Reviewer has chosen to be Anonymous

Overall Impression: Average

Content:
Technical Quality of the paper: Good
Originality of the paper: Yes, but limited
Adequacy of the bibliography: Yes

Presentation:
Adequacy of the abstract: Yes
Introduction: background and motivation: Good
Organization of the paper: Needs improvement
Level of English: Satisfactory
Overall presentation: Average

Detailed Comments:

Assumptions / Argumentation:

- Multiple studies provide evidence that LLMs return to previously erroneous decisions in their context, even when presented with correct information; see, e.g., work on ‘chain disloyalty’ [1]. These issues could generalize to the “deterministic output from a symbolic process” (despite the positive experimental results, which, in the studied setting, suggest otherwise). The authors should address this or discuss the issue as a potential limitation.

- The operability framework, per Definition 1, does not enforce causal use of prior feedback in any future decision. So claims, such as “each F_i acts as a hard constraint that prunes the probabilistic search space, forcing convergence toward valid outputs” and “genuine learning signal,” should be softened.

- I would argue against the usage of ‘mechanistic’ for the hallucination taxonomy, especially within machine learning and XAI communities. The taxonomy is causal (at least in the sense of some of the causes defined by Aristotle), but it does not try to identify the computational mechanics of hallucinations (which heads, circuits, and representations are responsible); in fact, computations relevant to some epistemic and training-induced hallucinations might be traced to the same part of the LLM.
Instead, the taxonomy identifies the abstract causes of hallucinations (architectural design for structural hallucinations, the global approximation/extrapolation properties of ML systems for epistemic hallucinations, and reward design for training-induced hallucinations).

- Minor note about classification within neurosymbolic AI: While the authors argue for a distinct dimension of their neurosymbolic proposal, the proposed framework technically follows (in terms of the closure of a feedback loop) the methodology of a neurosymbolic learning cycle [2]. Then it could be argued that the system falls into the Neural | Symbolic class of NeSy systems in Kautz’s taxonomy, since it (within an iteration) employs neural systems producing outputs that are then reasoned about symbolically.

Originality:

- The originality lies mostly in the theoretical framing of the operability framework. The framework describes an iterative system in which an LLM receives feedback on its output from a rigorous symbolic system. Similar systems have been described in prior work [3, 4]. While the work is original in its framing of the problem, the experimental part seems rather incremental. Could the authors provide some basic comparison/benchmarking to similar prior work?

Formality / Presentation:

- Some terms, such as the essential concept of operability or individual types of hallucinations (structural, epistemic, training-induced), should be defined earlier (operability is defined in Sec. 3.2, types of hallucinations in Sec. 3.1; while important arguments about them are made as early as in Sec. 1.3).

- There are some repetitive phrases without additional benefit, e.g., “not merely useful but necessary” (p. 7, 11), “whether it knows or interpolates” (p. 3, 11), etc.

Formatting:

- The paper does not distinguish between textual and parenthetical citations, overusing the textual version and making some sentences hard to read (see citations of Turpin et al. (2023) and Dherin et al. (2025) in the Abstract, Lewis et al. (2020) and Kadavath et al. (2020) in Section 1.1, and many more).

- The empty space on pages 6, 21, and 22 should be resolved. The usage of spacing around dashes should be made consistent. The layout/spacing of Table 1 should be improved.

- Figure 1 should be closer to its first (or at least any other) mention in the text.

- I can’t seem to find the affiliation number 2 (of the second author) anywhere.

Conclusion:

The presented framework is interesting, and the experimental results are solid. However, several mostly presentational/formatting weaknesses lower the overall impression of the work.

[1] Lu, Haolang, et al. "Auditing meta-cognitive hallucinations in reasoning large language models." Advances in Neural Information Processing Systems 38 (2026): 162500-162543.
[2] Towell, Geoffrey G., and Jude W. Shavlik. "Extracting refined rules from knowledge-based neural networks." Machine learning 13.1 (1993): 71-101.
[3] Quan, Xin, et al. "PEIRCE: Unifying material and formal reasoning via LLM-driven neuro-symbolic refinement." Proceedings of the 63rd Annual Meeting of the Association for Computational Linguistics (Volume 3: System Demonstrations). 2025.
[4] Jha, Sumit Kumar, et al. "Counterexample guided inductive synthesis using large language models and satisfiability solving." MILCOM 2023-2023 IEEE Military Communications Conference (MILCOM). IEEE, 2023.

Reviewer has chosen to be Anonymous

Overall Impression: Weak

Content:
Technical Quality of the paper: Weak
Originality of the paper: Yes, but limited
Adequacy of the bibliography: No

Presentation:
Adequacy of the abstract: No
Introduction: background and motivation: Limited
Organization of the paper: Needs improvement
Level of English: Satisfactory
Overall presentation: Average

Detailed Comments:

Major Issue 1: Abstract & Intro - The Central Claim Is Not Supported by Its Citations
The paper takes "hallucination is architecturally necessary" as its argumentative starting point. Neither class of citation offered supports a claim of that strength.

First class of citations: Kalai & Vempala (2024) and Kalai et al. (2025)
Domain mismatch. The Kalai & Vempala lower bound concerns arbitrary facts. The authors themselves distinguish such facts from those governed by learnable systematic regularities, for which the theorem does not hold: a model can learn the rule and remain both calibrated and non-confabulating. Operability, by contrast, requires executability, a deterministic oracle, and decidable ground truth. Maze-solving and job-shop scheduling fall squarely into the class the theorem explicitly exempts.
Type mismatch. More fundamentally, both Kalai papers are results about the statistical properties of output distributions. They bear no entailment relation to the paper's architectural claim.

Second class of citations: Turpin, Huang, Stechly, Khalifa, and Tighidet in §2.1
The evidence type cannot support an impossibility claim. These are empirical negative results: failures observed in particular models, on particular tasks, under particular prompting regimes. "Architecturally necessary" is an impossibility claim, which requires a constructive argument or a lower bound.

Major Issue 2: Tech quality - The taxonomy does not supply usable criteria for assigning a failure to a category

(a) Several of the paper's own failures satisfy both the Structural and the Epistemic definitions.
§3.1 places constraint forgetting under Structural hallucinations — "where the model loses track of conditions imposed at the beginning of the task" — and glosses the category as errors "propagating mechanically through a reasoning chain, where a correct earlier step would have prevented the later failure."
Applied to the reported failures, this criterion assigns them to Structural:
Door-without-key violations. Had the model collected key 'c' at the appropriate earlier step, the later constraint violation would not have occurred.
NL state-tracking loss. The paper's own description is that the model "forgets" which keys have been collected as complexity increases, which restates the constraint-forgetting definition.ASP omitted jobs. NL failures omit one or more jobs from the schedule entirely; an earlier correct step would have prevented the omission.
The paper classifies all of these as Epistemic and does not explain they don't belong to Structural.

(b) The criterion used in §5.1 to exclude Structural is the property §3.1 uses to define Structural.
§5.1: "The model's errors are not mechanical failures in reasoning chains (Structural) — each step is locally coherent."
§3.1, "Crucially, each step appears locally valid, making these errors undetectable by the model itself but highly responsive to external feedback."
Local step-validity is presented in §3.1 as a defining and diagnostically crucial feature of Structural errors, and in §5.1 as grounds for ruling them out.

Major Issue 3. Originality, tech quality, bib - The Operability framework: novelty, operationalization, and an untested distinction

(a) The framework overlaps substantially with LLM-Modulo (Kambhampati et al., ICML 2024), which is not cited.
The paper didn't cite LLM-Modulo, whose Generate-Test-Critique architecture is structurally the same loop: the LLM generates candidate solutions, external sound critics evaluate them, and a back-prompt controller pools the critiques into the next prompt. We suggest cite and compare against this work. Without that comparison the framework's claimed novelty is difficult to assess.

(b) The three conditions are not operationalized.
Definition 1 states the conditions but supplies no criteria for determining whether a given configuration satisfies them, and no way to measure the degree to which it does. For example, semantic persistence. The paper didn't provide an instruction of what meaning is preserved for different domains and how the degree of persistence should be measured.

(c) The distinction between Operability and execute-and-check is central to the framework and is not tested.
Table 2 identifies three levels of external interaction and locates the epistemic difference in the third: tool-use augments, execute-and-check evaluates, Operability appropriates. Only the third involves iteration on a persistent artifact with diagnostic feedback.
The experiments do not instantiate this comparison. Both domains contrast executed against mental, a closed loop against no external process at all. Neither contrasts a single-shot execution against an iterated one, nor diagnostic feedback against pass/fail on the same number of attempts. The comparison that would isolate the claimed epistemic difference is the one not run.

Major Issue 4: Tech quality - Bounded computation as an unexcluded alternative to the Epistemic account
The paper attributes every failure it observes to Epistemic hallucination. A more parsimonious hypothesis accounts for the same data: within bounded inference resources, the model cannot execute a search over a huge state space (e.g. of size O(n²·2^k)). Under this reading, the model writes correct BFS and then fails to produce the path not because it cannot tell verified state from interpolated state, but because simulating ~1152 states within a single forward pass (or a bounded chain of thought) exceeds its capacity.
The paper's own results fit the computational account better than the epistemic one. For example, CODE_mental reaches 100% on Maze Easy for both non-thinking models, then degrades as the state space grows from ~128 to ~400 to ~1152. Easy also requires tracking intermediate state and the models perform perfectly. What the data show is therefore not an absent capacity but a capacity with a ceiling that problem size exceeds. A failure that decays monotonically with scale is what resource exhaustion looks like; the paper does not explain why an architectural inability to track provenance would produce a difficulty gradient, or where the 100% on Easy comes from if the capacity does not exist.

Minor Issue 1: Unsupported assertions
Citations should be added to support claims. E.g. first paragraph of 1.1, similarly, first sentence of 1.2

Minor Issue 2: Presentation problems

Misattribution of Kalai et al. (2025)
"Kalai et al. (2025) further demonstrate that standard loss functions exacerbate this: cross-entropy penalizes explicit uncertainty relative to confident guessing, creating what they term the 'risk of hedging' — the model is trained to guess confidently rather than express doubt."
(a) The guessing/uncertainty argument belongs to post-training evaluation, not to the loss function.** The "reward guessing over acknowledging uncertainty" argument appears in Section 4 of that paper (_Post-training and hallucination_), and its object of analysis is benchmark grading, not the training objective. The role of cross-entropy in the cited paper is driving towards calibration instead of penalising uncertainty.
(b) "Risk of hedging" does not appear to be a term used by the cited authors.**

Reconcile the strength of the indistinguishability claim
§1.2 states the claim in its strong form: the same parameters encode verified facts and plausible interpolations, "making the two indistinguishable at inference time." §2.1 contains both a strong formulation ("LLMs cannot internally distinguish verified knowledge from plausible interpolation... an architectural limitation") and a weaker one ("models cannot access this distinction without external probing"). These are not equivalent: the first says the distinction is not represented, the second says it is represented but not available to the model's own output decision. We recommend that the manuscript standardise on the weaker formulation throughout.

Experiment result explanations
the quantitative attribution of ASP in RQ3: the paper credits the full 13% → 97% gap to the closed loop, but on its own account a substantial share of it is the effect of delegating computation, independent of iteration or feedback. The 47% → 97% segment is what the loop contributes.