  An isoperimetric inequality on the Hamming cube for exponents $\beta\ge
0.50057$ is proved, achieving equality on any subcube. This was previously
known for $\beta\ge \log_2(3/2)\approx 0.585$. Improved bounds are also
obtained at the critical exponent $\beta=0.5$, including a bound that is
asymptotically sharp for small subsets. A key ingredient is a new Bellman-type
function involving the Gaussian isoperimetric profile which appears to be a
good approximation of the true envelope function. Verification uses
computer-assisted proofs and interval arithmetic. Applications include progress
towards a conjecture of Kahn and Park as well as sharp Poincar\'e inequalities
for Boolean-valued functions near $L^1$.
