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I have a task of proving my software design to be right in a some scientific way. I asked a question about it (https://softwareengineering.stackexchange.com/questions/272932/software-architecture-design-basic-literature), but it was ill-formed and was putted on hold.

Can I rephrase it to something like "Scientific rationale for the choice of software architecture. How can it be done?" and re-ask?

Is there answers to such question? Or there is no silver bullet and that's it?

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There are no silver bullets in software engineering, only choices that are better or worse for some aspects of software architecture.
To top it off, even the best architecture can be ruined by a bad implementation, so even a scientifically validated architecture doesn't buy you that much.

What remains are a huge pile of trade-offs that you as architect have to make the best of.

Questions covering the whole architecture are too broad for the Stack Exchange Q&A format, but what you can ask about is what you need to take into account in your architecture for some aspect of it and how you might be able to measure that the final product has achieved sufficient quality in that aspect.

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