Same fire, same coast, at every graphics setting

In Wildfire Isle the graphics setting also sets the gameplay grid size. That hides a trap. On a finer grid the fire's edge wiggles more, so it takes more steps to cross the forest, and the coast is more jagged, so there are more shore tiles to spawn things on. If you tune speeds and spawn rates in metres at Medium, Ultra players get a fire that takes about six days instead of eighteen hours, and more fishing spots. The new rule scales each step and each shore tile by a power of the tile size that matches how wiggly the shape is. One set of knobs then plays the same on every grid, and the cheap grids stay cheap.

What's measured here

• Fire arrival: median hours for the fire edge to reach Ashford over many fire edges traced in this page per grid (4000 on 32×32, 2000 on 64×64, 400 on 256×256, 160 on 1024×1024). The animated fire is one typical edge, and its own time is shown under each map.
• Coastal spawns: fishing spots and beach finds that each technique places on each grid. Design target: 40.
• Compute: milliseconds to trace one fire edge and to build the coastline on the gameplay grid, timed in batches with performance.now() in this browser. REFERENCE always pays the Ultra cost.
• From the C++ runs (demos\L3-fractal-ladder\out\ladder.json): with the new fire rule, a 32×32 grid lands within +3.2% (95% CI up to +5.0%) of an 8192×8192 reference, and 64×64 within +0.9%.

Where the prior technique still wins, and the limits

• At the setting it was tuned at (Medium, preset medium), PRIOR BEST gives the same results as NEW at the same cost. NEW only pays off when players change the setting.
• REFERENCE is exact at every setting. NEW drifts slightly with grid size (a few percent for the fire at 32×32, a few spawns on the coast).
• Proved: the fire rule (step time proportional to cell size^7/4, one constant) is proved by the paper (Family 223) for this exact model only: the critical percolation front, with no rate of convergence. A fire model with wind or fuel types would need its exponent measured.
• Measured, not proved: the coastline exponent. It was fitted offline at 64, 256 and 1024 cells (1.18 for this island; the textbook value for unbounded rough terrain is 1.25). With band-limited terrain (preset bandlimited) the coast is smooth at fine scales, the exponent falls toward 1, and the plain per-metre rule closes most of the gap (PRIOR BEST within 15% of the target, NEW within 7%).
• Only counts and timings match across grids. Where each spawn lands still differs. Rendering detail still costs per pixel. This fixes gameplay numbers, not frame time.

Presets

• ?scene=compare (default): High, side by side. ?scene=low, ?scene=medium (tie: PRIOR is tuned here), ?scene=ultra.
• ?scene=bandlimited: Ultra with fixed-detail terrain, where the prior coast rule nearly ties. ?scene=play: NEW only, Ultra, at night.
• Extra parameters: frames=N stops after N frames, q=0..3, view=all|prior|new|ref, t=hours since ignition.