That can happen when two or more chemicals are mixed together. Many times that mixture creates something much more potent then the individual chemicals would be.
There may also may be plants that just won't tolerate that mix.

"Many times that mixture creates something much more potent then the individual chemicals would be."

Name two. Name one.

There might be an example or two, but I'll bet this is mainly simple myth. The "many" is certainly wrong. The notion is possibly derived from the very real drug interactions in animals. Or else from the rare combinations that have problems for plants but are physiological in origin, not via chemical reaction in the application mix, and would arise if the chemicals were applied separately too soon.

Making lime-sulfur is one example of making a much more toxic chemical from a mix but of course that is the intent there.

Try each of your ingredients separately and observe the results. I've had good success with a simple surfactant (dish soap) dealing with aphids on foliage plants, sodium bicarbonate (baking soda) may also help with the PM (so I've read but never tried).

The little black spots may well be diagnostic, I can't think of anything that would cause them though.

If I were in this situation, one possible thing I would try is using twice the water so the mix is only half as strong. Then if that still burns, dilute again until there is no burning but apply more often. Just an idea. Good luck on finding an answer to the problem.

We had several chemical companies here some years ago that dumped the waste from production on the ground and as they moved into the water table they mixed together and produced some very toxic stuff that you and I are still paying to clean up. Those chemicals caused numerous people to stop using their wells and connect to the municipal system because the ground water they were using, from their wells, was too contaminated with these toxic substances to even wash with much less drink.
The synergistic affect is well known and is why sometimes people have bigger medical problems when combiniing two medicines.

Here is a link that might be useful: synergistic affect

I hear mixing fuel oil and ammonium nitrate can lead to a rather hazardous mixture but then again the fuel oil probably wouldn't do plants any good. No point in posting any links, we all know what happened in Okalahoma City.

I like your idea too rockguy, see if the negative side effect goes away by diluting the mixture that is causing problems and then seeing if the mixture still does what is hoped for.

I agree that the rates per gallon of water seem high. More is NEVER better when it comes to mixing chemicals to spray on plants....no matter what kind of products.

Tank mixes need to be done with care. Professionals know to carefully read the label of whatever product(s) they wish to use to learn if a mix is recommended....or not.

With a home-brewed concoction such as this, Bill will never know (for certain) if it is the dosage or the combination that has caused the problem. Surely, one or the other.

PM can be controlled by simple sprays of plain water (so can aphids) or a light horticultural oil.

Soap and oil together can cause burn. Oil alone, even olive oil can easily burn plants in full sun as the oil acts like a magnifying glass. If you treat will oil give the plants some shade for a day after treatment.

I used dawn dish liquid 1 part with 10 parts water to spray for insects on my tomatoes, squash and peppers. The leaves are brown and crispy. Some fell off. Will the vegetables survive?
Will the plants revive?

Dawn dish "soap" is a detergent made from non renewable resources and is not an acceptable organic product. Any Insecticidal Soap, sprayed at the wrong time might also cause the same symptoms, however. Whether the plants will survive depends on many things. What were you spraying for?

Too many factors to know what is the cause. Sunlight, multiple artificial light sources, and a homebrew concoction that has multiple ingredients, including water.

Have you tried spraying at a time when the lights are off for a period of time after spray is applied? My first guess would simply be leaf burning due to water/oil spots on the leaf. Then I'd alter the mix one ingredient at a time until I could say for certain that this and not that is what the cause is.

And I for one am getting to where I don't care what your definition of organic is, and I'm not really certain who you are, or who you think you are. It's all a chemical, used for the purpose of pest control...so do you really think there isn't going to be the risk of something adverse? If you are telling me that the adverse effect is that someone who spends a lot of time on line is going to tell me I'm doing something wrong (according to him/her/it)...bring it.