Ah, yes, the good old Pascal's mugging[1].

I am aware that cold fusion experiments are both much cheaper, and if successful, considerably safer and cheaper than hot fusion. Yet, there is no plausible mechanism for fusion to occur at room temperature. For two protons (hydrogen ions) to fuse, they must be brought to within 10e-15 m of one another. This requires overcoming the Coulomb repulsion between the two positively charged protons. This in turn requires a great deal of energy, roughly 6 KeV even once quantum tunneling is taken into effect. This corresponds to velocities of roughly 10e6 m/s, or roughly 0.5% the speed of light. This is is called the Gamow energy or the Gamow peak[2]. In hot fusion, this is accomplished by heating a gas to something like 10e7 K, at which point the average energy of any given proton in the plasma is roughly 6 KeV. Not coincidentally, this is also roughly the interior temperature of stars. It has been shown time and again that fusion does occur in a tokamak reactor... just not quite fast enough to overcome the cost of creating and containing the plasma.

At room temperature, the fraction of protons traveling at 0.5% of the speed of light is zero for all intents and purposes. Thus, spontaneous does not normally occur at room temperature, which we can all agree on. For cold fusion to proceed, there needs to be some other mechanism capable of making up this difference, either by somehow accelerating protons to an extremely high velocity, or otherwise encouraging them to fuse, perhaps by lowering the Coulomb barrier by some unknown mechanism. For the Fleischmann–Pons experiment (the original "cold fusion" experiment in the 1980s) this was hypothesized to be achieved by the crystal structure of palladium[3]. However, after the failure to replicate the original experiment, this hypotheses appears to have been falsified. In fact, no experiment has ever shown a measurable rate of fusion occurring at low temperatures.

And yet, there is the precedent of the Gamow factor. The classical potential for the Coloumb barrier is 3.4 MeV. Therefore, in a purely classical model, it is literally impossible to get two protons to fuse unless they collide at close to the speed of light. Yet Gamow showed that fusion could occur, with some probability, at much lower energies, thanks to a well known phenomenon called quantum tunneling. Why could there not be some other way taking this further? Furthermore, there is the precedent with fission. In the 1930's, many notable scientists were fission could ever be used as a power source. They were aware that fission could be effected by alpha particle bombardment, but this did not seem "energy positive." Sound familiar? And yet, when Hahn and Meitner[5] discovered spontaneous nuclear fission occuring due to a chain reaction in uranium in 1938, it immediately became apparent it could be used as a massive source of energy, and Einstein immediately warned FDR and the Manhattan project began.[4] Why could not a similar story play out for fusion? Cold fusion is tantalizingly plausible and the experiments, as you say, can be conducted on a tabletop. Why not try?

My answer, which is of course a subjective judgement, is that good science happens by searching where the light is, exploring the implications and edges of existing theories, not out in the dark, trying things completely at random. Rutherford wasn't bombarding gold foil for the hell of it. Randomly trying things in an atheoretic way in the hopes that a previously undiscovered and unsuspected piece of new physics will drop out is closer to alchemy, and just about as likely to be successful. More than 2,000 years of randomly combining urine and lead resulted in not one ounce of gold. (I grant you that tabletop cold fusion experiments may very well find a new bit of chemistry.) But the argument against is simply this: 6 KeV. It's simply too much. No chemical or electrical process is going to get you that, unless it first turns your experiment into plasma, in which case you're back to hot fusion! It's like throwing pebbles at the moon and calling it the Apollo program. It's not a matter of just getting the right pebble. You can try quartz and obsidian, rough and smooth, for as long as you like, but you're not even beginning to address the invariant in the room, which is that you just can't impart enough kinetic energy to your pebbles to even get them out of Earth's gravity well, much less to the moon. That is why I believe that a dollar spent on cold fusion actually has lower expected payoff than a dollar spent on hot fusion.

[1]: https://en.wikipedia.org/wiki/Pascal%27s_mugging

[2]: https://en.wikipedia.org/wiki/Gamow_factor

[3]: https://en.wikipedia.org/wiki/Fleischmann%E2%80%93Pons_exper...

[4]: https://en.wikipedia.org/wiki/Einstein%E2%80%93Szil%C3%A1rd_...

[5]: https://en.wikipedia.org/wiki/Otto_Hahn#Discovery_of_nuclear...