Coastlines come from a 1 km global land mask, depths from Natural Earth bathymetry contours, and the satellite option draws NOAA's ETOPO1 natural-colour relief beneath the simulation, both block-averaged onto the chosen grid. The front advances by contact at the chosen speed (default 15 km/h, about 4 m/s: ordinary ice grows into water supercooled by 15–20 K at tens of centimetres per second, and the sea is 40–45 K supercooled relative to ice-nine), independent of the grid and the time step. Every freeze dumps its heat of crystallisation into its own column; lateral eddy mixing over tens of kilometres moves nothing on these timescales. The column then sheds heat through its surface at bulk air-sea rates (10 W/m²/K sensible, 22 W/m²/K latent at 20 °C doubling every 12 K, 5 W/m²/K net longwave) toward its climatology shifted by the air's anomaly; the turbulent part warms the air, the longwave part leaves directly, and the diurnal cycle modulates the loss by about ±20%. Water above 45.8 °C cannot take the lattice. Each column carries a solid fraction: once seeded it locks at up to half its depth per hour while it is below 45.8 °C, and if excess heat arrives it melts back only as far as that heat allows, taking the latent heat back from the water and the air. An enclosed sea therefore settles at 45.8 °C part-frozen and finishes freezing as the air carries heat away.
Depth. A column's heat capacity scales with its depth, so a 100 m shelf swings tens of degrees from heat that a 4 km abyssal column barely notices, and the shelf also cools back fast. A column keeps its own heat, so a shelf stalls at the melting point long before the abyss beside it. This stands in for the vertical dimension: ice-nine takes the whole column on contact, so what matters is how much water each column has to soak up the heat, not where in the column it sits. A full 3D model would add a thermocline and deep currents, which change the timing of the resumption, not its character.
Floats or sinks. The book never says whether ice-nine is denser than water, and it decides the ending. By default the lattice sinks: crystals fall through the column and grow as they fall, a column is slush top to bottom within hours, and one temperature per column is a fair description. With the toggle on, the crust floats: each ocean column is a 100 m mixed layer over a deep layer with its own temperature and salinity, and everything the front does happens in the mixed layer. The solid rejects salt, so the liquid under a growing crust becomes both warmer and saltier; when it is denser than the water below (UNESCO equation of state), the two layers overturn and mix heat and salt, and the deep water can freeze against the crystals before the crust is complete. Polar surface water starts slightly fresher, as in the real ocean, so the initial ocean is stable. The melting point follows the liquid's salinity and stops falling where the brine saturates. Without overturn, the layers exchange heat only by eddy diffusion (about 10⁻⁴ m²/s) or conduction through the crust (about 10⁻⁶ m²/s). Ocean frozen is by mass; Area locked is the surface; the line under it shows the share of the deep ocean overturning.
Snowfall. Spray lofts crystals from seeded water; they travel with the air and rain washes them out within about a week. One crystal freezes a drop, so rain falling through crystal-laden air below 45.8 °C lands as ice-nine: it whitens the ground (2 cm of water hides it), seeds lakes and rivers, releases its latent heat into the air, and melts back where the air is above the melting point. Precipitation over land is today's map (GPCC, 1° cells), scaled by how much the ocean evaporates, which rises steeply over a hot sea and falls as the ocean crusts over. White land raises the planetary albedo three quarters as much per square metre as white sea. The line under Rivers and lakes shows the land covered, the water locked on land as metres of ocean, and the rain rate against today's.
The atmosphere is a 1° grid of air columns, each holding 1/1550 of the heat of a 3700 m ocean column: the sea's turbulent heat loss warms it, it mixes with its neighbours (eddy diffusivity 10⁶ m²/s, scaled by the mixing slider) and drifts with zonal winds of up to 8 m/s, radiates to space through a Planck response of 3.3 W/m²/K weakened by the water-vapour greenhouse (1.3 W/m²/K today, growing as the air warms; a 35-day timescale at today's temperature, against 10⁷ J/m²/K), and warm air raises the temperature the sea beneath it cools toward, so a frozen Atlantic no longer heats the Pacific directly. Seawater's salt lowers the melting point by 2.2 °C at 45.8 °C (the cryoscopic constant scales with the square of the melting temperature), the brine left behind as a column locks lowers it as 1/(1 − solid fraction), and at about seven times seawater the brine saturates and the depression stops at 16 °C. Water above 100 °C boils off as steam, and locked columns hold about half the heat per degree that liquid water does. Ordinary sea ice is drawn where it would be for the date, with a longitude-dependent edge (open Barents Sea, frozen Hudson Bay and Okhotsk, a northward Weddell lobe) and a patchy marginal zone; it is a metre or two over a kilometre of water, so the lattice runs through its brine channels as through open water and the heat budget changes by a hair. The Greenland and Antarctic ice sheets are modelled as ice columns (300 m at the coast to 3 km inland) with their own temperature and half of water's heat capacity. Converting ordinary ice releases only the difference between the two latent heats (55 kJ/kg at the default), but over a 3 km column that is enough to warm the ice by some 26 °C. Through cold, dry-based ice the lattice advances only by solid-state transformation at the chosen rate; where the bed is wet (about 55% of Antarctica and 35% of Greenland, patchily) it runs through the basal water at a third of the liquid rate; and once a column is above 0 °C its remaining ordinary ice melts and converts at the full liquid rate, so a sheet that warms past freezing finishes itself. Ice shelves are kept as sea. About a third of the ocean freezes on its own heat within weeks; the rest is paced by radiation to space, so the complete freeze takes decades at the highest time step. A white surface reflects sunlight: 102 W/m² when the whole ocean is white (planetary albedo 0.3 to 0.6), so the end state is a snowball. Latent heat flux grows sharply with sea temperature and is the main channel from sea to air. Airborne seeding lets storms loft microcrystals downwind so the lattice can jump coastlines. Surface currents are built from the major gyres (narrow on their western sides, which is what makes the Gulf Stream, Kuroshio, Brazil and Agulhas currents strong), the Antarctic Circumpolar Current and the equatorial currents; they carry heat downstream and sweep crystal fragments off a part-frozen front, so a seed in the Caribbean rides the Gulf Stream toward Europe. Ice at the surface stops the flow, and each circulation weakens with its basin's open-surface share, so freezing the Florida Straits stalls the whole North Atlantic gyre. Rivers (Natural Earth centerlines) and lakes (water not connected to the ocean) are freezable channels: the lattice runs up a river from its mouth and into the lakes it drains, at the same contact rate as at sea, and inland water is tallied separately since its mass is a ten-thousandth of the ocean's. Not modelled: groundwater, deep circulation, and the thermocline. Ocean mass 1.4 × 10²¹ kg, area 361.9 M km². Nothing is compressed: the constants are the physical ones (derivations in the repository's agent_docs/physics.md).