Jacob, it's hard to build anything solid on the genetic differences data. Sure, we share 98% of our genes with chimps, but we also share 97% of our genes with mice. Genes are so highly conserved, that nothing useful can come of understanding what that 1% is doing.

If it is calculated that we have about 100,000 genes, then 1% is 1000 genes. That is a lot of genes, if you propose that many of them serve for human characteristics depending on size, body development, speech, and cerebral development. Interestingly, only recently did a group of scientists embark on identifying those genes!

Most of what makes humans different from chimps or mice is at the level of how the genes are organized relative to each other, so that their interactions are different--some protein products being amplified, others diminished, etc.

And mainly, the proteins and among them, the enzymes. That is, obviously, the gist of the genes' functions.

You said that 1% of our genome would be 700 genes. In fact, the current estimate of the human genome is about 300,000 genes, so 1% would be about 3,000.

My most recent information says between 70 and 100 thousands. You might be interested in knowing that I discovered one new protein (The Folate Binding Protein), permitting then the identification of the respective gene.

The rest of your posting hinges on "understanding." I can think of some ways of defining 'understanding' that would restrict to human beings (e.g. if, for example, you tied understanding to a sense of future or past).

Unclear.

I can also think of defining 'understanding' in a way that permits us to include some non-human animals (e.g., the ability to engage in a set of activities).

Not clear.

It seems that rather than stack the deck one way or another, we should consider what kinds of things admit of being understood.

Economy of language facilitates clear dialogue.

For example, we could begin by distinguishing between what the agent does, and what the agent recognizes as being done to the surroundings. On this basis we could speak, with perfect intelligibility, of a chimp understanding how to crack open a nut, and at the same argue that the chimp does not (and cannot?) understand how it is that nuts come to be lying on the ground.

A chimp learns to crack a nut, but it does not and cannot understand why the nut is cracked. It must learn basic elements of force acting on the resistant nutshell. While learning that, it must go one step further, which is, exulting: "Eureka! Now I understand what I learned..."
I asked some of my grandchildren around the age of three, why I and my wife are their grandparents. They expressed puzzlement, and I asked if they think that every child is assigned grandparents. Their faces seemed to betray that indeed, that was their belief. Then I explained the conundrum, and it took sometime to place the respective family position in their minds. Children will then, besides having learned that there are grandparents, bee able to understand why. I leave the interest in the intelligence of chimps and rats in other hands.

Along similar lines, some primatologists have argued that non-human primates cannot understand spatial or temporal processes as we do. We understand these processes in terms of causal relations, whereas the non-human primates understand these processes in terms of antecedent-consequent relations. So, for example, a chimp could not understand that a single cause can have many possible effects or that a single effect could have many possible causes.
A chimp could see the wind shake the branches of a tree and see the fruit fall down, and understand that there is a connection. The next time the wind blows, the chimp could then return to the tree to see if there's any more fruit. But the chimp could not, in all likelihood, realize that shaking the trees would accomplish the same effect as the wind would.

The word UNDERSTAND is not applicable to the chimp. It might learn that when wind shakes, fruit will be available in the ground. It may be taught to shake the tree for the same phenomenon to occur. And other tricks too. It may discover spontaneously tricks to solve easy problems, so that it, as so many animals, is capable of problem-solving. But it will not understand that the force applied to the object is stronger than the force attaching the object.

There's also a great deal of evidence that monkeys and apes can understand what's happening in their social environment much better than they can understand what's happening in their physical environment. My references here are: How Monkeys See the World, and Primate Cognition.

I can not dialogue on unread material.

Finally, I have to ask why one must offer "irrefutable evidence" of animal understanding. I know of no branch of science in which the standard of evidence is "irrefutable".

Science accepts only irrefutable evidence. For that purpose, it sets up experimental procedures. Positive results must be reproducible by the same researcher and by others. Negative results are not irrefutable. It had been found experimentally that there is no folic acid in milk, and that the addition of copper sulfate destroys the vitamin C in milk, causing folic acid deficiency in monkeys given such milk. This was meant to indicate that vitamin C plays an important role in the biological actions of folic acid. That experiment became a classic paper. Some years later, using a technique partially modified by me, I found that there is folic acid in milk. Copper destroyed it too, which explained the results in monkeys.
Since strict experiments are not available to test animal understanding, the results of researches in that area are suggestive, at best, that there might possibly be a certain faint spark of animal UNDERSTANDING, in its strict sense. Because, I say, sentences composed by the right words are needed to reach understanding, not to be confused with impressive emotive manifestations in animals. Without understanding why, animals can learn how to act in situations that demand prompt action.

(I exclude mathematics, which does indeed trade in certainties.)

Mathematics is not a science. It is the core of Physics, from which all sciences derive. The more mathematics in a science, the more exact it becomes.

The standard is rather one of plausibility relative to other explanations. To set the bar so high is to guarantee that it cannot be met.

Plausability is a condition for research. Science exists because scientists set the bar so high...