Drag a mass to move it. Any edit restarts the guarantee from the current state.
The mixing
Overlap: the share of attention every mass gives to all the others equally. Locality: how much a mass prefers near, heavy neighbours. Frame pull: the frame centre as one more mass everyone attends to. Hold: how much of itself each mass keeps per step.
Readings
Spread over time
Solid: spread Δ (largest distance between any two masses, log scale). Dashed: the proved ceiling Δ₀ ∏(1 − η). The solid line can never cross the dashed one.
Default gravity against overlap
The arrangement this run started from, with the current settings, run for 60 steps at each overlap value; the dot is the current slider. DGI is The Crit's Default Gravity Index formula (centering, lock, central mass, sector symmetry) computed on the rendered field.
What is going on
Each mass is a token. Every step, each one moves toward a weighted average of all the masses, its own included. The weights come from attention: nearer and heavier masses get more. That is the operation F(X) = A(X) X studied in Russell Foltz-Smith’s notes on transformer latent dynamics.
The notes prove (Lemma 1) that if every pair of masses shares at least η of their attention, the spread of the whole group shrinks by at least a factor of (1 − η) every step. The dashed line is that guarantee, computed from the attention actually used at each step.
Two ingredients, and they do different things. Overlap makes everything agree: push it up and the masses settle into one, but wherever the crowd's weight happens to be (try Counterweight). Frame pull decides where agreement lands: add it and the one mass moves to the middle and radiates from there. In this toy, the single central mass needs both.
With overlap near zero and a local attention, masses gather into separate groups that hold apart for a long time. Not forever: any overlap above zero guarantees they meet eventually, and the dashed line says how fast at most.
Pin a mass and it stops listening: it becomes an anchor. Two anchors share no attention with each other, so η is 0 and the guarantee goes silent. Anchors are where the contraction hypothesis fails, which is how a composition can hold.
This is pure attention mixing, the restricted system the theorem covers. Diffusion image models are not this system. The likeness between the collapse here and the AI default is an analogy to play with, not a finding about any image model.
And it was tested. On the same portraits, token spread inside DINOv2 ViT-S/14 rises through blocks 1 to 6, plateaus, and falls only at the final block. AI-default portraits appear to collapse more (AUC 0.73) only because they enter the network with more disagreement between patches; matched for input spread the difference falls to 0.62, 0.51 and 0.43 across terciles. So the mechanism this toy makes visible is not what separates the two corpora. The narrow vocabulary is a property of the pictures, not of a network averaging its tokens.