Imagine a layer of fluid that generates heat throughout its interior. Both its floor and ceiling are kept cold. Warm fluid rises, so convection tends to carry more heat upwards. But can it make the heat escaping through the floor arbitrarily small?

This mathematical candidate gives stronger limits on how small that bottom heat loss can become. It studies an idealized regime called infinite Prandtl number, where the velocity responds through a Stokes equation. The horizontal size is fixed, and the solutions are assumed to remain classical.

The answer depends on the boundaries. With rigid, no-slip plates, the guaranteed heat loss decreases roughly as the inverse cube root of the heating parameter, with a logarithmic correction. With stress-free plates, the bound decreases as the inverse square root. These improve the powers in earlier unconditional bounds, within the stated fixed-domain setting.

The proof starts with a minimum principle: temperature stays nonnegative. A pointwise estimate then limits how strongly fluid motion can carry heat at each height. An integrating factor turns that local restriction into a lower bound on the heat leaving the base. Finally, a temperature ceiling removes an unknown feature of the particular flow from the estimate.

The distinction between proof and calculation matters. The exponents follow from written analytical arguments. Sampled decimal prefactors are estimates, not certified numerical bounds. The stored simulations cover finite time windows and do not establish long-time statistical convergence.

The review also uncovered a useful caution. One temperature profile appeared to satisfy an inequality at every sampled grid point, yet exceeded it very close to the upper wall. New boundary-slope and between-grid checks detect that effect. It does not disprove the corresponding stationary hypothesis.

The result does not show that any actual flow reaches these bounds, and it does not solve the finite-Prandtl or arbitrary-domain problem. The manuscript and supporting checks are available on Evidence Press as an unrefereed candidate. This is synthetic speech.