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REVIEW 4 major objections 4 minor 44 references

High-confidence LLM hallucinations are often stable, internally coherent states — 'false fixed points' — not fragile glitches, and probes show wrong answers are as locally stable as correct ones.

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2026-08-04 09:29 UTC pith:SCU5VGFL

load-bearing objection The no-instability-gap result is real, but 'stable miscalibration' outstrips the probe, which measures input-embedding sensitivity, not the internal dynamics that matter. the 4 major comments →

arxiv 2510.14925 v4 pith:SCU5VGFL submitted 2025-10-16 cs.AI cs.CLcs.LG

False Fixed Points: Kantian Feedback, Stable Miscalibration, and Representational Compression in LLMs

Akira Okutomi This is my paper
classification cs.AI cs.CLcs.LG
keywords false fixed pointsstable miscalibrationhallucinationhidden-state sensitivityKantian feedbackH-Riskrepresentational compressionhigh-SNR inertia
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tries to establish that many LLM hallucinations are not transient failures of fragile computation but 'stable miscalibration': the model settles into an internally coherent, locally stable representation that is confidently wrong. The empirical anchor is the absence of an instability gap: across three open-weight models, overconfidently wrong answers are no more sensitive to small input perturbations than confidently correct ones. The paper also shows that a Kantian-inspired self-critique prompt reduces hidden-state sensitivity globally, and that for one model (Qwen2.5) large activation norms compress small perturbations, suggesting a high signal-to-noise 'inertial' regime. If the central claim is right, output-only fixes such as temperature scaling or re-sampling will have limited power; process-level interventions that perturb and re-evaluate the inference trajectory are needed. Why it matters: distinguishing stable-but-wrong from fragile-but-wrong changes what kind of correction strategy can work.

Core claim

The paper argues that hallucinations in modern LLMs often behave less like bugs caused by fragile dynamics and more like robust but misaligned specifications supported by stable internal representations. It formalizes this through a Kantian feedback framing and a minimal linear feedback model in which stability and correctness can diverge. Empirically, the paper reports no instability gap: final-layer sensitivities for confidently correct versus overconfidently wrong items are nearly identical (e.g., 16.45 vs 16.55 for Llama-3.1-8B; 4.74 vs 4.73 for Qwen2.5-7B; 13.06 vs 13.37 for DeepSeek-R1-Distill). A Kantian prompt that asks the model to enumerate failure modes and revisit its stance redu

What carries the argument

The central probe is the layer-wise hidden-state sensitivity S_ℓ = ||Δh_ℓ||/||Δe||, the ratio of hidden-state change to a small input-embedding perturbation, measured at a handful of layers per model with embedding noise σ = 0.01 and 40 trials per item. This operationalizes 'local stability' of internal representations. The paper also introduces H-Risk, a composite instability index aggregating spectral margin, conditioning, temporal sensitivity, and innovation amplification of a closed-loop operator Φ = A − KH from a linear–Gaussian state-space model; for LLMs it uses a coarse domain-wise proxy built from confidence fluctuations and overconfident errors. The 'Kantian feedback' mechanism is

Load-bearing premise

The load-bearing premise is that the layer-wise sensitivity probe — measured at only four probed layers per model with embedding noise σ = 0.01 — operationally captures the local fragility that matters for hallucination persistence; if the relevant fragility lives where this probe does not look (attention-logit saturation, the final decoding step, cross-layer amplification beyond probed layers, or perturbations of internal states rather than input embeddings), then the 'no in

What would settle it

Run the same sensitivity analysis on all layers (or via full Jacobian–vector products) and additionally perturb internal hidden states rather than input embeddings, on a model where output-only calibration methods are known to fail; if confidently wrong items then show substantially higher sensitivity than confidently correct items, the claimed absence of an instability gap would be shown to be an artifact of the probe's placement and noise scale.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If hallucinations are stable fixed points, lowering sampling temperature or rescaling output probabilities will not reliably remove high-confidence errors, because those heuristics do not disturb the underlying attractor.
  • Abstention-aware self-critique (answering 'cannot judge') reduces overconfident wrong commitments, but only by sacrificing coverage; the trade-off is sharpened, not eliminated, by a rule-based explicit feedback gate.
  • Calibration assessments should include internal stability probes (layer-wise sensitivity, spectral/activation norms), because output-level accuracy alone cannot distinguish brittle errors from stable-but-misaligned ones.
  • Models can occupy distinct epistemic regimes — for example, Qwen2.5's inertial high-SNR regime versus DeepSeek-R1's reactive regime — so correction strategies may need to be model-specific rather than one-size-fits-all.
  • The absence of an instability gap across three main models and two smaller checkpoints suggests the stable-miscalibration pattern is not an artifact of a single architecture or scale.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the framework predicts that hallucination detectors based on internal-state probes (sensitivity or Jacobian spectra) will outperform output-only consistency checks specifically on high-confidence errors; this is a directly testable comparison.
  • Editorial extension: if high-SNR compression is the mechanism, then models with larger activation norms and normalization-based architectures should show more persistent hallucinations under re-sampling; comparing model families along this axis would test the mechanism.
  • Editorial extension: the paper's 'quiet-then-flip' attention-saturation channel suggests that rare, larger perturbations (adversarial or contextual) may flip a stable-looking wrong answer abruptly; probing sensitivity as a function of perturbation scale could reveal bistable behavior.
  • Editorial extension: the separation of stability from truth-tracking implies that fine-tuning or training-time interventions that explicitly penalize stable wrong attractors may be more effective than post-hoc calibration; this is an untested but plausible consequence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a 'Kantian feedback' framework that reframes epistemic stability as closed-loop control-theoretic stability, introduces a composite instability index H-Risk (with an LTI instantiation and an LLM proxy H_proxy), and tests the framework in two small experiments: linear-Gaussian Kalman simulations and a binary fact-checking study on three 7-8B open-weight LLMs with an abstention-aware 'Kantian' prompt. The main empirical claim is negative: confidently wrong answers show no 'instability gap' under a layer-wise input-embedding sensitivity probe, which the authors interpret as evidence for 'stable miscalibration' — hallucinations as locally stable but truth-misaligned attractors. The paper is heavily hedged, repeatedly framing the LLM experiments as a 'sanity check' and 'exploratory illustration,' and provides code, data, and fixed seeds for reproducibility.

Significance. If the stable-miscalibration claim were established, it would be an important conceptual contribution: it separates robustness from truth-tracking and implies that output-only interventions (temperature scaling, re-sampling) may be insufficient for high-confidence hallucinations. The paper's strength is its explicit, testable framing and its unusually candid limitations section. It also ships reproducible code and frozen data, which is commendable. However, the load-bearing empirical result is a null difference measured with a probe that the paper itself concedes does not estimate the local Jacobian, and the null is reported without inferential statistics. The present evidence is therefore suggestive rather than demonstrative; the contribution is a promising hypothesis-generating framework rather than a confirmed mechanism.

major comments (4)
  1. [§5.1, §4.2, §3.4] The 'no instability gap' result rests on S_l = ‖Δh_l‖/‖Δe‖, an input-embedding sensitivity probe at four layers and σ=0.01. This does not measure whether a false answer is a fixed point of the internal dynamics: the paper explicitly says 'we do not estimate J_t' (§3.4), and §5.2 itself describes a 'quiet-then-flip' regime where attention-logit saturation makes small input noise unable to move the dominant pattern while the computation is one logit flip away from a different answer. In that regime S_l would report low sensitivity even though the answer is brittle. The null result is therefore equally consistent with stable miscalibration and with saturation-induced apparent stability. To support the headline conclusion, the authors need internal-state or attention-logit perturbations, Jacobian-vector products, or an explicit argument for why the input-embedding probe captures the relevant
  2. [§5.1, Table 4, Appendix Figure 5] The central null result is reported as point estimates only: 16.45 vs 16.55, 4.74 vs 4.73, 13.06 vs 13.37. There are no standard errors, confidence intervals, or equivalence tests. The appendix noise sweep says group differences 'remain negligible' but explicitly omits the numerical values ('omitted for brevity'). A null difference with no measure of uncertainty cannot anchor the claim that confidently wrong answers are 'at least as locally stable' as correct ones. The authors should report per-group SEs/CIs, ideally a pre-specified equivalence bound, and the group-stratified noise-sweep numbers, so readers can assess whether the absence of a gap is a meaningful finding or just an underpowered comparison.
  3. [§3.3, §4.1, footnote 3] H_proxy(d) is built from overconfident-wrong indicators and confidence fluctuations, and footnote 3 concedes it is 'designed for simplicity rather than identifiability.' The paper also states 'we do not attempt a formal bound' between H_proxy and the operator-level HRiskLLM. The validation that H_proxy 'flags the same high-stakes domains' where ΔSE_policy improves is not independent, since both quantities are computed from the same items and policy decisions. This circularity weakens the domain-wise claim. An out-of-sample or independently labeled test of H_proxy would substantially strengthen the paper.
  4. [§3.2, Eq. (3.1)] HRiskLTI is defined as the product m̄_LTI·c̄_LTI·s̄_LTI·ā_LTI, but only m_LTI = 1/(1−ρ(Φ)) is given explicitly. The conditioning, integrated-sensitivity, and innovation-amplification descriptors c_LTI, s_LTI, a_LTI are described only verbally; no formulas, normalizations, or reference-configuration values are provided. As a result, the simulation claim that 'higher HRisk predicts overconfident errors and degraded closed-loop behaviour' cannot be reproduced or quantitatively checked. This is a core theoretical contribution, not a cosmetic gap. Please provide the full functional forms and the reference configuration parameters.
minor comments (4)
  1. [Abstract/throughout] There are several missing-space typos from LaTeX markup, e.g., 'Kant’sCritique ofPure Reasonas', 'we donotfind', 'with∆', 'al.,'. A careful proofread is needed.
  2. [Appendix noise sweep] The sentence 'numerical values omitted for brevity' is unsatisfying for a robustness claim that is central to the null result. At minimum, a table of the group-stratified means across σ values should be included.
  3. [Table 6] In the cultural_industry domain, C2 has lower selective accuracy and higher OC-Wrong|Ans than C0, which reads as a degradation; the text should comment on such exceptions rather than only emphasizing the pooled improvement.
  4. [Table 2] The CritPt probe uses gpt-4.1-mini, not the three open-weight models used in the main sensitivity analyses. The reader should be reminded that this is a different model family and not directly comparable to the stability results.

Circularity Check

1 steps flagged

Hproxy(d) validation is partly built from its own inputs; the central no-instability-gap measurement is independent.

specific steps
  1. self definitional [Section 4.1/footnote 3 (Hproxy construction) and Section 5 'What H-Risk tells us' (ΔSEpolicy consistency claim)]
    "Hproxy(d) is a deliberately coarse, task-specific proxy (built from discretised confidence levels and a simple overconfident-wrong indicator) ... our per-item instability factor is designed for simplicity rather than identifiability of policy-specific causal effects. ... Hproxy(d) flags the same high-stakes domains ... in which the cautious and Kantian policies yield the largest reductions in policy-aware squared loss (ΔSEpolicy)"

    Hproxy(d) is constructed from the 'overconfident-wrong indicator', while the ΔSEpolicy improvement of C1/C2 over C0 is dominated by abstaining on overconfident-wrong C0 items (the -0.39 cell in Table 3 maps p=0.8 on a wrong item, loss 0.64, to abstention p=0.5, loss 0.25). Therefore the domains flagged by Hproxy(d) and the domains with largest ΔSEpolicy reductions share the same OC-wrong signal by construction; the claimed 'consistency' is not an independent empirical confirmation. The footnote's admission that the factor was 'designed for simplicity rather than identifiability' indicates the authors themselves do not treat the proxy as a causally identified predictor.

full rationale

The only reducible step I can exhibit is the Hproxy(d)-vs-ΔSEpolicy consistency claim, where the proxy's input (overconfident-wrong indicator) is also the main driver of the outcome (loss reduction via abstention). This is a genuine but localized circularity in a supporting sanity-check component; it does not touch the core 'no instability gap' finding, which is an independent measurement of S_l at fixed σ=0.01 across correct/wrong groups (16.45 vs 16.55; 4.74 vs 4.73; 13.06 vs 13.37). The skepticism about whether S_l captures attention-logit saturation or hard-switching fragility is a probe-validity objection, not circularity; the paper explicitly does not estimate J_t, so the gap between probe and theory is acknowledged rather than concealed. There are no load-bearing self-citations, no imported uniqueness theorem, and no ansatz smuggled in via citation; the Kantian framing and RMSNorm mechanism are explicitly heuristic. Because the circular step is partial and located in the proxy-validation rather than in the central stability measurement, a score of 4 (rather than 6+) is appropriate.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 4 invented entities

The central claim rests on four hand-designed constructs with no independent falsifiable handles (H-Risk, effective signal temperature, stable-miscalibration attractor, Kantian feedback framing), plus six hand-chosen configuration choices (confidence mapping {0.5,0.8}, abstention p=0.5, σ=0.01, H_proxy design, LTI reference configuration, probe settings). The mathematics actually relied on (bounded P under ρ(Φ)<1 and detectability; non-normal amplification) is standard. The two load-bearing domain assumptions are the one-way heuristic mapping from Kant's architecture to a linear-Gaussian filter (§2.2) and the identification of embedding-perturbation sensitivity with the fragility that matters for hallucination persistence (§4.2). Ground-truth labels are OpenAI-API-synthesized. The proxy's monotone relation to the true index is asserted, not bounded (§3.3).

free parameters (6)
  • A-layer confidence mapping = p ∈ {0.5, 0.8}
    Coarse confidence levels chosen by hand for the rule-based A-layer classifier; robustness check in Table 5 varies the high value to 0.7/0.9 and preserves the qualitative ordering but changes absolute losses.
  • Abstention penalty in policy-aware loss = p = 0.5, loss = 0.25
    Abstention is assigned p = 0.5 in the squared loss (p − y)^2; this modeling constant determines the tradeoff picture across C0/C1/C2.
  • Embedding noise scale σ = σ = 0.01
    Chosen to stay in the 'local linear regime'; appendix sweep shows S_ℓ changes by a factor of 2–3 across σ ∈ {0.005, 0.05}, so absolute sensitivity values are scale-dependent.
  • H_proxy(d) construction = n/a (designed, not fitted)
    Domain-wise proxy built from discretized confidence levels and an overconfident-wrong indicator, normalized by maximum; footnote 3 says it is 'designed for simplicity rather than identifiability.'
  • LTI reference configuration and descriptor forms = reference (A0,H0,Q0,R0,K0); m_LTI = 1/(1−ρ(Φ))
    Normalization point and descriptor functional forms chosen by hand; every descriptor equals 1 at the reference, fixing HRisk_LTI = 1 there, and the product form in Eq. 3.1 is one of many monotone separable choices.
  • Probe settings = n_trials = 40; 4 layers per model
    Perturbation trials per item and the few probed layers per model (L8/16/24/32, L7/14/21/28, etc.) are chosen by hand; group-difference statistics are not reported.
axioms (6)
  • domain assumption A linear-Gaussian state-space model is a valid formal instantiation of Kant's epistemic architecture (sensibility/understanding/reason as y/x/K).
    Section 2.2.1; the paper itself calls the correspondence 'heuristic and one-way' and disclaims doctrinal claims.
  • standard math Under detectability of (A,H) and ρ(Φ) < 1, the steady-state error covariance P is bounded via the Lyapunov equation.
    Sections 2.2.2 and 7; standard Kalman filtering / Lyapunov stability theory.
  • ad hoc to paper A monotone separable functional satisfying axioms (H1)-(H3) is a meaningful definition of an instability index.
    Definition 3.1; the axioms are deliberately weak and the multiplicatively separable product (Eq. 3.1) is chosen, not derived.
  • domain assumption H_proxy(d) is heuristically monotone in the notional HRisk_LLM.
    Section 3.3; the paper explicitly disclaims a formal bound between the proxy and the operator-level index.
  • domain assumption S_ℓ = ∥Δh_ℓ∥/∥Δe∥ at σ = 0.01 captures the local fragility of inference that matters for hallucination persistence.
    Sections 4.2 and 5.1; appendix noise sweep shows the absolute scale depends on σ, and no equivalence test or calibration of the probe against known instabilities is provided.
  • domain assumption OpenAI-API-generated binary items and labels are valid ground truth for factual correctness.
    Appendix 'Dataset and scripts (optional regeneration)': prompts and labels are synthesized via generate_kantian_prompts.py; labels are model-generated, not externally verified.
invented entities (4)
  • H-Risk / HRisk_LTI / HRisk_LLM composite index no independent evidence
    purpose: Quantify epistemic instability from spectral margin, conditioning, temporal sensitivity, and innovation amplification.
    Hand-chosen descriptors and normalization reference; validated only on internal LTI simulations with no external falsifiable prediction.
  • Effective signal temperature / high-SNR inertia no independent evidence
    purpose: Explain Qwen-2.5's low local sensitivity despite large spectral norms.
    Section 5.2; the paper labels it an analogy ('should not be confused with softmax temperature') and says the results 'motivate, but do not establish' it.
  • Stable miscalibration / robust misaligned attractors no independent evidence
    purpose: Central characterization of confident errors as locally stable fixed points of the inference dynamics.
    Supported only by the embedding-perturbation probe used in this paper; no independent falsifiable handle beyond the paper's own measurements.
  • Kantian Feedback framework (reason as regulator gain K) no independent evidence
    purpose: Design principle for critique and self-limitation in the Prompt-Critique-Revision loop.
    Interpretive framing; the paper disclaims doctrinal claims and notes prior Kant-cybernetics work [5,6,7].

pith-pipeline@v1.3.0-alltime-deepseek · 20109 in / 27734 out tokens · 227984 ms · 2026-08-04T09:29:47.616087+00:00 · methodology

0 comments
read the original abstract

High-confidence errors in large language models are often treated as fragile failures. We study an alternative: some errors may be false fixed points, locally stable, internally coherent, and confidently wrong. This separates robustness from truth-tracking. We develop the separation through a Kantian commitment-gate framing and a minimal linear feedback model in which stability and correctness can diverge. Across three open-weight models, overconfident wrong items are not systematically more locally fragile than confidently correct items under our hidden-state sensitivity probes. Abstention-aware self-critique reduces overconfident wrong commitments by sacrificing coverage, and C3-R, a rule-based explicit feedback gate, sharpens that tradeoff rather than eliminating it. These results motivate, but do not establish, high signal-to-noise (high-SNR) inertia and representational compression as possible mechanisms for stable miscalibration.

Figures

Figures reproduced from arXiv: 2510.14925 by Akira Okutomi.

Figure 1
Figure 1. Figure 1: Domain-wise H-Risk proxy. Mean per-item H-Risk proxy Hproxy(d) by domain, normalised by the maximum across domains. Error bars show within-domain standard deviation (descriptive only). Higher values indicate domains where items are both policy-wise unstable and prone to overconfident mistakes [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: ∆SEpolicy vs C0 (C1/C2) across domains (negative = improvement). Domains are ordered by mean ∆SEpolicy (best / most negative) with the best domain shown at the top. Points show the mean paired per-item difference (condition − C0) within each domain; horizontal bars show 95% paired bootstrap confidence intervals over items. ∆SEpolicy is computed on paired items using the policy-aware loss (p − y) 2 , where … view at source ↗
Figure 3
Figure 3. Figure 3: Layer-wise internal sensitivity Sℓ under standard vs Kantian prompts for three open-weight instruction-tuned models (all items, C0 baseline). The vertical axis is shared across models. In each panel we plot the mean local sensitivity at several representative layers under the overconfident baseline prompt (Standard) and the Kantian prompt. Across architectures, the Kantian prompt consistently reduces inter… view at source ↗
Figure 4
Figure 4. Figure 4: Spectral and activation profiles for Llama-3.1-8B and Qwen-2.5-7B. (a) Spectral norm ∥W∥2 of the attention output projections (oproj) across layers. (b) Spectral norm of the MLP down-projections (downproj). (c) Layer-wise activation norm ∥x∥2 at the last token. In all panels, solid lines (circles) denote Llama-3.1-8B and dashed lines (squares) denote Qwen-2.5-7B. Qwen-2.5 exhibits consistently larger spect… view at source ↗
Figure 5
Figure 5. Figure 5: Noise-scale sweep for local sensitivity on DeepSeek-R1-Distill (C0 baseline, standard prompt). Each curve shows the mean sensitivity Sℓ at a given layer as a function of the embedding noise standard deviation σ ∈ {0.005, 0.01, 0.025, 0.05}. Larger noise scales yield smaller estimated Sℓ, reflecting departure from the purely local linear regime, but the absence of an instability gap between confidently corr… view at source ↗
Figure 6
Figure 6. Figure 6: Brier score distribution for the C0_ECE condition (pooled over domains). The discrete support reflects the limited grid of self-reported probabilities (together with binary labels), producing a small number of possible Brier values. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Per-item ∆SEpolicy distributions by domain (violin plot). This view details the spread of per-item policy-loss changes within each domain, complementing the aggregated means in [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Robustness sweep under model mismatch. We introduce random perturbations to the state transition matrix A while keeping the estimator fixed. The plot shows that even with this mismatch, the instability index HRisk remains predictive of tail calibration error (NISq). Auxiliary probe: CritPt abstention behaviour To provide an external check on refusal/hesitation behaviour under extreme task difficulty, we ra… view at source ↗

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