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() > 5)
        throw std::runtime_error(
            "listunspent ( minconf maxconf  [\"addresses\",...] [include_unsafe] [query_options])\n"
            "\nReturns array of unspent transaction outputs\n"
            "with between minconf and maxconf (inclusive) confirmations.\n"
            "Optionally filter to only include txouts paid to specified addresses.\n"
            "\nArguments:\n"
            "1. minconf          (numeric, optional, default=1) The minimum confirmations to filter\n"
            "2. maxconf          (numeric, optional, default=9999999) The maximum confirmations to filter\n"
            "3. \"addresses\"      (string) A json array of bitcoin addresses to filter\n"
            "    [\n"
            "      \"address\"     (string) bitcoin address\n"
            "      ,...\n"
            "    ]\n"
            "4. include_unsafe (bool, optional, default=true) Include outputs that are not safe to spend\n"