I agree with you about the importance of (mathematical) modeling in understaning horns. Physical models are important too, but you can make many mathematical models in a relatively short period of time. Makes it much easier to narrow down your choices to target your efforts. That way, physical models can be made of the ones you think are closest to what you want.

This topic reminds me of the struggles faced by James Watson and Francis Crick when trying to discover the structure of DNA. They worked fairly closely with Maurice Wilkins and Rosalind Franklin, who shared their experimental data with them. Linus Pauling, on the other hand, was somewhat arrogant and self-promotional. A sort of competition sprung up between Watson and Crick and Pauling as a result. Linus Pauling published the structure of the alpha helix in 1951, a fundamentally important structural component of proteins. In 1953, he published an incorrect triple helix model of DNA. But Watson and Crick weren't convinced in the model, thinking it just didn't seem right and wouldn't hold itself together. Shortly afterward, they realized the double helix structure, which seemed to fit perfectly. They published this now famous structure and were ultimately awarded the Nobel Prize for it.