Category: Physics

Summary: Finding the prestress gradient needed to recapture a soft shell mode after disorder has spread it into the bulk.

Shell-like mechanical structures can guide soft vibrational activity onto specific regions, but disorder can destroy that confinement. This experiment asks whether a smooth prestress gradient can reverse that loss and pull the softest mode back onto the shell after it has spread into the bulk.

The script constructs dense symmetric shell-core stiffness matrices with fixed disorder and then bisects a gradient parameter that varies prestress from inner to outer regions. By repeating the same threshold logic at larger sizes, it estimates when relocalization of the soft mode becomes robust.

That makes the problem a rescue experiment rather than a simple fragility test. It is aimed at understanding whether smooth mechanical tuning can restore spatial control once disorder has already degraded it.

Method: Dense symmetric eigensolve with iterative deepening and bisection on a smooth prestress-gradient parameter.

What is measured: Critical relocalization gradient, shell localization of the softest mode, minimum eigenvalue, and bracket width.