pip: ERROR: Could not find a version that satisfies the requirement / No matching distribution found

ERROR: Could not find a version that satisfies the requirement X followed by ERROR: No matching distribution found for X means pip searched the package index and found no file it can install for that name and version on this Python and platform. The part in parentheses lists the versions pip did see: (from versions: none) usually means a wrong name or an index without the package; a list without your version means the version or the Python version does not match.

Reproduce it

pip download resolves exactly like pip install but only saves the files, so it shows the same error without installing anything:

pip download --disable-pip-version-check --no-deps -q -d dl errcodex-no-such-package
echo "exit status: $?"

ERROR: Could not find a version that satisfies the requirement errcodex-no-such-package (from versions: none)
ERROR: No matching distribution found for errcodex-no-such-package
exit status: 1

A version that does not exist:

pip download --disable-pip-version-check --no-deps -q -d dl "six==9.9.9" 2>&1 |
  sed -E 's/\(from versions: [^)]*\)/(from versions: ...)/'

ERROR: Could not find a version that satisfies the requirement six==9.9.9 (from versions: ...)
ERROR: No matching distribution found for six==9.9.9

The real message lists every published version of six; it is shortened here.

Diagnose

Typical fixes

python -m pip install "requests>=2.31"        # a version that exists
py -3.12 -m pip install somepkg               # Windows: pick another installed Python
python3.12 -m venv .venv                      # or create a venv with a supported Python