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I'm sure this wasn't his intention at the outset, but here's my understanding of the history of GR so you can see where I'm coming from:

The Einstein-Grossmann "Entwurf" version of the theory used the Ricci tensor in place of the Einstein tensor, and it was only when Einstein finally conceded the necessity of general covariance that he realized he needed a tensor with zero divergence. And so, in some sense, the real breakthrough in November 1915 was the discovery of a mathematical identity -- the existence of a curvature tensor that is inherently conserved, just like matter and energy. Thus, the field equations seem to have offloaded an important part of the role of Newton's laws, into the intrinsic nature of the Lorentzian manifold itself. That is, once you accept that spacetime is Lorentzian, there automatically has to be this locally conserved entity, and the field equations "merely" say that matter (whatever that may be) must follow its flow.

The other law of GR is the principle of geodesic motion, but, as quoted here, Einstein did not believe that was an independent postulate, but rather that it could be derived from the vacuum field equations. And although that claim may be disputed technically, he was not philosophically ambivalent about it:

One of the imperfections of the original relativistic theory of gravitation was that as a field theory it was not complete; it introduced the independent postulate that the law of motion of a particle is given by the equation of the geodesic.

And since we know that Einstein believed in a unified field, presumably the only ultimate law of physics he would have been willing to admit would be the field equation restricting the behavior of that field. However, as argued above, his own field equation for GR outsourced much of that restrictive authority to the (pseudo-)Riemannian geometry of spacetime itself. And perhaps it is not a coincidence that the change in the model of spacetime which enabled that transition was itself the removal of a mathematical "law", viz., Euclid's fifth postulate!

So I see this trend in the development of GR, of laws being stripped away. From a certain point of view, you might even say that GR contains no laws of physics, in that it doesn't even try to specify the nature of matter (the right hand side of the field equation). Once again, it outsources to theories of the structure of matter, such as quantum theory, the job of explaining why the inherently conserved Einstein tensor evolves in the particular way it does. Almost in the same way that Newton outsourced to his readers the job of explaining the remote influence of gravity. An explanation that was ultimately given, not in terms of new entities or phenomena, but in the mathematical terms of the continuum proper.

Hence my question: is there any indication that Einstein believed the laws of physics would someday "work themselves out of a job" -- that their explanatory role would be assumed by the mathematical identities inherent in the continuum, in conjunction with the particular types of substructures of that continuum proposed to correspond to the known elements of matter (e.g. topological singularities for particles, with mass/charge being some kind of topological invariant)?

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    $\begingroup$ I am having hard time distinguishing between "laws of physics" and "mathematical identities". First of, the identities are not purely "mathematical", the quantities in them have physical (in particular, experimental) interpretations. And second, any typical law can be expressed as an identity (or inequality), and the tendency to do that hardly started with Einstein. Already Newton expressed general mechanical laws as "mathematical identities", not just gravity. So what is that law "essence" that is being "stripped away"? Philosophizing about the "nature of matter" in the style of Aristotle? $\endgroup$
    – Conifold
    Commented Oct 21, 2022 at 1:43

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I believe you are making a contradiction between mathematical identities and physical laws that Einstein would not make. For him, mathematical identities are the laws of nature, and having them derive from the Unified Field Equation is the explanation.

As he wrote in 1934 in "On the Method of Theoretical Physics":

Our experience up to date justifies us in feeling sure that in Nature is actualized the ideal of mathematical simplicity. It is my conviction that pure mathematical construction enables us to discover the concepts and the laws connecting them which give us the key to the understanding of the phenomena of Nature.

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To justify this confidence of mine, I must necessarily avail myself of mathematical concepts. The physical world is represented as a four-dimensional continuum. If in this I adopt a Riemannian metric, and look for the simplest laws which such a metric can satisfy, I arrive at the relativistic gravitation-theory of empty space. If I adopt in this space a vector-field, or in other words, the antisymmetrical tensor-field derived from it, and if I look for the simplest laws which such a field can satisfy, I arrive at the Maxwell equations for free space.

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It is essential for our point of view that we can arrive at these constructions and the laws relating them one with another by adhering to the principle of searching for the mathematically simplest concepts and their connections. In the paucity of the mathematically existent simple field-types and of the relations between them, lies the justification for the theorist's hope that he may comprehend reality in its depths.

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