Nothing. It is not one of the three claim types US law defines, and the category it does belong to needs no approval from anyone before it appears on a bottle.

The short version: "clinically proven" is not a regulated term. It is not one of the three claim types US law defines for food and supplement labels, so nothing sets a threshold it has to clear. The claims that are defined split into one type that needs FDA authorisation and one that needs no approval from anyone — and nearly everything on a supplement bottle is the second kind.
The FDA sets out three categories of claim that may appear on food and dietary supplement labels: health claims, nutrient content claims, and structure/function claims. (FDA, Label Claims for Conventional Foods and Dietary Supplements)
They are not equally hard to make.
A health claim links a substance to reduced risk of a disease, and the gate is real. Under the Nutrition Labeling and Education Act, the FDA authorises these by regulation after reviewing the scientific literature against a "significant scientific agreement" standard. That is why the small number of authorised health claims sound so careful — "adequate calcium throughout life may reduce the risk of osteoporosis" is the FDA's own example.
There is a second route for foods, added by legislation in 1997, that allows a claim based on an authoritative statement from a US government scientific body. It does not apply here: the FDA states plainly that this method cannot be used for dietary supplements.
A structure/function claim describes a substance's role in maintaining normal structure or function of the body. "Supports healthy testosterone levels." "Promotes restful sleep." "Helps maintain normal muscle function."
The FDA is explicit about what stands behind them: structure/function claims are not subject to premarket review and authorisation by FDA. The manufacturer decides the wording, prints it, and sells the product. Nobody at an agency approved it first.
This is why almost every claim you read on a supplement is phrased that way. It is not clumsy writing. It is the only sentence shape available without a gate.
Turn any US supplement over and you will find a version of this, usually in the smallest type on the package:
"This statement has not been evaluated by the Food and Drug Administration. This product is not intended to diagnose, treat, cure, or prevent any disease."
That disclaimer is the legally required companion to a structure/function claim. It is the label telling you, in the words the law requires, that the sentence above it went through no review at all.
Because no rule defines it, the phrase carries only what the reader brings. In practice it usually means a study exists. It does not tell you:
Nothing on this site scores a product for what it claims. Every ranking is built from what the label discloses, how that compares to the dose used in published research, what buyers report, and what a serving costs — the method is public in How We Rate.
A product can be entirely legal, accurately labelled, honestly marketed and still a poor way to spend forty dollars. That gap is not a regulatory failure. It is the space the regulation leaves open, and it is most of what our rankings measure.
This describes US labelling rules and is general information, not legal advice. Rules differ in other countries.