---
slug: order_parameter
name: Order Parameter
one_liner: "The emergent macroscopic observable that is zero in the disordered/symmetric phase and non-zero in the ordered/broken-symmetry phase, so that its value quantifies how much order the system has and serves as the coordinate along which a phase transition is described."
type: prime
schema_version: 2
status: density_v2_draft
categories: [structural, physical, complex_systems]
origin_domain: statistical_physics
alternate_origin_domains: [complex_systems, condensed_matter_physics, nonlinear_dynamics, social_dynamics]
aliases: [broken_symmetry_measure, phase_distinguishing_observable, degree_of_order, collective_order_variable, landau_order_parameter]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-08 threshold/phase, flavor 8a) — the emergent macro-observable that measures the DEGREE of order and distinguishes the phases either side of a transition. Op-08 was RETIRED AS SATURATED (the threshold/phase region is heavily covered; near-zero pilot yield); `order_parameter` is one of only 2 genuine domain-general gaps op-08 surfaced. Discovered/confirmed absent 2026-07-03. MODERATE hold (established scientific name — Landau theory of phase transitions; verified absent from corpus)."
- "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 `symmetry_breaking` (the qualitative event this quantifies) and `phase_diagram` (whose regions this observable labels), plus `tipping_points_or_phase_transitions`, `emergence`, and collective-behavior primes; confirm it is the genuine distinct observable (the emergent scalar that measures degree of order and distinguishes phases) and not a facet of `symmetry_breaking` or `phase_diagram`; if kept, draw related edges to `symmetry_breaking` and `phase_diagram`; 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: 4
  structural_abstraction: 3
  transfer_evidence: 4
  reasoning: 'The bare structure — an emergent macroscopic quantity that is exactly zero in the symmetric/disordered regime and becomes non-zero as the system enters the ordered/broken-symmetry regime, so that its magnitude quantifies the degree of order and its onset locates the transition — is a genuinely substrate-neutral form. It recurs across structurally-unlike collective-ordering systems: magnetization in a ferromagnet, density difference at a liquid-gas transition, condensate amplitude in a superfluid/superconductor, alignment in an isotropic-to-nematic transition, the Kuramoto phase-coherence measure in synchronizing oscillators, mean heading in a flock, and the majority fraction in opinion polarization. This spread earns domain_breadth 4. Structural abstraction is held to 3 because the concept, though portable, still leans on a physics-derived scaffolding — a well-defined broken symmetry, a critical point, and a notion of "order" measured on a macro-observable — which travels but carries some of its statistical-mechanics home with it. Transfer evidence is strong: the construct is independently and canonically named in condensed-matter physics, nonlinear dynamics (Kuramoto), and active-matter/flocking literatures.'
similarity_to_nearest_existing_prime:
  slug: symmetry_breaking
  similarity: 0.62
structural_framed:
  label: structural
  aggregate: 0.36
  criteria: {vocab_travels: 0.7, evaluative_weight: 0.15, institutional_origin: 0.1, human_practice_bound: 0.15, import_vs_recognize: 0.45}
  rationale: 'Reads structural. The core claim — an emergent macro-observable that is zero in the disordered phase and non-zero in the ordered phase, quantifying the degree of broken symmetry — is recognized rather than imported, and travels cleanly into synchronization (the Kuramoto order parameter), flocking (mean alignment), and opinion polarization without carrying condensed-matter substance. There is no evaluative or institutional residue: an order parameter is a descriptive measurement, not a norm or a human-practice construct. The frame residue that keeps import_vs_recognize off the floor is the physics scaffolding it presupposes — a definite broken symmetry, a critical point, a macro/micro split — which is portable but must be re-recognized in each new domain rather than being fully substrate-blank.'
phase_b_passes_completed: []
---
# Order Parameter

## Core Idea

An order parameter is the emergent macroscopic observable that distinguishes the phases of a system on either side of a transition: it is *zero* in the disordered, symmetric phase and *non-zero* in the ordered, broken-symmetry phase, so that its value measures *how much* order — how much broken symmetry — the system has. It is the coordinate along which a phase transition is described. Where `symmetry_breaking` names the qualitative event ("the system spontaneously chose one of several equivalent ordered states"), the order parameter is the *quantity* that says by how much and in which direction: the magnetization that rises from zero as a paramagnet cools into a ferromagnet, the density difference that opens up across a liquid-gas transition, the condensate amplitude of a superfluid, the alignment of a nematic. It is the single number that tells you which phase you are in and how deeply you are in it.

The load-bearing anatomy has three parts. First, the order parameter is an *emergent collective variable*, not a microscopic detail: it is a property of the aggregate that becomes the relevant descriptor precisely at the transition, invisible or irrelevant in the interior of either phase. Second, it takes a *definite value that distinguishes the phases* — identically zero on the symmetric side and non-zero on the ordered side — and it changes characteristically at the critical point: rising continuously from zero in a second-order (continuous) transition, or jumping discontinuously in a first-order transition. Third, its *onset marks spontaneous symmetry breaking*: the moment the order parameter departs from zero is the moment the system commits to one among a family of equivalent ordered configurations. These three features make it the natural coordinate of Landau's theory of phase transitions — the emergent scalar whose behavior encodes the whole story of the transition.

Read structurally, the same shape recurs wherever a large population undergoes collective ordering, and the instances are genuinely unlike. In *synchronization*, the Kuramoto order parameter measures the phase coherence of a population of coupled oscillators: zero when phases are scattered, non-zero and growing as they lock together. In *collective motion*, the mean alignment (average heading) of a flock or swarm is zero in the disordered milling state and non-zero once the group polarizes into coherent motion. In *social dynamics*, the majority fraction or mean opinion is an order parameter for polarization: near-zero in a well-mixed population, swinging to a large magnitude as consensus or factional order sets in. Across all of these the shared abstract form is one emergent macro-observable, identically zero in the symmetric/disordered regime and non-zero in the ordered regime, whose magnitude quantifies the degree of order and whose onset locates the transition — the physics of magnetization is one species of the genus, and the synchronization, flocking, and polarization siblings show the genus is the bare structure.

## Notes

**Provenance.** Produced by operator 08 (threshold/phase), flavor 8a, as the emergent macro-observable that measures the *degree* of order distinguishing the phases either side of a transition. Op-08 was retired as saturated — the threshold/phase region of the corpus is already heavily covered and the pilot yield was near zero — and `order_parameter` is one of only two genuine domain-general gaps the operator surfaced. Confirmed absent from the corpus on 2026-07-03. Established scientific construct: the order parameter of Landau's theory of phase transitions, with canonical instances including magnetization (para→ferromagnet), density difference (gas→liquid), condensate amplitude (superfluid/superconductor), and alignment (isotropic→nematic). Substrate-neutral siblings: the Kuramoto order parameter in synchronization, mean alignment in flocking, and the majority/mean-opinion measure in polarization. Manifest: MODERATE hold.

**For owner evaluation (keep/cut).** The case FOR: `order_parameter` is a *canonical, named scientific construct* (Landau), verified absent, with a clean and distinctive anatomy — an emergent scalar that is zero in the symmetric phase and non-zero in the ordered phase, quantifying the degree of broken symmetry and serving as the coordinate of the transition — and it has genuine cross-domain reach (magnetization, Kuramoto synchronization, flocking alignment, opinion polarization). It fills a real gap: the corpus has `symmetry_breaking` (the qualitative event) and `phase_diagram` (the map of regions), but not the emergent observable that *measures the degree of order* and thereby distinguishes the phases. The RISKS Kurt should weigh: (1) **closeness to `symmetry_breaking`** (proposer similarity ~0.58–0.66) — the discriminating claim is observable-vs-event: `symmetry_breaking` describes *that* the system broke symmetry, `order_parameter` is the *quantity* that measures *how much* and along which coordinate; they are complementary, not redundant, with the order parameter being the observable that makes the qualitative event measurable. (2) **Risk of reading as physics-specific** rather than domain-general — the counter is the synchronization/flocking/opinion generalizations, where the order-parameter form is independently and canonically named (the Kuramoto order parameter especially). (3) **Partial overlap with `phase_diagram`** — an order parameter *labels the regions* of a phase diagram and measures the degree of order within them, but is not the diagram itself; they attach by a related edge, not merger. Sharp keep/cut question: *is "the emergent phase-distinguishing observable that measures the degree of order" a distinct prime, or is it a facet of `symmetry_breaking` / `phase_diagram`?* — the proposer's answer is that the quantitative emergent measure, decoupled from both the qualitative event and the region map, is a distinct construct the corpus currently lacks.

**At incorporation.** Re-run similarity against `symmetry_breaking` (nearest neighbor / complementary event), `phase_diagram` (whose regions this observable labels and quantifies), `tipping_points_or_phase_transitions` (the transition itself, which the order parameter is the coordinate *of*, not the transition), `emergence`, and collective-behavior primes to confirm no collapse. If kept: draw related edges to `symmetry_breaking` (observable ↔ event) and `phase_diagram` (observable labels regions), and add a missing-edge note flagging the complementary observable/event relationship with `symmetry_breaking` and the labels-regions relationship with `phase_diagram` explicitly. Recommend evaluating `order_parameter` alongside `symmetry_breaking` and `phase_diagram` as a cluster. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
