---
slug: catastrophe
name: Catastrophe
one_liner: "The geometry of a discontinuous jump: how a SMOOTH change in one or more control parameters drives a system's equilibrium state to fold back on itself and jump abruptly, classified into a small set of stable shapes (fold, cusp, ...) with named signatures — bimodality, hysteresis, divergence, inaccessibility."
type: prime
schema_version: 2
status: density_v2_draft
categories: [structural, mathematical, dynamical]
origin_domain: dynamical_systems
alternate_origin_domains: [ecology, psychology, economics, structural_engineering]
aliases: [catastrophe_theory, cusp_catastrophe, fold_catastrophe, discontinuous_jump_geometry, control_surface_fold]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-08 threshold/phase, flavor 8a gradual→tipping). Op-08 was RETIRED AS SATURATED — the threshold/phase region is one of the most-covered in the corpus and the pilot yielded near-zero; `catastrophe` is one of only 2 genuine domain-general gaps op-08 surfaced. Discovered/confirmed absent 2026-07-03. MODERATE hold. STATUS: HOLD pending owner keep/cut — the heavy overlap with `tipping_points_or_phase_transitions` is the dominant concern and is flagged prominently below."
- "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 `tipping_points_or_phase_transitions` FIRST (est. ~0.62–0.70, HIGH) to settle whether `catastrophe` is a distinct prime, a child/refinement, or a fold; also screen vs `hysteresis`, `bifurcation` (corpus aliases this to tipping_points), `regime_change`, `metastability`, `symmetry_breaking`. If kept, draw `tipping_points_or_phase_transitions` → `catastrophe` (child/refinement). 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 — an equilibrium surface that folds back on itself over a smooth control-parameter space, so that the state becomes a multi-valued function of the controls and a smooth push in a control forces a sudden discontinuous jump — is a genuinely substrate-neutral piece of geometry. It recurs, with the same named signatures (bimodality, hysteresis, divergence, inaccessibility), across structurally-unlike domains: phase transitions and structural buckling in physics/engineering, population collapse in ecology (the spruce budworm cusp), fight/flight attitude switching in psychology (Zeeman''s dog), and market crashes in economics. domain_breadth scores 5 on this spread of unlike siblings. structural_abstraction is 4 because the core is a pure claim about the topology of an equilibrium manifold over a control space — no material substance required — but stays short of 5 because the vocabulary still leans on "control parameter" and "state," a mild dynamical-systems residue. transfer_evidence is 5: catastrophe theory is a canonical named framework (Thom 1972; Zeeman 1977) explicitly applied and re-derived across all these domains.'
similarity_to_nearest_existing_prime:
  slug: tipping_points_or_phase_transitions
  similarity: 0.66
structural_framed:
  label: structural
  aggregate: 0.35
  criteria: {vocab_travels: 0.75, evaluative_weight: 0.1, institutional_origin: 0.1, human_practice_bound: 0.15, import_vs_recognize: 0.4}
  rationale: 'Reads structural. The core — an equilibrium surface folding over a control space to produce a discontinuous jump with fixed qualitative signatures — is recognized rather than imported, and travels cleanly into physics, ecology, psychology, and economics without carrying any one domain''s vocabulary. There is essentially no evaluative or institutional residue: the object is a piece of manifold geometry. The only framing weight is import_vs_recognize and vocab_travels off the floor because the canonical presentation leans on the dynamical-systems terms "control parameter" and "state variable," but the underlying shape is substrate-neutral. Comparable framing to its parent `tipping_points_or_phase_transitions`.'
phase_b_passes_completed: []
---
# Catastrophe

## Core Idea

A catastrophe is the *geometry* of a discontinuous jump. Catastrophe theory (Thom 1972; Zeeman 1977) describes how a *smooth* change in one or more **control** parameters can drive a system's equilibrium state to a *sudden, discontinuous jump*, and classifies the small number of stable shapes this geometry can take — the elementary catastrophes: fold, cusp, swallowtail, and a few higher forms. What it adds beyond a bare threshold is a *manifold*: over the space of controls, the equilibrium state folds back on itself, so that the state is a **multi-valued** function of the controls. Crossing the fold — a smooth trip in control space — is what produces the jump. A plain tipping point names the *fact* of an abrupt change at some critical value; catastrophe names the *shape of the surface* that makes the change abrupt.

The load-bearing anatomy is threefold. (1) A **control-parameter surface**, often of two or more controls, over which the equilibrium folds — the cusp catastrophe, the workhorse case, has two controls (a "normal" or splitting factor and a "bifurcation" or asymmetry factor) whose interaction opens a wedge-shaped region of the control plane where two stable states coexist. (2) A set of named **qualitative signatures** that betray the fold even before it is drawn: *bimodality* (two stable states available at once), *hysteresis* (the jump-up and jump-down thresholds differ, so the state depends on history not just present controls), *divergence* (nearby control paths end in qualitatively different states), and *inaccessibility* (a middle, unstable branch the system can never occupy). (3) The **cusp point** itself, where two fold lines meet — the organizing center from which the bimodal region opens. It is this geometry, not merely the existence of an abrupt change, that catastrophe carries.

Read structurally, the same fold appears across genuinely unlike domains, always with the same signatures. In *physics and engineering*, it is a phase transition or the buckling of a loaded strut — the state jumps as a smooth load crosses the fold. In *ecology*, it is the spruce-budworm cusp: a slowly rising forest resource smoothly pushes a pest population until it jumps to outbreak, and cannot be returned by simply reversing the resource (hysteresis). In *psychology*, it is Zeeman's aggression model — a dog's smoothly varying fear and rage place it on a cusp surface whose fold produces a sudden switch between flight and fight. In *economics*, it is a market crash: sentiment and fundamentals move smoothly, then the price jumps. Across all of these the shared abstract object is one folded equilibrium manifold over a control space; the domain content is interchangeable, and the fold geometry is the invariant.

## Notes

**Provenance.** Produced by operator 08 (threshold/phase), flavor 8a (gradual → tipping), on 2026-07-03. Op-08 was *retired as saturated*: the threshold/phase region is one of the most heavily-covered parts of the corpus (`tipping_points_or_phase_transitions`, `hysteresis`, `metastability`, `regime_change`, `symmetry_breaking` are all present), and the op-08 pilot yielded near-zero new domain-general primes. `catastrophe` is one of only *two* genuine domain-general gaps the operator surfaced. Established framework: René Thom, *Structural Stability and Morphogenesis* (1972); E.C. Zeeman, *Catastrophe Theory: Selected Papers* (1977); cusp / fold catastrophe. Verified absent from the corpus as a named prime. Manifest: MODERATE hold.

**For owner evaluation (keep/cut) — the tipping_points overlap is the dominant concern, stated candidly.** This is a *close refinement* of an existing prime, and the keep/cut turns on that overlap. The case FOR: catastrophe theory is a canonical, named, cross-domain framework with genuinely richer structure than a bare threshold — it adds (a) the *control-surface geometry* (a folded, multi-valued equilibrium manifold over 2+ controls), (b) the *elementary-catastrophe classification* (fold, cusp, swallowtail, ...), and (c) the four named *signatures* (bimodality, hysteresis, divergence, inaccessibility). The corpus has `tipping_points_or_phase_transitions` but does not name any of this geometry. The case AGAINST is real and should be weighed seriously: similarity to `tipping_points_or_phase_transitions` is estimated HIGH (~0.62–0.70), and a reasonable owner may judge that a tipping point *already presupposes* this geometry and that `catastrophe` is therefore a child/refinement rather than a peer — or may fold it entirely, treating the fold-geometry as an elaboration to write into the `tipping_points` body. Note too that one of catastrophe's four signatures, `hysteresis`, is *itself already a prime*, and that `bifurcation` — arguably the mathematical heart of the fold — is aliased by the corpus into `tipping_points`; this raises the risk that catastrophe is a *bundle of already-present pieces* rather than a new atom. Sharp keep/cut question: **is "the folded control-surface geometry of a discontinuous jump" a distinct prime, a child of `tipping_points`, or a fold into it?** The proposer's lean is *child of tipping_points* — distinct enough to name (the geometry and classification are absent), but subordinate to the threshold/phase-transition prime rather than a peer.

**At incorporation.** *Re-run similarity against `tipping_points_or_phase_transitions` FIRST* — this single check should drive the keep/cut, and if it lands at or above the proposer estimate, treat the child/refinement or fold options as live before the distinct-prime option. Then screen vs `hysteresis` (present — one of catastrophe's signatures; confirm catastrophe is not merely a restatement of it), `bifurcation` (corpus alias → `tipping_points`), `regime_change`, `metastability` (bistability without the control-surface geometry), and `symmetry_breaking`. If kept as a distinct prime, draw a subsumption/refinement edge `tipping_points_or_phase_transitions` → `catastrophe` and composition edges from `hysteresis` (and, if surfaced, `bistability`/`metastability`) into `catastrophe` as component signatures. Add a missing-edge note recording the `hysteresis`-as-signature and `bifurcation`-alias relationships explicitly so a later owner does not re-litigate the overlap. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
