The Verification Venue · pointed at a number nobody can look up

A Hundred Billion Walks, One Number

Polymer physics courses draw a handful of tangled conformations and then quote the connective constant of the square lattice from the literature. This page does not quote it. It counts every self-avoiding walk, one at a time, up to length 22 on the square lattice and 32 on the honeycomb, on your machine, in front of you, and measures the constant from its own census.

Every one of the 883 immersive pages in this corpus computed what one JavaScript thread could compute while a reader waited, and questions whose honest answer cost more than about ten million operations were never proposed, because they could not be shown. This page runs a C engine compiled to WebAssembly across every core you have, with a slow, deliberately obvious second implementation standing beside it, and it refuses to show you a number until the two agree on small cases. Drag the slider first: something true is on screen within a second. Then press Run the census and watch the estimate settle.

A self-avoiding walk is a path on a lattice that never visits a site twice. Let c(n) be the number of such walks of length n starting at the origin, with c(0) = 1: the empty walk is a self-avoiding walk. The counts grow geometrically, c(n) ≈ A·μⁿ·nᵍ, and μ, the connective constant, is the thing the whole field wants. Nobody has a closed form for it on the square lattice. What this page has is a census, and a partition that makes the census parallel without cheating: every walk of length n has exactly one prefix of length d, so handing prefixes to workers and letting each complete its own walks counts every walk exactly once, whatever d is and however many chunks there are.

The symmetry is spent honestly. The engine fixes the first step East and multiplies by 4, because the four rotations act freely on walks: the first step distinguishes them, so no walk is ever counted twice. The same argument gives 3 on the honeycomb. That fixing buys a theorem the engine was not written to produce: with the first step fixed, mirror-pairing leaves exactly one walk unmoved per length (the straight ray), so the fixed-first count is odd and c(n) ≡ 4 (mod 8) for every n ≥ 1. If the census ever violates that, the enumerator is broken, and the check panel below tests it at full size.

The constant is not the count, and getting from one to the other is where the honesty lives. Three estimators run on the same computed counts: the raw ratio c(n)/c(n−1); that ratio corrected pointwise for the power-law prefactor, with the exponent fitted from this page's own counts rather than quoted from a book; and a joint fit that estimates μ and the exponent together. At any finite n they converge to different places, and their spread is the honest uncertainty, not the last digit of the last ratio.

Then the estimator itself gets tested. On the honeycomb lattice the connective constant is exactly √(2 + √2), proved by Duminil-Copin and Smirnov (announced 2010). This page does not reproduce that proof and does not depend on it: it uses the exact value as a target for the estimator, the one place in this piece where a right answer exists to be missed. How far the honeycomb estimators sit from the exact value at n = 32 is a direct measurement of how much you should trust the square-lattice fit at n = 22, where no such target exists.

Census parameters, pinned for the verifier: square lattice to length 22, honeycomb to 32; prefix depth 8 with 256 chunks (main pass) and depth 5 with 97 chunks (check pass); symmetry factors 4 and 3; one-step walks: 4; the line lattice: exactly 2 at every length; honeycomb target √(2+√2) = 1.8477590650225735.