Does the problem of time series forecasting have or will it ever have a solution? In a strict mathematical sense. Will there ever be an algorithm that predicts a time series (almost)perfectly? Possibly with an acceptable lower bound for forecast error.

I know about classical time series forecasting methods (like ARIMA, ETS, TBATS, etc.), about machine learning methods (linear, boosting) and deep learning (MLP, RNN, LSTM, even Transformer). But none of these methods make any assumptions about the data. In my understanding, it's impossible for the same model to predict energy consumption, stock prices, a sequence of prime numbers (that's also a time series, right?), and, for example, a sequence of squares of prime numbers equally well. Or am I wrong? No Free Lunch Theorem states what, without assumptions about data, there can't be a model that predicts perfectly (although everyone uses the same models for completely different data). In my opinion, it follows from this that perfect forecasting is impossible. On the other hand, Nearly Perfect Prediction Theorem states what perfect forecasting is possible (or at least its continuous analog). How can this be?

I also very, very rarely see a strict formulation of the problem of forecasting a time series, from which it will follow whether the series is forecast correctly or not. Therefore I will give a statement of the problem that seems legitimate to me.

Formally, a time series is a sequence of values $y$, measured at constant time intervals: $$ y_1, ... , y_T, ..., \quad y_t \in \mathbb{R} $$

Formally, the problem is posed as a search for the function $f_T$: $$ y_{T+d} \approx f_T(y_T, ..., y_1, d) \equiv \hat{y}_{T+d|T} $$ where $d \in \{ 1, ... , D \}$ - forecast delay, $D$ - is the forecast horizon.

If it is impossible to forecast the series precisely, I would like to know whether it is possible to determine the theoretical lower bound(and maybe upper bound) of error for the predicted values and for a quality metric. Ideally, this error should be estimable from the available data and represent an irreducible (unavoidable) error.

I have not seen a single book or article in which everything is done from start to finish: a forecasting problem is set, a model is built, its parameters are estimated, time series is forecast, accuracy metrics are computed, and it is proven that it is impossible to predict better.

I would like to see something similar to the Gauss-Markov theorem.