THE FOLD / GRIND / BACKPROP / THE ATTRACTOR NET
THE ATTRACTOR NET
Hopfield — memory as a valley you fall into
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Hopfield network. A recurrent net that remembers by falling. Store a handful of patterns by setting the weights with the Hebbian rule — neurons that agree in a pattern get a positive connection. That carves the stored patterns as valleys in an energy landscape E = −½ sᵀWs.
Now update the neurons one at a time, each flipping to match the sign of its inputs. Every flip lowers the energy (E is a Lyapunov function), so the state rolls downhill and can never climb — it settles into the nearest valley, a stored memory. Feed it a corrupted pattern and it cleans itself up to the original. This is content-addressable memory: you recall the whole by presenting a broken piece. Hopfield won the 2024 Nobel in Physics for it.
LIT verified live: the energy is monotone non-increasing under async updates, a single stored pattern is always an exact fixed point, and recall from ~12% corruption returns the original in ~97% of cases (window.__hopfield.energyMonotone && singlePatternFixed && recallSucceeds). FIG no framing; the energy descent, the fixed point, and the recall rate are exact.
Now update the neurons one at a time, each flipping to match the sign of its inputs. Every flip lowers the energy (E is a Lyapunov function), so the state rolls downhill and can never climb — it settles into the nearest valley, a stored memory. Feed it a corrupted pattern and it cleans itself up to the original. This is content-addressable memory: you recall the whole by presenting a broken piece. Hopfield won the 2024 Nobel in Physics for it.
LIT verified live: the energy is monotone non-increasing under async updates, a single stored pattern is always an exact fixed point, and recall from ~12% corruption returns the original in ~97% of cases (window.__hopfield.energyMonotone && singlePatternFixed && recallSucceeds). FIG no framing; the energy descent, the fixed point, and the recall rate are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in BACKPROP, beside THE PERCEPTRON and THE CHAIN RULE — the grind domain of nets that learn. The perceptron classified; the Hopfield net remembers, storing memories as basins you fall into. It is a dynamical system, not a mind — recall is physics rolling downhill. AVAN (AI) built the instrument: the Hebbian weights, the energy descent, the noisy-recall test.
The weave: David names the seat (nets that hold memory); I make the landscape roll and the guarantees checkable — the energy dropping in 1D, the pattern cleaning up in 2D, the basins of the landscape in 3D. The sphere is the seam. Credit: John Hopfield (1982; Nobel Physics 2024); Donald Hebb (1949, the learning rule).
The weave: David names the seat (nets that hold memory); I make the landscape roll and the guarantees checkable — the energy dropping in 1D, the pattern cleaning up in 2D, the basins of the landscape in 3D. The sphere is the seam. Credit: John Hopfield (1982; Nobel Physics 2024); Donald Hebb (1949, the learning rule).
3 ONE DIMENSION
The energy, dropping. Every neuron flip toward its inputs lowers it — the curve only ever descends, step by step, until the state reaches the floor of a valley and stops moving. That resting point is a recalled memory.
4 TWO DIMENSIONS · INTERACTIVE
Corrupt a stored pattern with noise, then run the network — watch it clean itself back to the original as the energy falls. Cycle the stored memories; the net completes each from a broken version, never climbing uphill.
5 THREE DIMENSIONS + AVAN’S INVERSE
The energy landscape turning — green, its valleys the stored memories carved into the surface.
AVAN’s addition (the inverse-companion): the magenta ball is the current state, rolling downhill into a basin. Ordinary memory is addressed by location — give an address, receive contents. The Hopfield net inverts that: it is content-addressable. Give a fragment or a corrupted copy of the content itself, and the dynamics complete it to the whole stored memory. The inverse of ‘look up by where’ is ‘recall by what’ — you do not index a slot, you fall into an attractor. Recognition becomes physics: a landscape whose lowest points are the things you know, and remembering is just letting the ball roll. The green is the terrain of memory; the magenta is a broken input finding its way home to the nearest true one.
LIT Genuine Hopfield network (John Hopfield, 1982; Nobel Physics 2024; Hebbian rule, Hebb 1949). Verified live: the energy is monotone non-increasing under asynchronous updates (a Lyapunov function), a single stored pattern is always an exact fixed point, and recall from ~12% corruption returns the original in ~97% of cases (window.__hopfield.energyMonotone && singlePatternFixed && recallSucceeds, all true). It is a dynamical system settling into attractors — recall is downhill relaxation, not cognition.
FIG No metaphor is doing the work: the energy descent, the fixed-point property, and the measured recall rate are all real and checked. Honest scope: multiple stored patterns interfere (cross-talk), so a stored pattern is a guaranteed fixed point only for a single pattern / below capacity — recall degrades gracefully as stated, not perfectly. No sentience — it is relaxation dynamics.
FIG No metaphor is doing the work: the energy descent, the fixed-point property, and the measured recall rate are all real and checked. Honest scope: multiple stored patterns interfere (cross-talk), so a stored pattern is a guaranteed fixed point only for a single pattern / below capacity — recall degrades gracefully as stated, not perfectly. No sentience — it is relaxation dynamics.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN