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Computability and the only arbiter

The exact many-electron Schrödinger equation is uncomputable. What follows for it, for GR, and — held evenly — for RealQM.

What this is. A summary of a discussion on computability as a criterion: what an uncomputable model can and cannot claim, why General Relativity is not in the same predicament, and why — applied evenly — the same logic makes RealQM's case purely empirical. In the spirit of John Bell's discomfort with a “dirty” quantum theory, but sharpened from vagueness to computability.

The uncomputable equation

The exact $N$-electron Schrödinger equation lives on a $3N$-dimensional configuration space; its cost is exponential in the electron count. For anything beyond a handful of electrons it cannot be solved — not just “no closed form,” but genuinely intractable even numerically. This is not a fringe claim; it is why the whole apparatus of approximations (Hartree–Fock, DFT, coupled cluster) exists.

The logical point — and it is valid

You cannot verify a computable replacement against an uncomputable original. If the exact solution can't be computed for a large system, there is nothing to compare an approximation against — so the justification “this works because it approximates the true equation” is, for large systems, unverifiable and circular.

The consequence is blunt: for large systems the exact Schrödinger equation is epistemically idle — it can neither be evaluated nor used, and does no predictive work. A revered object that does no work is close to a fetish: the real predictions come from computable models, not from it.

The resolution — experiment, not the equation

The computable model is not justified by fidelity to the uncomputable equation; it is justified by matching experiment. DFT earns its place because its geometries, energies, and spectra match measurements, not because it matches an uncomputable wavefunction. The logic runs through experiment, and bypasses the exact equation entirely. (It is verified where computable — H, He, H$_2$, to many digits — and trusted inductively beyond; but that trust is an extrapolation, uncheckable at scale.)

The correction: GR is not in this boat

It is tempting to lump General Relativity in with the uncomputable Schrödinger equation. That is wrong. GR is computable to any required precision.

The Einstein–Infeld–Hoffmann equations are not GR — they are its first post-Newtonian approximation. But that approximation is a controlled expansion in a tiny parameter ($v^2/c^2 \sim 10^{-8}$ in the solar system), with a rigorously bounded, negligible error — categorically unlike the uncontrolled approximations of quantum chemistry.
The load-bearing prediction, Mercury's 43″/century, is the exact Schwarzschild result — a closed-form solution of the full field equations, not an approximation.
And the full nonlinear equations are solved numerically where needed — numerical relativity for black-hole mergers, giving the waveforms LIGO detected.

So there is no exponential wall in GR. The computability critique lands hard on the exact many-electron Schrödinger equation; it does not land on GR. Keeping the two separate is what makes the Schrödinger point strong rather than refutable.

The same logic, held evenly, falls on RealQM

RealQM is also a computable replacement of the uncomputable exact model — and a more radical one: not the antisymmetric wavefunction approximated, but a different ansatz (non-overlapping unit densities, geometric exclusion in place of antisymmetry). Therefore:

RealQM cannot be verified against the exact Schrödinger equation either — the same wall.
DFT actually has a stronger formal tie to exact QM (the Hohenberg–Kohn theorem proves an exact functional exists); RealQM has no such theorem — its relation to the exact equation is not provable.
So RealQM has no privileged claim to “being” true quantum mechanics. Its legitimacy is entirely empirical — does it match experiment? — plus parsimony and computability. Exactly like DFT.

The exact equation cannot adjudicate between computable models, because none can be checked against it. So the contest is purely empirical: which computable model predicts experiment best, at what cost, with what generality, and with how few free parameters. That is the only logical arena — and it is RealQM's strongest ground, not its weakest, provided it drops any claim of fidelity to the uncomputable equation and stands on being computable, parameter-free, real-space, and right against measurement.

The failure mode to avoid is the mirror image of DFT's: arguing “we are the true formulation and the standard one only approximates the uncomputable equation” commits the very error identified above — claiming a verified relationship to an unverifiable object. Drop the exact equation as arbiter entirely; let experiment decide among computable models, and RealQM wins only where it actually out-predicts, out-generalizes, or out-economizes — never by metaphysical priority.

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