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(
            "signmessage \"address\" \"message\"\n"
            "\nSign a message with the private key of an address"
            + HelpRequiringPassphrase(pwallet) + "\n"
            "\nArguments:\n"
            "1. \"address\"         (string, required) The bitcoin address to use for the private key.\n"
            "2. \"message\"         (string, required) The message to create a signature of.\n"
            "\nResult:\n"
            "\"signature\"          (string) The signature of the message encoded in base 64\n"
            "\nExamples:\n"
            "\nUnlock the wallet for 30 seconds\n"
            + HelpExampleCli("walletpassphrase", "\"mypassphrase\" 30") +
            "\nCreate the signature\n"
            + HelpExampleCli("signmessage", "\"1D1ZrZNe3JUo7ZycKEYQQiQAWd9y54F4XX\" \"my message\"") +