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This is not a problem with cosmological constant in models of universe. The "cosmological constant problem" is a bad name for a particular (bad) interpretation of a certain calculation procedure in QFT. You can find the usual account of this calculation e.g. in this answer by Nihar Karve: https://physics.stackexchange.com/a/626958/31895

This is not a problem with cosmological constant in models of universe. The "cosmological constant problem" is a bad name for a particular (bad) interpretation of a certain calculation procedure in QFT that leads to severe contradiction to observation. You can find the usual account of that calculation e.g. in this answer by Nihar Karve: https://physics.stackexchange.com/a/626958/31895

Having all eigenvalues infinite breaks calculations. The correct conclusion from this is that we have a defective (wrong) Hamiltonian. This is not a big deal, nobody expects canonical quantization to always give the correct Hamiltonian. We have a system with infinite number of degrees of freedom, not a finite set of harmonic oscillators, so maybe the Hamiltonian is something non-trivially different from what we have for a finite set of simple harmonic oscillators. So we should at least look for a better Hamiltonian (if not for a completely different formulation of the theory).

Having all eigenvalues infinite breaks calculations. The correct conclusion from this is that we have a defective (wrong) Hamiltonian. This is not some serious insurmountable difficulty - we shouldn't expect canonical quantization to always give the correct Hamiltonian. We have a system with infinite number of degrees of freedom, not a finite set of harmonic oscillators, so maybe the Hamiltonian is something non-trivially different from what we have for a finite set of simple harmonic oscillators. So we should at least look for a better Hamiltonian (if not for a completely different formulation of the theory).

Non-zero cosmological constant term is really a concept from GR and macroscopic observations; there is no real logical necessity or need in physics to have this concept linked to field ground state Hamiltonian eigenvalue in QFT. I should say many people don't agree on this last point, they think these things should be linked, but I suspect the motivation is problematic. And none of them know how to formulate a functional QFT of gravity where the above procedure produces a reasonable value of cosmological constant close to observation. All known variations that are free of doctoring the parameters produce wrong results. So I think it very likely the whole enterprise is ill-conceived, and no revelation on the observed value of cosmological constant will be coming from these QFT-based, or even any microscopic theory-based attempts.

Non-zero cosmological constant term is really a concept from GR and macroscopic observations; there is no real logical necessity or need in physics to have this concept linked to field ground state Hamiltonian eigenvalue in QFT. I should say many people don't agree on this last point, they think these things should be linked, but I suspect the motivation is problematic. And none of them know how to formulate a functional QFT of gravity where the above procedure produces a reasonable value of cosmological constant close to observed value. All known variations that are free of doctoring the parameters produce wrong results. So I think it very likely the whole enterprise is ill-conceived, and no revelation on the observed value of the cosmological constant will be coming from these QFT-based, or even any microscopic theory-based attempts.