The original note, as written at the time
Imagine a material whose tensile strength increases when its temperature rises. Instead of the distances between its molecules increasing, the entanglement of its polymers becomes denser.
Ongoing development
A material whose tensile strength increases as its temperature rises — because instead of the distance between its molecules increasing, the entanglement of its polymers becomes denser.
Almost every material we build with gets weaker when it gets hot. Steel loses stiffness, concrete spalls, polymers soften, and the whole discipline of fire engineering exists because of it. A material that strengthens with heat would invert a foundational assumption.
The mechanism you describe is a topological one rather than a chemical one, and that is why it is plausible. Thermal expansion pushes things apart. Entanglement pulls them together. If the second dominates as temperature rises, the material tightens under heat, and strength follows density of entanglement.
The uses are almost all protective. A cable that holds better in a fire. A containment vessel that seals harder as it heats. A roof that does not sag in the afternoon. Each of these is a place where the current material behaviour is the thing being designed around.
The problem is the same property in reverse. A material that tightens when heated also stores energy, and a material that stores energy can fail suddenly. Every benefit is a stress that has to go somewhere.
There is also a manufacturing question that decides everything: can it be made in a sheet, a bar and a cable, or only one of them? A material that only exists as a film is a curiosity.
Materials that behave contrary to expectation are how engineering moves. This is a good one to chase.