If I unlock an encrypted wallet for 359711509437336256 or less seconds, I can sign transactions with it:

    $ bitcoin-cli walletpassphrase "$x" 359711509437336256
    $ echo $?
    0
    $ bitcoin-cli signrawtransaction $tx | grep complete
      "complete": true

But if I unlock it for 359711509437336257 or more seconds, it appears to have unlocked but I can't sign any transactions:

    $ bitcoin-cli walletpassphrase "$x" 359711509437336257
    $ echo $?
    0
    $ bitcoin-cli signrawtransaction $tx | grep complete
    error code: -13
    error message:
    Error: Please enter the wallet passphrase with walletpassphrase first.

I don't know the significance of that number. It's nowhere near a power of 2.

I was trying to unlock the wallet effectively "forever", so put a big number. If I use too big a number (2^63 or more) I get an error message `JSON integer out of range` but for numbers between 359711509437336257 and 2^63-1 inclusive I get no error, but no actual unlocking.
{
    CWallet * const pwallet = GetWalletForJSONRPCRequest(request);
    if (!EnsureWalletIsAvailable(pwallet, request.fHelp)) {
        return NullUniValue;
    }
    if (request.fHelp || request.params.size() > 2)
        throw std::runtime_error(
            "listaccounts ( minconf include_watchonly)\n"
            "\nDEPRECATED. Returns Object that has account names as keys, account balances as values.\n"
            "\nArguments:\n"
            "1. minconf             (numeric, optional, default=1) Only include transactions with at least this many confirmations\n"
            "2. include_watchonly   (bool, optional, default=false) Include balances in watch-only addresses (see 'importaddress')\n"
            "\nResult:\n"
            "{                      (json object where keys are account names, and values are numeric balances\n"
            "  \"account\": x.xxx,  (numeric) The property name is the account name, and the value is the total balance for the account.\n"
            "  ...\n"
            "}\n"
            "\nExamples:\n"