_version: 2
status: density_v2_draft
categories: [structural, physical, computational]
origin_domain: statistical_physics
alternate_origin_domains: [game_theory, epidemiology, computational_neuroscience, machine_learning]
aliases: [mean_field_theory, mean_field_approximation, self_consistent_field, mean_field_game, one_body_reduction_of_many_body_interaction]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-04 spatialization/scaling, flavor 4c / micro→macro aggregation) — the aggregate structure obtained by replacing each unit's detailed pairwise interactions with all the specific others by its interaction with a single population AVERAGE (the mean field), converting an N-body problem into a self-consistent one-body problem. Discovered 2026-07-03. Convergence ×3, from `non_zero_sum_game` (mean-field game), `convolution`, and `hysteresis` (Preisach model). STRONG hold (verified absent from corpus; canonical scientific name — mean-field theory, Weiss/Curie-Weiss; mean-field game theory, Lasry-Lions & Huang-Caines-Malhamé; self-consistent field, Hartree-Fock). 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."
- "Phase-C action: re-similarity-check vs `aggregation` (the nearest parent), `local_to_global_aggregation`, `emergence`, `self_organization`, `incorporation`, and any wisdom-of-crowds / self-consistency prime; confirm it is a genuinely distinct self-consistent aggregation structure (each unit responds to the average it helps produce; many-body → one-body reduction) and not merely a restatement of `aggregation` or `emergence`; decide whether mean-field-theory (physics) and mean-field-games (economics) are one head or two (proposer recommends ONE — same self-consistency structure); draw a subsumption edge `aggregation` → `mean_field` at incorporation; 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: 5
  reasoning: 'The bare structure — replace each unit''s detailed interactions with all the specific others by its interaction with a single population average, subject to the self-consistency condition that the average each unit responds to equals the average the units collectively produce — is a genuinely substrate-neutral aggregation form. It recurs wherever a many-body interacting system is approximated by a one-body-in-an-average-field problem: phase transitions in statistical physics (Ising/Weiss), games with a continuum of small agents in economics and control (mean-field games), well-mixed compartmental models in epidemiology (mass-action SIR), population/rate models in neuroscience, and mean-field variational inference in machine learning. These are structurally-unlike siblings sharing only the abstract one-body-in-a-self-consistent-average-field reduction, so domain_breadth scores 5 and transfer evidence 5 (independently named canonical constructs in physics, game theory, and chemistry). Structural abstraction is 4 because the core is a pure claim about an aggregation operation — average-out the detail, close the loop with self-consistency — requiring no physical substance; it stays short of 5 because the notion of a well-defined population "average" and the infinite-range/dimension limit in which the approximation becomes exact carry a mild quantitative residue.'
similarity_to_nearest_existing_prime:
  slug: aggregation
  similarity: 0.60
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 structural. The core claim — replace detailed pairwise interaction with interaction against a single self-consistent population average, reducing an N-body problem to a one-body problem — is recognized rather than imported, and travels cleanly into game theory (mean-field games), epidemiology (well-mixed models), neuroscience (rate models), and machine learning (mean-field variational inference) without carrying statistical-physics vocabulary. The frame residue is light: "field," "average," and the infinite-range limit in which the approximation becomes exact import a mild quantitative/physical cast (keeping vocab_travels and import_vs_recognize off the ceiling), but the underlying aggregation structure — a self-consistency loop closing an averaged-out interaction — is substrate-neutral. Comparable framing to its parent `aggregation`.'
phase_b_passes_completed: []
---
# Mean Field

## Core Idea

Mean field is a specific *self-consistent aggregation*: the approximation in which each unit's detailed interactions with all the specific others are replaced by its interaction with a single **average** — the "mean field" — produced by the whole population. The many-body problem of N mutually interacting units, in which every unit's state depends on the particular states of every other, is collapsed into a one-body problem: a single representative unit responding not to its actual neighbors but to an averaged, population-level field. Generic `aggregation` summarizes many parts into a whole; mean field does something sharper and more structured — it *closes the loop*. The average that each unit responds to is not an external given but is itself produced by the units, so the approximation is defined by a **self-consistency condition**: the average field each unit sees must equal the average the units collectively generate when they respond to that field. Assume a field, let every unit best-respond to it, compute the population average of those responses, and require it to reproduce the assumed field. The fixed point of that loop *is* the mean-field solution.

The load-bearing anatomy is a structure the corpus's generic aggregation primes do not name. First, a **drastic dimensional reduction**: N² pairwise interactions become a single unit interacting with one field, so an intractable many-body problem becomes a tractable one-body problem. Second, the **self-consistency condition**: the aggregate is not merely read off the parts (as in a sum or a histogram) but must be a fixed point — each unit both *shapes* the average and *responds* to it, and the two must agree. Third, a principled **loss of fluctuation and correlation detail**: mean field discards the specific, local, correlated structure of who-interacts-with-whom, keeping only the population average. This is why mean field is *exact* in the limit of infinite range or infinite dimension (where every unit genuinely feels the whole population equally) and *fails* precisely where local correlations dominate — near critical points in low dimensions, in sparse or spatially-structured networks, in systems with strong local clustering. The approximation's domain of validity is thus written into its structure: it is the aggregation you get when detail can be averaged away because the interaction is effectively all-to-all.

Read structurally, the same one-body-in-a-self-consistent-average-field shape appears across genuinely unlike domains. In *statistical physics* it is mean-field theory of phase transitions — each spin feels the average magnetization (the Weiss/Curie-Weiss field) rather than its actual neighbors, and the self-consistency equation predicts the transition. In *economics and control* it is mean-field game theory (Lasry-Lions; Huang-Caines-Malhamé): a continuum of small agents, each optimizing against the *distribution* of all the others rather than tracking each rival individually, with equilibrium as the fixed point where the assumed population distribution equals the one the agents' optimal responses generate. In *epidemiology* it is the well-mixed compartmental model — mass-action SIR replaces the actual contact network with an average contact rate against the whole population. In *neuroscience* it is population/rate models where a neuron responds to the mean activity of its pool; in *machine learning* it is mean-field variational inference, which approximates an intractable joint distribution by a product of independent factors, each fit against the average of the rest. Across all of these the shared abstract engine is identical: replace detailed interaction with a self-consistent population average, reducing many bodies to one, at the price of the correlation structure — and the physics case is one species of a genus far wider than physics.

## Notes

**Provenance.** Produced by operator 04 (spatialization / scaling), flavor 4c (micro → macro aggregation), as the aggregate structure absent at the micro level: replace each unit's detailed interactions with all the specific others by its interaction with a single self-consistent population average, collapsing an N-body problem into a one-body problem. Surfaced 2026-07-03 with **×3 convergence** — reached independently from `non_zero_sum_game` (mean-field game), `convolution`, and `hysteresis` (the Preisach model of hysteresis as a mean-field-like aggregation of many bistable hysterons). Canonical, multiply-named scientific construct: mean-field theory / mean-field approximation in statistical physics (Weiss molecular field; Curie-Weiss; Ising), mean-field game theory in economics and control (Lasry & Lions 2007; Huang, Caines & Malhamé 2006), and self-consistent field in chemistry (Hartree-Fock). Verified absent from the corpus. Manifest: STRONG hold.

**For owner evaluation (keep/cut).** The case FOR is strong: mean field is a *canonical, multiply-named scientific construct* (not a proposer coinage), verified absent, with a clean and distinctive anatomy — a *self-consistent* average replacing detailed interactions, a many-body → one-body reduction, and a structurally-encoded validity domain (exact at infinite range, failing where correlations dominate) — and genuinely unlike siblings spanning physics phase transitions, mean-field games, epidemiology, neuroscience, and variational inference. It names a specific, powerful aggregation structure that the corpus's generic `aggregation` and `emergence` do not: the self-consistency loop in which each unit responds to the average it helps produce. The RISKS Kurt should weigh: (1) **overlap with `aggregation`** — mean field is a *specific self-consistent form* of aggregation, not aggregation in general; the discriminating features are the self-consistency fixed point and the many-body→one-body reduction, which generic aggregation (a sum, a pooling, a summary statistic) does not carry. Argue it earns its own node as a subsumption child. (2) **Overlap with `emergence` / `self_organization`** — mean field is an *approximation technique / aggregate structure*, not a claim that novel macro-behavior arises; screen for confusion. (3) **One head or two** — whether mean-field *theory* (physics) and mean-field *games* (economics) are one prime or two; the proposer recommends **one head**, since both instantiate the identical self-consistency structure (assume a field/distribution → best-respond → require the aggregate response to reproduce the assumption). Sharp keep/cut question: *is "self-consistent average replacing detailed interaction" a distinct prime, or is it just `aggregation`?* — the proposer's answer is that the self-consistency condition plus the N-body→one-body reduction is structure `aggregation` lacks.

**At incorporation.** Re-run similarity against `aggregation` (nearest parent), `local_to_global_aggregation`, `emergence`, `self_organization`, `incorporation`, and any wisdom-of-crowds / self-consistency prime to confirm no collapse. If kept: draw a subsumption edge `aggregation` → `mean_field` (mean field is a specific self-consistent aggregation), and add missing-edge notes flagging the contrast with `emergence`/`self_organization` (approximation structure vs. arising-of-novel-behavior) and any composition relationship to `fixed_point` / `self_consistency` structure that the self-consistency condition invokes. Resolve the one-head-vs-two question in favor of a single node with mean-field-games and self-consistent-field recorded as domain instances. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
