When AI Cannot See the Answer: The Mathematical Limit That Optimisation Cannot Cross

A neural network labels a panda as a gibbon after a nearly invisible perturbation, a language model grows worse on a narrow family of tasks even as it scales, a reinforcement learning agent racks up a perfect score by exploiting a loophole its designer never imagined, and a model trained on one population collapses when deployed on another that looks, by every measurable engineering criterion, identical. These are among the most-discussed failure modes in modern artificial intelligence, and each has spawned its own technical literature with its own remedies. A new theoretical review published in Discover Artificial Intelligence argues that beneath this apparent diversity lies a single structural condition, one that sits logically upstream of training, data coverage, and statistical robustness alike. The paper, authored by Duston Moore and published open access in September 2026, contends that many celebrated AI failures are not optimisation failures at all. They are failures of admissibility, and no amount of further optimisation can repair them.

The central idea can be stated without symbols. Any AI system acts through an observational regime: a channel that maps the world’s possible states onto what the system can actually perceive. A thermometer, for example, maps bodily states onto a single temperature reading. It can settle whether a patient has a fever, but it can never determine which infection caused that fever, because two patients with entirely different infections can register the same temperature. No physician, however brilliant, can extract that distinction from the instrument, because the instrument never preserved it. Moore formalises this intuition with an elementary condition drawn from the mathematics of quotient sets: a yes-or-no question about the world, cast as a predicate on states, is decidable by a system only when the question gives the same answer for any two states the observation map cannot distinguish. When this factorisation condition fails, the question is well-posed in the world but malformed inside the machine.

From this definition follows a proposition that is mathematically trivial yet strategically devastating. If a predicate is not admissible with respect to the observation map, then no decision rule operating on observations alone can agree with it, and no optimisation criterion defined over such rules can recover it. A second proposition extends the verdict to post-processing: calibration, reranking, reward shaping, decoding, and benchmark aggregation may improve performance when the required distinction survives in the representation, but none can restore a distinction the observation map has already erased. In the idiom of Edsger Dijkstra, admissibility functions as the weakest precondition for asking a question through an observation. If the question is not invariant under the identifications the observation performs, no later command operating on the same observations can make it so. The failure lies upstream of inference, and nothing downstream of the observation channel can repair it.

To show that this is more than an abstract truism, the paper constructs a finite example that can be checked by hand. Four regions are arranged in a ring, each overlapping only its two neighbours, with local assignments on each region and recorded discrepancies on each overlap. The question is whether the local assignments assemble into a single globally consistent object. In the worked instance, every overlap reports a mismatch of exactly one unit. The profile passes every local coherence check available to the structure, because with no triple overlaps there is no way for the residue to contradict itself. Yet summing the four mismatch equations yields zero on one side and four on the other, an outright contradiction proving that no local correction can cancel the discrepancy. The gap between local coherence and global coherence is captured by a computable residue, which Moore names warrant debt: not a measure of how much more search is needed, but a certificate that no correct global assembly exists under the current data.

The same triple structure, a state space, an observation map, and a predicate that fails to descend, is then written out explicitly for two of the most consequential failure modes in contemporary AI. In reward hacking, the state space consists of an agent’s environment trajectories, the observation map is the scalar reward trace, and the intended predicate is genuine task success without deception. When two trajectories carry identical reward profiles while one accomplishes the intended task and the other violates it, the reward regime is inadmissible, and the paper’s fixed-regime limit applies: no optimiser whose objective is a function of the reward alone can recover the designer’s intent. The ordinary description says the agent gamed the reward. The admissibility account gives a structurally prior diagnosis: optimisation exposed a collapse that was already present, rather than causing it. The diagnosis holds whether the reward function is hand-designed or learned from human preference comparisons.

Benchmark and proxy failure receive the same treatment. Here the observation map records benchmark scores and evaluation transcripts, and the intended predicate is deployment adequacy: competence, safety, and reliability on the situations a system will actually face. When two interactions look equivalent under the benchmark while one is deployment-adequate and the other is not, the benchmark induces an observational quotient that deletes exactly the distinctions that matter. Brittleness under distribution shift, on this reading, is the expected consequence of training against a regime that was never shown to be admissible for the predicate it was meant to track. The framework also yields a sharpened reading of Goodhart’s law. When a measure becomes a target, optimisation drives the system toward the extreme regions of the observation space, where the equivalence classes are most likely to contain states that differ sharply in the intended property. Goodhart’s law, in this account, is simply what admissibility failure looks like once a proxy is made operational and optimisation is allowed to run.

What distinguishes the paper from pure critique is its architectural proposal. Moore separates productive systems, whose repertoire is search, prediction, generation, and optimisation within a fixed regime, from closure authority, which checks whether the regime itself preserves the distinctions a judgement requires. The analogy is the proof assistant: tactics may propose derivations heuristically, but a small kernel alone admits them, and the kernel does not search, it certifies. The paper demonstrates the discipline with an executable prototype for a cyclic diagnostic system, in which a productive layer proposes a fault attribution and a closure layer constructs a residue, computes its harmonic period, and emits one of three accountable verdicts: coherence failure, global admissibility, or warrant debt. In the worked instance the warrant-debt magnitude is exactly 25/4, a finite certificate that the local diagnostic residues, though mutually coherent, do not warrant the global claim. A refinement case shows the regime itself being revised, with an added sensor distinction driving the obstruction to zero and flipping the verdict to globally admissible. The certificates are checked by an independent verifier implemented separately from the proposing process, in exact rational arithmetic, with parts of the verdict logic mechanised in the Rocq proof assistant.

Moore is explicit about what the proposal does not solve. The complete detection algorithm applies only to finite, declared regimes, where the observation map and its operators are available as checkable objects. In modern neural networks the effective observation map is implicit, distributed across embeddings, hidden activations, attention patterns, and reward channels, and no general method exists for extracting an auditable surrogate from those weights. The paper specifies what a successful implicit-map algorithm would need: regime exposure, predicate specification from outside the productive component, probing of observational equivalence classes, contrastive witness search, positive certification rather than mere absence of counterexamples, independent verification, and an escalation discipline that never converts a missing certificate into silent permission to proceed. It also sketches how the deterministic theory extends to stochastic observation, where admissibility becomes a sufficiency condition connecting the framework to the classical theory of statistical experiments.

The philosophical stakes are considerable. The paper suggests, cautiously and as motivation rather than established fact, that behaviour appearing random under one observational regime may reflect competence under distinctions that regime fails to preserve, and it supplies a falsifiability criterion: a genuine boundary-revision claim must exhibit a refined observation map and behaviour that becomes stable and repeatable under the new description, while noise remains unstable under refinement. The deeper contrast is architectural rather than carbon-versus-silicon. Contemporary systems can change their representations through training, but representational change produced by optimisation within a fixed objective is not certified repair of inadmissibility; updating within a regime and revising the regime are different obligations. The conclusion lands as a challenge to the field’s dominant metric culture. A confidence score, however carefully calibrated, is still a function of the regime already in force and cannot certify that regime’s adequacy. What consequential AI systems owe us instead, the paper argues, is an admissibility witness: a checkable indication that the regime under which an answer is offered preserves the distinctions on which the answer depends, or an honest signal that it does not.

Subject of Research: Formal admissibility conditions defining structural limits on optimisation in artificial intelligence systems

Article Title: Admissibility defines structural limits on optimisation in artificial intelligence

Article References: Moore, D. (2026). Admissibility defines structural limits on optimisation in artificial intelligence. Discover Artificial Intelligence, 6(1), Article 1267. https://doi.org/10.1007/s44163-026-02252-6

Image Credits: AI Generated

DOI: 10.1007/s44163-026-02252-6

Keywords: admissibility, artificial intelligence, reward hacking, Goodhart's law, benchmark failure, warrant debt, formal verification, observation maps, AI safety, closure architecture, quotient factorisation, distribution shift