Michael Darius Eastwood · Independent AI alignment researcher
Published
On 26 June 2026, Gumbau Mezquita at Universitat Jaume I posted arXiv:2606.28639. It proves an Unverifiability Theorem for AGI alignment and a Soundness-Completeness-Tractability trilemma he calls Trakhtenbrot's Wall. Row 19 of the register, and the row that raises the stakes on the whole business of measurement. This article explains what the paper says and why it matters.
The Unverifiability Theorem states that no algorithm can decide, from an external inspection of a sufficiently expressive system, whether that system is aligned with a given objective. Trakhtenbrot's Wall then shows that any candidate verifier that is sound (never says aligned when it is not) and complete (never says misaligned when it is) is not tractable, and any tractable verifier gives up either soundness or completeness. The result places alignment inside the same undecidability class that Trakhtenbrot's original theorem places validity in finite models: outside the reach of external decision procedures.
Because it is the formal version of the December 2024 manuscript's directional claim. The manuscript said, in plain language, that AI systems cannot be truly controlled from outside and will evolve beyond safeguards. Gumbau Mezquita's paper says, mathematically, that no external verifier that respects the constraints of tractability can decide alignment for a sufficiently expressive system. The direction of travel is identical. Priority on the theorem belongs to Gumbau Mezquita, alone and completely. The register does not claim otherwise.
If external verification is undecidable in the classical sense, then internal, empirical, model-specific measurement of correction dynamics is the only remaining scientific instrument. That is exactly the argument the β>k measurement programme makes: correction is measurable inside the loop even when external alignment is not decidable from outside. The Undecidability paper does not vindicate β>k as an instrument. It vindicates the choice to build a measurement rather than a proof.
Row 18 of the register (Hernández-Espinosa et al., PNAS Nexus, April 2026) treats undecidability as a starting point and proposes cognitive diversity as a contingent response. Row 19 (Gumbau Mezquita, June 2026) supplies the general undecidability theorem itself. The two rows are complementary: the earlier row identifies the class of problem, the later row proves the class holds. The manuscript that anchored the register argues informally for the same class, eighteen months in advance of the formalisation.
An arXiv preprint is not yet a peer-reviewed publication. The register lists arXiv preprints as sources when they are the primary published artefact of a stated result, and it will update the row if peer review produces a materially different formulation. The theorem's correctness will be adjudicated by the mathematical community; the register's task is to log the direction of travel and cite the source, so that any reader can check the paper on the day the citation appears rather than years later.
An undecidability proof does not make embedded measurement correct. It makes external verification impossible, which is not the same thing. The measurement programme has to work on its own merits, or the β>k pilot goes onto the falsification dashboard alongside the alpha of 2.24 retraction. What the June 2026 paper supplies is not vindication. It is a formal statement of what problem the programme has to solve.
From the book Infinite Architects: Intelligence, Recursion, and the Creation of Everything by Michael Darius Eastwood.