Neutrino-catalyzed nuclear decay? Is it possible that some nuclear decay that we think of as 'spontaneous' is actually catalyzed by an incoming neutrino?

I am imagining that there is some radioactive decay, let's just say a beta decay that appears to happen spontaneously:
$$ \rm ^n_mX \to {}^n_{m+1}X + e^- + \nu_e + \gamma $$
But it is actually catalyzed (somehow) by a solar neutrino, which comes in and out with the same energy/momentum:
$$ \nu_e + \rm ^n_mX \to {}^n_{m+1}X + e^- + 2\nu_e  + \gamma $$
My key assumption is that the 'catalyzing' neutrino appears on both sides of the reaction with (nearly) the same energy/momentum, so it wouldn't be obvious that anything was missing from the spontaneous formula above.
Unless we were specifically looking for this effect, it could be very hard to detect because you can't shield your experiments from neutrinos and the neutrino flux is (to my knowledge) relatively constant at all locations on the earth.

*

*Has this behavior been excluded by experiment?

*Is this prohibited by theory/symmetries somehow?

*Is this a real thing?

*What happens if we sub the neutrino for dark matter candidate particles?

 A: You are talking about  the start of a process called "neutrino lasing" , for example:

We present a calculation of a neutrino decay scenario in the early Universe. The specific decay is $\nu_{2} \to \nu_{1} + \phi,$ where $phi$ is a boson. If there is a neutrino mass hierarchy, $m_{\nu_{e}} < m_{\nu_{\mu}} < m_{\nu_{\tau}}$, we show that it is possible to generate stimulated decay and effects similar to atomic lasing without invoking new neutrinos, even starting from identical neutrino distributions. Under the right circumstances the decay can be to very low momentum boson states thereby producing something similar to a Bose condensate, with possible consequences for structure formation. Finally, we argue that this type of decay may also be important other places in early Universe physics.

There are also lasing calculations for neutrinos in the sun, closely related to your question:

Applying the phenomenon of neutrino lasing in the solar interior, we show how the rate for the generic neutrino decay process $\nu -> fermion + boson$, can in principal be enhanced by many orders of magnitude over its normal decay rate. Such a large enhancement could be of import to neutrino-decay models invoked in response to the apparent deficit of electron neutrinos observed from the sun. The significance of this result to such models depends on the specific form of the neutrino decay, and the particle model within which it is embedded.

Note the date, 1994. At present  mainstream physics accepts that neutrino oscillations, which have been seen in the lab also, explain the deficit of electron neutrinos, so this model is not validated,
The reason there is no attempt to study  a lasing mechanism in the laboratory is due to the weak interaction of neutrinos with matter. To get a lazing action there should be a high probability for the secondary neutrinos to keep raising the level of new atoms,as with electromagnetic  lasing,  but the weak coupling constant is so much smaller than the electromagnetic that this cannot happen with weak interactions in the dinensions  of matter on earth. The sun and the  cosmological early times are the field for this study.
The weak coupling constant is also the reason why the decay times for unstable nuclei cannot be measurably affected by a neutrino starting a lasing inversion. The probability of a neutrino interacting with a nucleus is very small.
edit:

Has this behavior been excluded by experiment?

It cannot  be excluded by experiment because of the very low probability of the interaction ( weak coupling constant. It might in the future be useful to model in cosmological observations.

Is this prohibited by theory/symmetries somehow?

no

Is this a real thing?

Real in physics means measurable. see answer to 1

What happens if we sub the neutrino for dark matter candidate particles?

it has been done by various models. example .
