Huh, I've seen something vaguely similar in a physics textbook: the authors "derive" the Schrodinger equation by describing the properties that an equation has to have in order to describe an object (such as a single photon) that "interferes with itself" in the double slit experiment. Another textbook I've read simply says that "derivations" of the Schrodinger equation are basically bogus; the Schrodinger equation is an empirical formula that is chosen because it matches observations, and doesn't actually have any more justification than that.
The best discussion you are likely to find is in Ballentine. If you accept (empirically) Galilean invariance, the STRUCTURE of the Schroedinger equation falls out of group representation theory quite naturally.
The actual specifics of a problem involved picking a potential to use in the problem, and this is empirical. So if you ask the question:
What equation does an electron in an atom obey? That is empirical.
If you ask:
Given Galilean invariance and a 1/r potential, what equation does an electron in an atom obey? This doesn't need any more empirics.
There are a great many ideas which don't have enough carefully-measured evidence to be sufficiently confirmed as scientific fact and accepted as such by the scientific community (a recent joke was "While the Higgs Boson has not been discovered yet, its mass is 125 GeV"), but don't have enough carefully-measured evidence to be ruled out yet, either. Do any of the tools of the LW community help narrow down which ones are more worthy of consideration than others?
Eg:
* Cryonics as an arguably reasonable bet for its cost: proto-science
* Cryonics as a surefire way to achieve immortality: nigh-certainly pseudoscience (unless it's the method by which your Everett Immortality keeps you alive)
* Using math to demonstrate that taking classical physics and adding determinism results in MWI-style quantum physics: proto-science.
* Using math to demonstrate that quantum physics proves Christianity is true, from a certain point of view: pseudo-science
* Tubulin might self-organize into microtubules capable of computation on a sub-neuron scale: Possibly proto-science
* Tubulin architecture is 'quantum' in nature and that is what gives rise to consciousness: Probably pseudo-science
* 'Quantum consciousness' means anything is possible: Downright silly
* The E8 Lie group can provide a system for organizing the properties of subatomic particles: Proto-science, perhaps
* Heim theory is useful for predicting particle masses: Pseudo-science, probabilistically
* Using the Bullet Cluster to claim that dark matter is a better theory than Modified Newtonian Dynamics: proto-science
* Claiming that dark matter is made of 'anapoles': Proto-science, perchance
* Suggesting that dark matter is actually gravitational leakage from MWI 'parallel universes': You tell me. (But if it's true, then since I can't seem to find any previous serious discussion of this idea, I get to name part of it after myself, right? :) )
These may not be the best examples, but they're the closest ones I can think of to the boundary. If you know of any better ones, feel free to comment with them.