I read some of your other posts in the other thread. I will attempt to help you as to why the toxin is not present in homebrew. In order for the bacteria to produce the toxin, it needs to be in its niche. Suppose only two conditions are important for the organism's niche. For simplicity, we will make these conditions measurable on a continuous scale. We plot one condition on the x-axis and one on the y-axis. This will form a plane on the graph which is the organisms niche. Example, the values for condition 1 are 5-10, and for condition 2 also 5-10, this will form a square flat region on the XY plane. Now, add in a third condition and a z-axis and for simplicity again the range is 5-10. Now you have a cube. Now we can't really draw another dimension but each condition that is important will have its own axis and a hyperdimensional region of space, a hyperdimensional cloud will describe the niche of the bacteria. Homebrew does not fall in that cloud. Your mistake is considering the various conditions separately or perhaps somewhat in context to the other conditions but not all at once. The full niche is where the bacteria can reproduce, including creating the toxin. Given the number of times people have reproduced the experiment of creating beer, it does not seem at that homebrew is any where near the niche.

I have considered this. I think in the other post, I mentioned the 'combination' of factors that prohibit growth, so like your graph cloud of different niche's.

For simplicity sake, let's just leave this at 2 dimensions and we can add further dimensions later. If the substance is above 4.6 pH, it technically can be in the cube, but it might not have right Y value to call into the cube. So it also needs less than 6% alcohol to fall into that cube. If both of those values are met then it falls into the cube. But let's say that's variable. We do know that 4% abv *begins* to stunt the growth of the bacteria (not entirely) so the y axis is probably not a straight line or there is a gray area of probability. So above 4% and below 6% alcohol, you'll have a grayer area of probability. The bacteria could grow, but it is less likely to do so the higher it goes (so a slope of sorts). Now from my research, I'm not sure if probability slopes as pH goes up or if there is a cap of probability. But I think the bacteria prefers base and not acid so I would assume it begins slope. Below 4.6 though completely prohibits growth and 4.7 doesn't, but maybe it is a lot less probable than when pH is 5.5.

So about probability, I completely agree. The combination of different factors probably decreases likelihood to very low probabilities (or complete inhibition). Like if beer has 5.0ph and 5.5% alcohol, it is way less likely than something that has 5.0ph and 3.9% alcohol. And something 4.4pH and 6% alcohol is also very very unlikely.

One of the scientific journals I read noted that if even one inhibitory factor was removed, the likelihood would go up. (They noted something like pH being >4.6pH) Thus getting us a lot closer to the theoretical cloud. I'm not saying we ever reach that cloud, but as you noted, experimentation suggests that we don't ever get to the cloud.. I'm sure a low abv (1-2%) high pH beer is a heck of a lot closer to the threshold than normal beer.

But yeah I'm totally on board with that concept that there's a lot of variables that push it away from the cloud. I'm more curious what all those variables are (like what is the threshold IBU that completely inhibits growth). Because I'm not sure it's *just* alcohol and pH.

For instance, the FDA only considers one variable (pH) when making recommendations for home canning. Now canned foods aren't typically alcoholic, but if they were, I'm sure they would include that as a factor and if they did they would probably suggest that you either have at least 6% alcohol or below 4.6 pH. Maybe at that point they'd even have a combination range like as long as you have 4% and less than 5.5pH you're good or something like that.

That could be the case but there's not much research on that or combination factors.

All of this would depend on whether there was a slope of likelihood and cut offs. Like from the articles I've read we know below 4% abv, the bacteria completely resists (and grows) but above 4% it begins to struggle. I'm unsure if it's the same for pH or if there's a cut-off. Like above 4.6pH, it could have no struggle at all and likelihood would be the completely same whether it was 5.3 or 6.8.

So under 4%abv, the inhibitory factor is completely removed as a variable because it has no effect on the bacteria

Okay new info: I have read a small amount and it appears likelihood remains close to the same around 5-7pH and from 4.6-5 there is a drop in probability and requires incubation for longer.

Then below 4.6 it is pretty much completely inhibited. So there is the curve from 4.6-5 and then it kind of levels out at 5. So there would be negligible difference in probability between something at 5.3 or 7.0pH, it appears