What you consider proven or not may speak to you more than the evidence, but much is written on it. I prefer books, and think you should visit a library. In in the meantime:
“We have moved from a society in the 1950s and 1960s, in which race was more consequential than family income, to one today in which family income appears more determinative of educational success than race,” said Sean F. Reardon, a Stanford University sociologist. Professor Reardon is the author of a study that found that the gap in standardized test scores between affluent and low-income students had grown by about 40 percent since the 1960s, and is now double the testing gap between blacks and whites.
In another study, by researchers from the University of Michigan, the imbalance between rich and poor children in college completion — the single most important predictor of success in the work force — has grown by about 50 percent since the late 1980s.
Education Gap Grows Between Rich and Poor, Studies Show - NYTimes.com
The isolation of the prosperous, he said, means less interaction with people from other income groups and a greater risk to their support for policies and investments that benefit the broader public — like schools, parks and public transportation systems. About 14 percent of families lived in affluent neighborhoods in 2007, up from 7 percent in 1970, the study found.
http://www.nytimes.com/2011/11/16/us/middle-class-areas-shrink-as-income-gap-grows-report-finds.html
More controversially, unequal societies appear to have higher levels of social ills, from teenage pregnancy to violence and obesity, that affect quality of life across the board (New Scientist, 16 April 2011, p 50).
Now there is another reason to decry growing inequality. Greater wealth correlates with selfishness and lack of empathy (see "Why money can't buy you love: the unexpected price of wealth"), which might help explain why the divide persists and the rich seem so reluctant to close it.
Rich suffer as well as the poor in unequal society - opinion - 26 April 2012 - New Scientist
Income distribution and mortality: cross sectional ecological study of the Robin Hood index in the United States | BMJ
You've made a huge leap from suggestion to proven, why is that? Maybe that suggests more about you than the evidence that you have presented.
Let's look what you presented. The first piece has this: "family income appears more determinative of educational success than race"
In the second piece: " the imbalance between rich and poor children in college completion — the single most important predictor of success in the work force — has grown by about 50 percent since the late 1980s" references nothing about income distribution, just how wealth impacts college completion. From the National Center for Education Statistics: "Enrollment in degree-granting institutions increased by 11 percent between 1990 and 2000. Between 2000 and 2010, enrollment increased 37 percent, from 15.3 million to 21.0 million." Could it be with so many more people entering college that more are unprepared for the experience? I don't know, but it seems like it could be a legitimate factor.
The third piece: Less interaction. I would think that with all these people going to college, there would be more interaction. Policies that support the broader public. Would that include encouraging hard work, two parent households, or even keeping ones neighborhood clean? It sounds more like, if you won't use public money for what we think is public good then you don't care about people.
The fourth piece: "More controversially, unequal societies appear to have higher levels of social ills," From an opinion piece. I don't tend to equate "appear" with proven. As for the idea that wealth makes people more selfish and less empathetic. I think that the more people want to take your stuff, the more you want to hold on to it. I'm not even sure how to quantify selfishness and empathy. Heck, distance also has a factor. Someone in my neighborhood probably registers more in me than 10,000 people dieing in Japan. Is that selfish or a lack of empathy?
Further, From the National:
Our findings provide some support for the notion that the size of the gap between the wealthy and less well off—as distinct from the absolute standard of living enjoyed by the poor—matters in its own right. This finding in no way diminishes the importance of measures to alleviate the burden of poverty. None the less, in an affluent society such as the United States, reliance on trickle down policies may not be enough—society must pay attention to the growing gap between the rich and the poor.
Supporting notions, that doesn't sound like something being proven to me.
Lastly, the correlation of the Robin Hood Index to mortality. This sounds serious. The Robin Hood Index: "The Robin Hood Index is conceptually one of the simplest measures of inequality used in econometrics. It is equal to the portion of the total community income that would have to be redistributed (taken from the richer half of the population and given to the poorer half) for the society to live in perfect equality."
This is an interesting use of statistics with an interesting agenda. I didn't know that perfect equality was an economic term and not one of "equal under the law". It is a bit cliche but equal opportunity does not guarantee equal outcome. Furthermore, "Some economists argue that the size of the gap between the wealthy and less fortunate seems to be related to mortality." Argue and seems don't sound like "proven" to me either.
My conclusion: You mean well in wanting to help others, but you seem to only view that help in economic terms. You look at studies and bless them them with the word "proven" and assume that others should just accede to your conclusion. Does one of those book things in something you call a library include a dictionary containing the word proven?
See how irritating being dismissive of others can be to the receiver?