The 3439 page Intel® 64 and IA-32 Architectures Software Developer’s Manual is a specification. It just isn't a formal specification suitable for use in proving properties of programs. ARM on the other hand, released a machine readable architecture specification for ARMv8 but that architecture isn't as old or as large as x86_64. Nor is Intel/AMD in the business of selling architecture licenses where having a machine readable architecture specification would be useful for verifying implementations and proving properties. Indeed they're kind of in the business of preventing that from happening.
For example the Herd tool kit can take small concurrent programs with some constraints and a memory model and then answer whether any executions are possible.
http://diy.inria.fr/doc/herd.html
Alternately there is the online tool for the same purpose for ARMv8:
The software they developed is called Sandsifter.
His other talks are quite interesting, as well!
[0]:https://github.com/xoreaxeaxeax/movfuscator [1]:https://www.youtube.com/watch?v=2VF_wPkiBJY
https://static.docs.arm.com/ddi0487/da/DDI0487D_a_armv8_arm....
which is mostly written in plain English for the common programmer, but frequently suffers due to lack of specificity or general clarity.
Also, sometimes you want to leave things undefined. This killed the attempt to create a formal specification for ECMAScript 4 back in the day. For example, it was hard to formally specify things like "Array.sort calls the sorting order callback in such a way as to make the array sorted" without mandating a particular sorting algorithm.
Note that formal specification of ECMAScript 5 does exist (JSCert), although it is not de jure and it does not handle the standard library.