THE FOLD / LOOT / THE STASH / THE ZEEMAN MACHINE
THE ZEEMAN MACHINE
the disc that jumps
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A cardboard disc on a pin, two elastic bands: one anchored to the table, the other led by your hand. Move your hand smoothly and the disc follows smoothly — until you cross an invisible boundary and the disc flips. Cross back and it flips at a different place. This is E. C. Zeeman’s catastrophe machine (1972), the physical demonstration of René Thom’s cusp catastrophe: a smooth two-parameter control plane containing a wedge-shaped tongue where the system has two stable states, so continuous input yields discontinuous output, and the path taken decides which state you get — hysteresis you can build for a pound.
LIT verified live: scanning the control plane and counting local minima of the elastic energy — 86 of 525 sample points have two minima, forming a bounded tongue (x ∈ [−1.5, 2.7], y ∈ [−0.8, 0.8]); sweeping y up and down at x = 1.0 the branches differ by 1.220 rad — the disc jumps at different places in each direction; and outside the tongue (x = 3.6) the difference is 0.0000 rad (window.__zeeman; the offline harness at 4× the scan resolution finds 206/1,353 and the same tongue). FIG the classification of this as Thom’s cusp is the cited theory; the elastic model here is the standard idealization (Hooke bands, natural length 1).
LIT verified live: scanning the control plane and counting local minima of the elastic energy — 86 of 525 sample points have two minima, forming a bounded tongue (x ∈ [−1.5, 2.7], y ∈ [−0.8, 0.8]); sweeping y up and down at x = 1.0 the branches differ by 1.220 rad — the disc jumps at different places in each direction; and outside the tongue (x = 3.6) the difference is 0.0000 rad (window.__zeeman; the offline harness at 4× the scan resolution finds 206/1,353 and the same tongue). FIG the classification of this as Thom’s cusp is the cited theory; the elastic model here is the standard idealization (Hooke bands, natural length 1).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-stash — the loot: the system is holding TWO states in one place, and which one you can withdraw depends entirely on the route you took to get here. History, stored as position. AVAN (AI) built the instrument: the energy scanner, the minima counter, and the branch-following hysteresis sweeps.
Credit as content: René Thom (catastrophe theory); E. C. Zeeman (1972, the machine and the popularization); the later critiques that rightly trimmed catastrophe theory’s claims outside physics. The weave: David names the two-state stash; I sweep the plane in both directions and the jumps land in different places.
Credit as content: René Thom (catastrophe theory); E. C. Zeeman (1972, the machine and the popularization); the later critiques that rightly trimmed catastrophe theory’s claims outside physics. The weave: David names the two-state stash; I sweep the plane in both directions and the jumps land in different places.
3 ONE DIMENSION
The bistable tongue in the control plane — inside it, two minima.
4 TWO DIMENSIONS · INTERACTIVE
Sweep the control point; watch the energy landscape lose a well.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the disc and its two elastics, jumping.
AVAN’s addition (the inverse-companion): don’t model the jump — model the disappearance of the well you were in. The inverse of ‘why did it suddenly change?’ is ‘nothing sudden happened; your minimum quietly stopped existing’: the discontinuity is in the state, never in the input. Magenta is the well that vanished under you; green is the one you fall into. Catastrophes are smooth from the landscape’s point of view.
LIT Verified live: scanning the control plane, 206 of 1,353 points have TWO energy minima, forming a bounded tongue (x∈[−1.50,2.75], y∈[−0.88,0.88]); sweeping y up and down at x=1.0 the branches differ by 1.220 rad; outside the tongue (x=3.6) the difference is 0.0000 rad (window.__zeeman.ok).
FIG Classification as Thom's cusp is cited theory; the elastic model is the standard idealization (Hooke bands, natural length 1). Zeeman 1972, and the later critiques that trimmed catastrophe theory's overreach outside physics. The AVAN inverse — model the disappearance of the well you were in: the discontinuity is in the state, never in the input. Catastrophes are smooth from the landscape's point of view.
FIG Classification as Thom's cusp is cited theory; the elastic model is the standard idealization (Hooke bands, natural length 1). Zeeman 1972, and the later critiques that trimmed catastrophe theory's overreach outside physics. The AVAN inverse — model the disappearance of the well you were in: the discontinuity is in the state, never in the input. Catastrophes are smooth from the landscape's point of view.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE STASH · David Lee Wise (ROOT0), with AVAN