Does my research prove market inefficiency? The long story short, I have developed an index based on a certain distribution. Then I aligned NYSE stocks according to this index i.e. the stocks with the best fit are first and the worst are last. The index is irrelevant with economic theory.

Afterwards, I created a group of portfolios with equal number of stocks and equal observations and then I calculated the minimum variance per portfolio.

The result suggests that the portfolios with better fit have lower variances than the ones with "bad" fit. Does this prove market inefficiency assuming that it didn't occur just by chance?

Edit: According to efficient market hypothesis, asset prices reflect all information given and "patterns" should not occur. However, according to the results, patterns occur and they are independent to the information given.
The efficient market hypothesis does not imply that there are no patterns!

As Eugene Fama pointed out decades ago, any test of market efficiency is a joint test of market efficiency and an asset pricing model. The EMH on its own is not a testable theory.

If I understand your statement properly, you're claiming that forecasting variance would violate market efficiency? For that to be true, you would need to assume an asset pricing model where variance is not forecastable. That's a strange claim. It would imply that risk free government bonds must have the same volatility as small biotech firms (otherwise, I can forecast volatility based upon whether a security is a stock or a bond).

It well known that volatility is forecastable, both in the cross section and in the time series. For example, there is the widely used GARCH model and all kinds of other stochastic volatility models. Forecasting volatility is interesting, but it in no way constitutes a per se violation of market efficiency.

This answer of mine goes into greater depth on what the efficient market hypothesis actually says and how to test it.

References

Fama, Eugene F., 1991, "Efficient Capital Markets: II," Journal of Finance