The leap. Underwater, the fish accelerates over the path length available in the pool, s = D / sin(theta). With an average acceleration a it can reach sqrt(2 a s); the exit speed is the smaller of that and its muscle-limited burst speed, which the lab scales with water temperature. Once out of the water the flight is a projectile: y(x) = x tan(theta) - g x^2 / (2 v^2 cos^2(theta)), apex H = (v sin theta)^2 / 2g, range R = v^2 sin(2 theta) / g. The lab integrates the flight with RK4 including quadratic air drag F = 0.5 rho C_d A v^2 (about 0.1 percent effect for a 3.5 kg fish) and compares it with the closed forms in Check 1.
Best angle. The height at horizontal distance d is largest for tan(theta*) = v^2 / (g d), giving y_max = v^2/2g - g d^2 / 2 v^2. Check 2 finds the same angle by scanning numerically. The consequence: starting d metres back costs height; an exit speed of 7 m/s allows at most about 2.5 m straight up but only about 1.8 m at d = 2 m.
The ladder. In a vertical-slot ladder the water drops dh between pools and passes through a slot as a jet with speed v_j = C sqrt(2 g dh) (C about 0.85 here). The fish needs to swim at a speed U above v_j for the time t = L_slot / (U - v_j), and endurance falls with speed: this lab interpolates log(time) linearly between a sustained speed (0.4 of burst, 200 minutes) and burst speed (20 s), a common shape of fish fatigue curves with illustrative end points. The second test is the turbulence in the pool: volumetric power dissipation eps = rho g Q dh / V with flow Q = v_j w d must stay under a design guideline of order 150 to 200 W/m3 (typical, check local standards), otherwise fish are disoriented.
Temperature. Muscle performance is scaled by exp(-((T-15)/12)^2), an illustrative bell curve peaking at 15 C, not a measurement. Oxygen saturation uses the cubic fit 14.62 - 0.3898 T + 0.006969 T^2 - 0.00005896 T^3 mg/L for fresh water at sea level, checked in Check 3 against textbook table values.
Honest simplifications. A single fish, a vertical straight launch plane and a flat wall; no spray, current in the pool or fish tail kicks at the surface (real fish sometimes get a second push from the tail in the fall), and no behaviour such as searching for the best jump pool. Burst speeds, acceleration, endurance end points and the dissipation limit are typical orders of magnitude, not species-specific data: spot-check them against the fisheries literature before relying on any number.