Lab
138 explorable, interactive pieces — every concept we write about, made tangible.
Train a model on its own output, generation after generation. Each step draws a finite sample of size M and refits — so the tails starve first and the curve collapses toward a spike. Toggle replace vs accumulate: replace drives σ̂ toward zero, accumulate holds it. A copy of a copy of a copy.
A glowing marker performs literal gradient descent over a 3D loss landscape — analytic gradients, momentum, and a fading trail. Drag to orbit the terrain; click anywhere on it to drop a new starting point.
Explore how Uniswap v4's 14-bit permission bitmap maps to pool lifecycle callbacks. Toggle bits or select preset patterns to see which hook functions fire for each operation.
The KZG polynomial commitment scheme, stepped through. Pick an evaluation point z on the cubic P(x) = x³−2x²−x+2 to watch the opening proof π materialize, see the pairing check pass, then corrupt π to make it fail — and read why 50,000 gas ($0.067) is all Ethereum needs to verify any such claim.
An interactive Merkle tree: click any leaf to watch its hash cascade hop by hop to the root, with the sibling hashes that form the inclusion proof highlighted along the way — four hashes prove one of sixteen leaves.
Interactive diagram of the five-step LLM IP protocol: embed a statistical fingerprint, commit it with Poseidon, register on-chain, prove ownership with a ZK proof, enforce via smart contract. Select an attack scenario to see which steps are vulnerable.
Two random vectors in high dimensions are almost always near-orthogonal. Slide dimension d from 3 to 8192 and watch a sphere of random directions collapse onto the equator — the geometry behind DiFR's random-projection activation fingerprints.
Forward passes through a small feed-forward network, animated: signals ripple layer to layer, weights set the glow, and activations heat the neurons they reach.
Semaphore's double-signal defense, explorable: send transactions to spend month-scoped nullifiers PBH-style, replay a spent proof to watch the gate revert with the real contract errors, and roll the month to see the external nullifier change reset the quota.
Bradley-Terry vs. majority voting over pairwise comparison rounds. Five nodes with hidden quality scores compete; a malicious node is down-ranked by BT while majority voting is captured. Reference: Fortytwo GPQA Diamond — BT 85.90%, majority 68.69%.
Interactive calculator: watch how deposit-borrow loops compound leverage toward the 1/(1−LTV) asymptote. Adjust LTV and loop count to see effective leverage, the liquidation trigger, and the loss from a single price move.
Story Protocol's two royalty policies, runnable: tap any IP asset in a derivative chain to pay 100 WIP into it and watch LAP give every ancestor its absolute cut until the stack hits 100% and mints revert, while LRP compounds hop by hop and dilutes the original to dust.