"The application "Crossway.app" can't be opened." (a plain dialog with only an OK button)

The zip most likely reached your Mac through a chat app (WhatsApp, Telegram, and similar). macOS marks files saved by those apps so they can never run, no matter how the app is signed. Paste this into Terminal, then open the app again:

xattr -r -d com.apple.quarantine /Applications/Crossway.app

To avoid this entirely, download the zip in your browser instead of receiving it through a messenger.

"Crossway is damaged and can't be opened."

The zip was modified on its way to you. Delete the app and re-download it.

"Crossway can't be opened because Apple cannot check it for malicious software."

Same cause — a re-zipped or mangled copy. Re-download in a browser.

"Crossway requires macOS 14.0 or later."

Your Mac runs an older macOS. Crossway needs macOS 14 (Sonoma) or newer; there is no workaround.

"Crossway was not downloaded from the App Store."

Your Gatekeeper setting allows App Store apps only. In System Settings → Privacy & Security, set "Allow applications downloaded from" to App Store and identified developers (on newer macOS: "App Store & Known Developers").

Once installed: click the Crossway menu-bar icon → Settings… — your version is shown at the bottom of the Settings window. Or select /Applications/Crossway.app in Finder and press ⌘I (Get Info).

To see if you're current: click the menu-bar icon → Check for Updates… — Crossway compares your build against the newest release and offers the download if there's something newer.