The science
Construction. The castle is six stages. A stage needs W builder-hours and a fixed bill of timber, stone and gold, spent in proportion to the work done. The work rate is crew x efficiency during daylight only (06:00-18:00); efficiency = 1 + 0.35 (crane) + 0.25 (mason's yard) + 0.06 per smithy + 0.10 per master mason + 0.15 (pulleys, research). If a store runs dry, the building pauses and the limiting resource is shown. Stage time with unlimited stock is W / (crew x eff) x 24/12 hours, which the checks verify.
The crane. Raising a block of mass m through h needs work m g h; with a treadwheel of efficiency eta the human power is P = m g v / eta. A 250 kg block at 0.05 m/s with eta = 0.6 needs about 204 W, roughly three walkers at an assumed 75 W each; this is why a medieval crane lifted only a few blocks an hour. Accelerating and braking add and subtract equal work, so the total stays m g h (checked numerically).
Masonry. The pressure at the foot of a solid wall of height H is sigma = rho g H (rho = 2400 kg/m3, typical limestone). A 20 m wall gives 0.47 MPa, about 1/40 of a typical crushing strength of 20-60 MPa, so a thick stone tower fails by overturning or cracked mortar long before crushing. The game does not simulate collapse.
Population. Logistic growth dP/dt = r g P (1 − P/K) + m (1 − P/K), with K the housing capacity (cottage 4, timber house 6, manor 12, plus 6 per finished castle stage), r = 0.012 per hour, g a happiness factor and m = 0.25 settlers per hour; over capacity people leave, with no food the town shrinks. Closed form t(K/2) = ln((K − P0)/P0)/r.
Harvest and food. Farm output follows f(d) = 0.8 + 0.35 cos(2π(d − 200)/360), peaking in mid-July (mean 0.8 over the year). Unmilled grain loses 20 %; stored food spoils at 0.15 % per hour divided by (1 + 0.8 x granaries). Fishermen on a dock add food. Each citizen eats 0.125 units/h (1 unit ≈ 1 kg, illustrative).
Wind. A windmill converts P = ½ ρ A v3 Cp (rho = 1.225 kg/m3, blade radius 9 m, Cp = 0.30, below the Betz limit 16/27 = 0.593).
Labour and money. Workers are 62 % of the population minus guards, split by your sliders into builders, farmers, woodcutters, stonecutters and artisans; a workplace is staffed to min(1, assigned/jobs). Tax per citizen scales with the tax slider and happiness; market prices follow stock with elasticity 0.5 (price = base (ref/stock)0.5).
Honest limits. All prices, costs, yields, wages, the raid model and the expedition rewards are illustrative game constants, not medieval statistics; real castles took decades and thousands of workers. No fighting is shown; the defence rating is a simple comparison with a rising threat. The foreground king is an invented character. Buildings, the crane, scaffolds, ships and the king were modelled in Blender; the economy, construction and seasons run in the page.