The phrase "Scientific Proof" is a high standard to acheive. In science, a phenomena is considered proven if it has met the standard of multiple independent replications, as determined through meta-analysis of all the data available. And certain experiments demonstrating psi phenomena have easily met that standard of proof.
The first question to answer is "what is the meaning of replication". A simple naive belief in replication is that it refers to a phenomenon which can be demonstrated at the 95% confidence level in every single experiment.
Unfortunately, in science involving huge numbers of uncontrollable variables such as human beings, this sort of replication almost never happens. Instead, replication is a statistical phenomenon.
In order to illustrate this phenomenon, Dean Radin selects the example of studies on aspirin as a preventative for second heart attacks in his seminal book on the meta-analysis of psi, The Conscious Universe. Today, everyone knows that aspirin is an effective preventative treatment for heart attacks. Why is this an accepted scientific fact? Because a large meta-analysis of multiple studies comparing aspirin to placebo showed an overall significant effect far beyond the chance expectation. See Radin's figure 4.2 below. Note that the vertical line with horizontal endpoints in these charts shows the 95% confidence interval with the actual measurement value in the center.
Notice that only 5 of the 25 individual studies actually returned statistically significant results on their own. If we relied on statistical significance of individual studies, we would say "aspirin's effects on heart disease can't be replicated" because of all these individual "failed studies". In fact, 3 of the 25 studies showed a (non-significant) negative effect from aspirin versus placebo! That is why we need to use a meta-analysis of studies from multiple independent researchers. The combined meta-analysis clearly shows us that aspirin has a statistically significant effect in preventing heart attacks. Aspirin therapy has gone up against the most rigorous examination possible and come out with the scientific seal of approval.
So what happens when we examine the evidence for psi phenomena? Certain categories of psi experiments have been extensively conducted at independent institutions by seperate research teams. These psi phenomena have all been subject to meta-analysis by Dean Radin and other independent meta-analysists, including skeptics such as Ray Hyman. And the effects are astronomically significant. For certain types of experiment such as the Ganzfeld and auto-ganzfeld, the time, effort and expense means that most of the data which could have been collected has been included in these meta-analyses, so no possible "file drawer" effects can even exist.
Below I have reproduced Radin's charts for meta-analysis of dream telepathy experiments, the 1985 Ganzfeld meta-analysis by Hyman and Honorton, an updated Ganzfeld meta-analysis, high-security ESP card tests, RNG PK experiments and dice-rolling PK experiments. Although all of these meta-analyses include data from trials showing non-significant effects, the overall meta-analysis is clear. These phenomena all show enormous, often astronomical deviations from the null hypothesis.
So the answer is clear. Certain psi phenomena have gone up against the most rigorous examination possible and come out with the scientific seal of approval. So why do so many scientists and "rationalists" think that psi is "nonsense", "without a shred of real evidence"? I'm afraid that is more of a sociological question than a scientific one.