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OBSERVATION ROUTE

CURRENT: 76 CATASTROPHE SURFACE OBSERVATORY

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10 EQUATION
26 PHASE
48 ROOT BRAID
76 GROWTH / CATASTROPHE EQUILIBRIA

Catastrophe Surface Observatory

Follow real equilibrium branches as folds, cusps, and hysteresis emerge from a quartic potential.

catastrophe / equilibria Observation Model

ROUTE 76 / GROWTH / CATASTROPHE / EQUILIBRIA

REAL EQUILIBRIUM SURFACE

Where stable forms fold

Move through the control plane (a,b) and read the real equilibria of a quartic potential. Stable and unstable branches meet at folds; the cusp is the point where the fold curve closes.

READY

Click the control plane to set (a,b). Focus the canvas: arrow keys adjust parameters, +/− changes the sweep, Space runs or pauses, S steps, and R resets.

Ready. The initial CUSP reading is finite and deterministic.

a control: —
b control: —
discriminant D: —
equilibria: —
stable: —
unstable: —
branch: —
finite: —

EQUILIBRIUM CONTROLS

Tune the potential

CUSP

PRESET

5 models

Every preset is deterministic. The real roots are sorted before stability is read.

CONTROL PLANE

a 0.00 · b 0.00

a quadratic control0.00 b tilt control0.00

The fold curve is 4a³ + 27b² = 0. Parameters have no physical units.

OBSERVE

local only

LENS

SURFACE

KEYBOARD / 44PX TARGETS

←↑↓→ adjust a,b · +/− sweep step · Space runs or pauses · S steps · R resets. Reduced motion keeps the chamber static; hidden tabs pause the sweep.

EQUILIBRIUM READOUT

Read every real branch

FINITE

BRANCH	x	V(x)	curvature κ	stability	residual

A neutral root marks a fold or cusp. It is not labelled stable or unstable. The model is a scalar potential contract, not a thermodynamic measurement.

DETERMINISTIC SELF-TEST

Numerical acceptance

PASS

5 FIXTURES / CUSP POTENTIAL / REAL ROOTS

CHECK	VALUE	LIMIT	RESULT

Observation report

OBSERVATION CONTRACT

Real equilibria, not complex braids

The primary object is the real equilibrium surface of a quartic potential. The fold curve and cusp are singularities in the control plane. This is not a complex-root continuation, a braid, an ODE trajectory, a particle model, or a thermodynamic measurement.

WHAT TO WATCH

Branches appear and disappear

ONE WELL has one stable minimum.
DOUBLE WELL has two stable minima separated by one unstable maximum.
FOLD shows a neutral double root on the discriminant.
HYSTERESIS follows a stable branch during an up-and-down b sweep.

MODEL NOTE

A two-control cusp potential

V(x;a,b) = x⁴/4 + ax²/2 + bx
Vₓ = x³ + ax + b = 0
D = −4a³ − 27b²

Curvature κ = 3x² + a classifies each real equilibrium. The discriminant separates one-root and three-root regions; the cusp is at (a,b) = (0,0).

Formula note

For fixed controls (a,b), solve x³ + ax + b = 0 and sort the real roots. A root is stable when κ > 0, unstable when κ < 0, and neutral when |κ| is within the deterministic tolerance. The fold locus is 4a³ + 27b² = 0; its two arms meet in the cusp.

Hysteresis is a quasi-static branch-following reading. The previous stable root selects the nearest current stable root. A jump is recorded only when that branch has disappeared. No physical time or noise is implied.

OBSERVATION GUIDE

Five lenses, one model

Start at CUSP, choose DISCRIMINANT, then move a negative. Choose DOUBLE WELL and sweep b to see the stable branches separate.

Use FOLD to inspect a neutral double root and its small residual.
Use HYSTERESIS, then SWEEP B. The report records branch jumps and loop area.
Use REPORT or PNG after any parameter change. RESET clears the sweep history and report output.

Runs inside the browser with no upload and no registration.

OBSERVATION POLICY

This chamber runs a deterministic real equilibrium model in your browser. Parameters, roots, reports, and images stay local. It does not measure visitors, physical systems, or thermodynamic quantities.

Route

This chamber belongs to Growth & Thresholds.

GROWTH Back to Growth & Thresholds Route.