Measured by the lab
0.592914
Is there a sharp threshold at which random, independent site occupation suddenly connects across a whole lattice — a phase transition made of pure geometry, with no energy, no dynamics, no temperature?
Units: dimensionless occupation probability (square-lattice site-percolation threshold; no closed form — Jacobsen 2015 graph-polynomial value 0.59274605079210(2))
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A phase transition with no dynamics at all: random occupation of a square lattice snaps into edge-to-edge connectivity at p_c = 0.592914 recovered vs 0.59274605 known (|Δ| = 1.7e-4) — by SELF-BOOTSTRAPPED finite-size scaling: the transition WIDTH shrinks as L^(−1/ν) giving ν̂ = 1.367 (known 4/3, never plugged in), and that measured ν̂ extrapolates the crossing points to L = ∞. The decisive pair: the SAME pipeline on the triangular lattice recovers the EXACTLY known p_c = 1/2 (Sykes–Essam star-triangle duality) to 1.1e-4 with the SAME exponent (ν̂_tri = 1.370) — the threshold moves with the lattice geometry, the exponent does not (universality). Below p̂_c the largest cluster VANISHES with L (frac 0.091 → 0.016 from L = 24 → 96: only finite islands); above, it is intensive (0.679 → 0.699: an 'infinite' cluster). A 1-D chain's threshold escapes to 1 (N/(N+1) exactly) — dimension, not randomness, makes p_c < 1.
Every site of an L×L lattice gets an independent uniform threshold u_i; sites are added in increasing-u order into a union-find forest with two virtual nodes (TOP glued to row 0, BOTTOM to row L−1). The u of the site whose addition first connects TOP to BOTTOM is the EXACT spanning threshold of that realization — one O(N log N) pass per lattice, no p-sweep (the threshold form of Newman–Ziff 2000). Across 9,600 realizations per size at L = 16…128: p̄(L) = mean spanning threshold, σ(L) = its width. The recovery is SELF-BOOTSTRAPPED: ν̂ is measured from the log-log slope of σ(L) vs L, then the crossing points p̄(L ≥ 24) are extrapolated linearly in L^(−1/ν̂) to L = ∞. Neither 0.5927 nor ν = 4/3 appears in the recovery — both are loaded from the reference only to score (asserted on the recovery source itself). Controls: the triangular lattice (one diagonal added per cell; p_c = 1/2 EXACT by Sykes–Essam 1964), the order parameter's L-dependence on either side of p̂_c, a 1-D chain (spanning needs every site), and a headless replica of the module's own on-screen Π = ½ sweep estimator at L = 72.
spanning-threshold distribution concentrates at p_c with width ∝ L^(−1/ν); crossing points p̄(L) → p_c as L → ∞
0.000168
| L | P bar | Sigma |
|---|---|---|
| 16 | 0.5891 | 0.06327 |
| 24 | 0.591379 | 0.04762 |
| 32 | 0.591722 | 0.038544 |
| 48 | 0.591988 | 0.028175 |
| 64 | 0.591997 | 0.023211 |
| 96 | 0.5923 | 0.017178 |
| 128 | 0.592612 | 0.013871 |
true
recovered p_c by finite-size scaling fed only by its own measured exponent; p_c and 4/3 are loaded from the reference only to score, gate G asserts the recovery source contains neither, and a tamper self-test (known_value → 0.55) makes the derisk EXIT 1 (gates A/B fail at 145 SE) while the recovered 0.592914 is unchanged. All 8 derisk gates pass in ~37 s.
Broadbent–Hammersley percolation (1957): occupy each site of a lattice independently with probability p; connectivity undergoes a sharp phase transition at a critical p_c — the simplest system in statistical mechanics with a genuine critical point, made of pure geometry (no Hamiltonian, no temperature, no dynamics). This world VALIDATES the square-lattice site threshold p_c = 0.59274605 (Newman–Ziff 2000, Jacobsen 2015) and the exact 2-D correlation-length exponent ν = 4/3 (den Nijs 1979; proven via SLE₆ by Smirnov–Werner 2001), and it is arranged to be NON-CIRCULAR and SELF-BOOTSTRAPPED: the per-realization spanning threshold is exact (sites added in random-threshold order through union-find), the exponent is measured from the width scaling σ(L) ∝ L^(−1/ν), and that measured ν̂ — not the textbook 4/3 — extrapolates the crossing points to the thermodynamic limit. The triangular-lattice control lands on the EXACTLY known p_c = 1/2 (Sykes–Essam 1964) with the same ν̂ — the cleanest possible demonstration that thresholds are non-universal geometry while exponents are universality-class properties, the same structure the logistic world found in Feigenbaum's δ (same constant for the sine map, different constant for a quartic maximum). The order-parameter gate shows what 'infinite cluster' means operationally: below p_c the largest cluster's lattice fraction vanishes with system size; above, it is intensive. The 1-D control (threshold = max of N uniforms → 1) is the connectivity analogue of the absence of 1-D phase transitions. Percolation's anatomy — order parameter, diverging correlation length, universal exponents — is the same as the Ising magnet's (?world=ising) with every dynamical ingredient stripped away; it is the lab's cleanest critical point, and the union-find spanning machinery is shared with nothing else in the catalogue. Distinct from ?world=dla (growth) and ?world=sandpile (self-organized criticality): here criticality is TUNED by p, not self-organized.
npm run derisk -- percolation (scripts/percolation-derisk.mjs)scripts/oracles/percolation.reference.jsonS. R. Broadbent & J. M. Hammersley, 'Percolation processes I. Crystals and mazes', Proc. Camb. Phil. Soc. 53, 629 (1957). M. E. J. Newman & R. M. Ziff, 'Efficient Monte Carlo algorithm and high-precision results for percolation', Phys. Rev. Lett. 85, 4104 (2000) — p_c = 0.59274621(13). J. L. Jacobsen, J. Phys. A 48, 454003 (2015) — p_c = 0.59274605079210(2). M. F. Sykes & J. W. Essam, J. Math. Phys. 5, 1117 (1964) — triangular site p_c = 1/2 exact. M. P. M. den Nijs, J. Phys. A 12, 1857 (1979) — ν = 4/3. S. Smirnov & W. Werner, Math. Res. Lett. 8, 729 (2001) — exponents proven for triangular site percolation.