Disclaimer: I don’t believe I have actually solved “the paradox”, but it is fun to think about.
The information paradox is resolved through the application of computational irreducibility and naive computationalism. The resolution indicates that information is genuinely lost, at least to us, and to any other possible means of computing, even theoretical. Yet the laws of physics remain deterministic. There is no need to evoke mysterious interactions nor rely on quantum randomness.
Naive Computationalism
First, what is meant by naive computationalism? The idea that our current framework of physics from relativity to quantum mechanics is NOT fully comprehensive and that the equations we “know” to be true are merely sufficient estimates.
In information terms, we might call these equations lossless compressions. In any equation that is said to govern the trajectory or evolution of some system there are terms that are considered negligible and can be essentially zeroed. When calculating the trajectory of a golf ball, even to the precision of 99.99999%, we would not need to know the exact gravitational pull of each blade of grass on the course. Or even the forces exerted by far away galaxies. You might even consider these terms “built into” the equations of motion we already use, within the variable g. However, this does not mean these factors are exactly zero or that they don’t influence the ball or that they add up to be exactly this value of g. They merely do not influence it enough to matter within the precision of human-scale observation or necessity or that g is “close enough”.
Conversely, we can see how extremely small imprecisions in measurements can vastly influence the calculated trajectories in a game of billiards. Calculations have shown that in order to accurately predict the trajectory of a ball in real life, after about 6 or 7 collisions, you would need to know the weight and location of the people around the table, as their mass begins to exert a non-negligible effect on the molecules of the ball during the course of the trajectory. This has been demonstrated largely on supercomputers which have failed to model actual ball trajectories beyond the practical limit of 5 to 6.
Mathematically, these additional factors can be expressed as terms. When an equation is reduced to only a few terms, there are nearly an infinite number of additional terms, each represented by the forces exerted from every other particle in the universe. Of course they are so weak and far apart that it is negligible on our scale, that is evidently not the case inside a black hole. Inside a black hole, there is an incredible number of particles exerting very dynamic forces on each other. While information is not technically “lost”, since the information itself is measured by…itself. It IS completely irretrievable to us humans, sadly. The reason for this lies with computational irreducibility.
Computational Irreducibility
What is meant by computational irreducibility? Stephen Wolfram describes this concept well. In a system of any size n, to track the information in that system we would need to hold a minimum of n bits when the system is at maximum entropy. In a system such as the black hole, every particle must be tracked, which implies a computer that can store as many bits as there are particles in a black hole, which is an overwhelmingly large computer.
In fact, the necessary computing device is exactly as large as the black hole. But even if such a computer could theoretically be built, it would need to somehow extract or record the information from the original source which in turn affects the trajectory of the system. We could say that the blackhole is its own computer, recording itself and all its trajectories. But it can never BE recorded by another computer of sufficient computational abilities without disturbing the trajectory of the system in an incalculable way, as the interaction between the two systems (computer and blackhole) would thus require an additional bit of information that must traverse the interface of the 2 without disturbing either, a physical impossibility.
In this sense, the information is not lost and does not necessarily violate any notions of physics, yet it does indicate the pursuit of reconstructing the flow of information in a blackhole completely and utterly impossible. Thus, the information paradox is resolved.