How to set cookies for the entire CNC site?

Question

PHP

I tried to set cookies using JS and PHP, now the code is as follows:

setcookie("auth", $text, time() + 3600*24*7, "/"); // (в принципе на javascript то же самое)

but there is a problem .
The user is authorized through the form in the header , that is, you can enter from any page of the site. This is the incomprehensible situation. Suppose if a person comes from the site.com /user/1 page, then cookies will not be visible on the site.com main page , but will be set for /user. Moreover, the most mysterious thing is that /user/ is not another directory, but the CNC, the user.php file and index.php lie side by side. How is it that despite the set path "/" the cookie is still one directory higher?