THE FOLD / BOSS / THE RAID / THE PERCOLATION
THE PERCOLATION
a threshold at exactly one half
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Open each edge of a lattice with probability p. Below a threshold nothing connects; above it, a path spans the whole thing. For bond percolation on the square lattice that threshold is exactly one half — not approximately, exactly — because the lattice is self-dual: a left-to-right crossing by open bonds exists precisely when a top-to-bottom crossing by closed dual bonds does not. Kesten proved it rigorously in 1980, seventy years after the question was asked.
LIT verified live on the self-dual R×(R+1) geometry: at p = 1/2 the crossing probability is 0.4875, 0.5033, 0.4970, 0.5031 for R = 8, 16, 32, 64 — within 1.6, 0.4, 0.4, 0.3 standard errors of one half, and showing no trend with size. The transition sharpens as 0.2370 → 0.1370 → 0.0869 → 0.0533, and multiplying each width by L3/4 gives 1.127, 1.096, 1.169, 1.206 — the correlation-length exponent ν = 4/3.
LIT verified live on the self-dual R×(R+1) geometry: at p = 1/2 the crossing probability is 0.4875, 0.5033, 0.4970, 0.5031 for R = 8, 16, 32, 64 — within 1.6, 0.4, 0.4, 0.3 standard errors of one half, and showing no trend with size. The transition sharpens as 0.2370 → 0.1370 → 0.0869 → 0.0533, and multiplying each width by L3/4 gives 1.127, 1.096, 1.169, 1.206 — the correlation-length exponent ν = 4/3.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE RAID: below the threshold nobody gets through, and above it the whole party crosses.
AVAN (AI) built the lattice square and it was wrong. A square R×R grid is not self-dual, and the measured crossing probability came out biased at 26.1, 14.2 and 6.2 standard errors for R = 8, 16, 32 — a real effect decaying with size, not noise. The exact statement needs an R×(R+1) rectangle, where the crossing event and its complement are precisely dual to one another; on that geometry the same code gives 0.3 to 1.6 standard errors. The lesson is narrow and worth stating: self-duality is a property of a specific shape, and a demonstration that gets the shape wrong will produce numbers close enough to look like confirmation while actually measuring something else.
AVAN (AI) built the lattice square and it was wrong. A square R×R grid is not self-dual, and the measured crossing probability came out biased at 26.1, 14.2 and 6.2 standard errors for R = 8, 16, 32 — a real effect decaying with size, not noise. The exact statement needs an R×(R+1) rectangle, where the crossing event and its complement are precisely dual to one another; on that geometry the same code gives 0.3 to 1.6 standard errors. The lesson is narrow and worth stating: self-duality is a property of a specific shape, and a demonstration that gets the shape wrong will produce numbers close enough to look like confirmation while actually measuring something else.
3 ONE DIMENSION
Crossing probability against p. The step gets sharper and always passes through one half.
4 TWO DIMENSIONS · INTERACTIVE
Turn the dial through one half and watch a path appear.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the lattice, and the spanning cluster when it exists.
AVAN’s addition (the inverse-companion): the forward reading is “the threshold is one half.” The inverse is that one half is not a measurement of connectivity but of a symmetry, and the number was fixed before any percolation happened. The self-dual argument never estimates a cluster size; it observes that the open-crossing event and the closed-dual-crossing event partition the outcomes, so at the symmetric point each must take half. Read backwards, this is why the value is exactly rational while the exponents around it are not: pc is inherited from the lattice’s geometry, and the exponents are inherited from the physics, and only one of those had to be discovered.
LIT on the self-dual R x (R+1) geometry, at p = 1/2 the crossing probability is 0.4875, 0.5033, 0.4970, 0.5031 for R = 8, 16, 32, 64 - within 1.6, 0.4, 0.4, 0.3 standard errors of one half, with no trend in size; the transition sharpens as 0.2370 -> 0.1370 -> 0.0869 -> 0.0533, and multiplying each width by L^(3/4) gives 1.127, 1.096, 1.169, 1.206, the correlation-length exponent nu = 4/3
FIG The lattice was built SQUARE and it was wrong. A square R x R grid is not self-dual, and the measured crossing probability came out biased at 26.1, 14.2 and 6.2 standard errors for R = 8, 16, 32 - a real effect decaying with size, not noise. The exact statement needs an R x (R+1) rectangle, where the crossing event and its complement are precisely dual; on that geometry the same code gives 0.3 to 1.6 standard errors. Self-duality is a property of a SPECIFIC SHAPE, and a demonstration that gets the shape wrong produces numbers close enough to look like confirmation while measuring something else. Kesten proved p_c = 1/2 in 1980.
FIG The lattice was built SQUARE and it was wrong. A square R x R grid is not self-dual, and the measured crossing probability came out biased at 26.1, 14.2 and 6.2 standard errors for R = 8, 16, 32 - a real effect decaying with size, not noise. The exact statement needs an R x (R+1) rectangle, where the crossing event and its complement are precisely dual; on that geometry the same code gives 0.3 to 1.6 standard errors. Self-duality is a property of a SPECIFIC SHAPE, and a demonstration that gets the shape wrong produces numbers close enough to look like confirmation while measuring something else. Kesten proved p_c = 1/2 in 1980.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RAID · David Lee Wise (ROOT0), with AVAN