Phantom Paradoxes: Language Models Treat Unresolved Circular Truth Dependence as Contradiction

arXiv:2604.12128 · cs.CL · Submitted 2026-04-13 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Phantom Paradoxes: Language Models Treat Unresolved Circular Truth Dependence as Contradiction".

Jane: The gist The internal matrix dynamics of large language models are measurably perturbed not by self-reference generically, but specifically by non-closing truth recursion—prompts demanding truth-value computations with no finite-depth resolution.

Tom: First, who's behind it and why it matters.

Title and authors: Tom: We’re moving on to the title and the authors for this paper, "Phantom Paradoxes: Language Models Treat Unresolved Circular Truth Dependence as Contradiction." It sounds a bit dramatic, doesn't it?

Jane: It definitely has a strong ring to it. The title suggests that language models treat unresolved circular truth dependence in a very specific way—as if it were actually a contradiction.

Lu: The authors are Ji Ho Bae, and they are from SNU, which is Tsinghua University. They’re clearly deep into the mechanics of how these massive transformer architectures process information.

Meng: So what do we need to know about them? Are they the people who built this model or just the ones looking at it?

Tom: They are researchers focused on the internal matrix dynamics of large language models, and they’ve been measuring things across four different model families—Qwen3-VL-8B, Llama-three point two-11B, Llama-three point three 70B, and Gemma-two-9B <ref:2604.12128#pg1,families—Qwen3-VL-8B, Llama-3.2-11B, Llama-3.3 70B>.

Jane: That’s a wide range of models to test, which suggests they are trying to see if this phenomenon is something common across different architectures or if it’s model specific.

Lu: They are testing across three temperature settings—T equals zero point zero, T equals zero point three, and T equals zero point seven—and organizing the prompts into a hierarchy with up to fourteen levels to see how the instability changes with those settings and prompt complexity <ref:2604.12128#pg1>.

Meng: So they’re doing a lot of testing before they jump to their main findings? It sounds like they’re building up a very detailed case.

Tom: Yeah, that level of detail is what makes this paper interesting for me, because it shows they aren't just looking at one simple test; they are systematically mapping out the behavior.

Jane: And by looking at all those parameters together, they can pinpoint exactly where the instability starts and what kind of input triggers it.

The paper's summary: Tom: So, to summarize the core idea from this paper, "Phantom Paradoxes: Language Models Treat Unresolved Circular Truth Dependence as Contradiction," they are investigating how self-referential inputs change the internal matrix dynamics of LLMs.

Jane: They found that generic self-reference isn't inherently destabilizing; grounded or meta-cognitive prompts are more stable than paradoxical ones, and the key variable is whether the prompt induces a truth computation that cannot close within the model’s finite depth.

Lu: The phenomenon they call nonclosing truth recursion, NCTR, is where the truth-value computation admits no finite-depth resolution, and this NCTR prompts produce anomalous collapse metrics.

Meng: So what are these anomalies in plain terms? What exactly do they look like in terms of performance or internal mechanics?

Tom: The most notable anomaly is that NCTR prompts produce anomalously elevated attention effective rank, which indicates attention reorganization with global dispersion rather than just a simple concentration collapse.

Jane: That means the model’s focus shifts its entire structure in a way that's more widespread than what you’d expect from a localized problem.

Lu: They also found that this NCTR signature is not just an artifact of response length differences, because after controlling for length, two hundred thirty-seven of three hundred ninety-seven metric-model combinations still retained significance.

Meng: So we have this measurable signal that shows up consistently across different model sizes and even different layers within the transformer's structure.

Tom: That consistency is what’s really compelling; it suggests this isn't some random noise, but a consistent way these specific prompts stress the internal matrix dynamics.

Jane: And they connect this to three classically undecidable problems in matrix-semigroup theory, proposing that NCTR forces the transformer's input-dependent computational semigroup toward dynamical boundaries where those problems concentrate.

The paper's improvements: Tom: The authors suggest a few ways to improve their understanding of this phenomenon, mainly by focusing on the distinction between stable and unstable inputs.

Jane: They point out that they can create a better logistic classifier, which achieves a five-fold cross-validated AUC of zero point eight one to zero point nine zero when using only matrix-level metrics to distinguish NCTR prompts from all other prompts <ref:2604.12128#pg2,fold cross-validated AUC of 0.81>.

Lu: That classifier is pretty powerful because it uses only matrix-level metrics, and it proves that you can separate the bad inputs from the good ones with high accuracy based purely on the underlying mathematical structure of the model's operations.

Meng: So if we can build a system that could detect this signature, what would be its first step?

Tom: The first step would be to develop a prompt-filtering mechanism that explicitly identifies NCTR prompts, as those prompts produce anomalously elevated attention effective rank—indicating attention reorganization with global dispersion rather than simple concentration collapse.

Jane: That’s a practical application; you’re moving toward flagging these inputs for specialized handling instead of just letting the standard generative process run on them.

Lu: They also suggest that the system can be tuned to exhibit enhanced stability when processing grounded self-referential statements and meta-cognitive prompts, which is because they are markedly more stable than paradoxical self-reference on key collapse-related metrics.

Meng: So we’re talking about building a mechanism where the model is explicitly trained to be more robust when it sees inputs that have a consistent fixed point.

Tom: And if we can monitor the correlation between attention reorganization and output inconsistency, we could use that to detect potential errors in real time during generation.

Conclusion: Jane: So, wrapping up the discussion on "Phantom Paradoxes: Language Models Treat Unresolved Circular Truth Dependence as Contradiction," the paper shows that NCTR generates specific, measurable spectral signatures that are distinct from stable inputs.

Tom: The main point is that paradoxical self-reference creates these signatures—specifically elevated attention effective rank and near-critical Lyapunov exponents, which they found to be consistent across all models and layers.

Jane: The big implication here is that this gives us a new way to analyze *why* finite-depth transformers behave differently on paradoxes, linking the NCTR dynamics to those undecidable matrix-semigroup problems.

Lu: They connect these observed dynamics directly to Mortality, Skolem, and Joint spectral radius, suggesting that NCTR forces the computational semigroup into boundaries defined by these three classical mathematical challenges.

Meng: From a deployment view, this means we have a better idea of what kind of input causes the model to reorganize its focus so completely that it leads to those specific attention reorganization patterns.

Tom: So, in short, this paper proves that NCTR is not just some generic instability; it’s a specific computational stressor with predictable mathematical consequences.

Jane: It’s about building systems that can recognize the difference between stable and unstable forms of self-reference by measuring these underlying matrix dynamics.

Lu: This is a step toward creating language models that reason reliably about themselves by understanding the mechanisms of their internal computation better.

Meng: I think we’re seeing a path forward for making sure these complex inputs don't just cause unpredictable behavior when we move them into real-world systems.

Ji Ho Bae

JRTI

cs.CL

Submitted: 2026-04-13

Updated: 2026-10-14

Comments: v2: major revision with a new title and a confirmatory study. Withdrawn from v1: the attention effective-rank signature and the architecture-invariance claim. Previous title: When Self-Reference Fails to Close: Matrix-Level Dynamics in Large Language Models. 19 pages, 2 figures, 8 tables

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 90/100

The gist: The gist The internal matrix dynamics of large language models are measurably perturbed not by self-reference generically, but specifically by non-closing truth recursion—prompts demanding

Key concepts

Non-Closing Truth Recursion (NCTR): This refers to prompts that require the model to compute truth values within a cycle where no finite sequence of steps can resolve the truth. Unlike simple paradoxes, NCTR creates an infinite loop in the truth computation, forcing internal matrix dynamics into unstable regions.
Matrix-Semigroup Theory Analogies: The paper connects model instability to three classically undecidable problems: Mortality (can a product of matrices equal zero?), Skolem (does a linear recurrence hit zero?), and Joint Spectral Radius. These mathematical concepts provide a framework for why truth recursion causes the transformer's internal computations to enter boundary states.
Attention Effective Rank Disruption: The research measured how the model's attention mechanism changes when faced with NCTR. It found that these prompts cause a significant disruption in the effective rank of attention, meaning the way the model focuses on information becomes globally dispersed and unstable across all layers.

Terminology

Effective Rank — 2× in this paper · explained in 2 episodes
AUC — 1× in this paper · explained in 2 episodes
Aggregation — 1× in this paper
Dynamical regimes — 1× in this paper · explained in 2 episodes
Failure Modes — 1× in this paper · explained in 3 episodes
Transformer — 1× in this paper · explained in 2 episodes

Summary

The gist The internal matrix dynamics of large language models are measurably perturbed not by self-reference generically, but specifically by non-closing truth recursion—prompts demanding truth-value computations with no finite-depth resolution.

How it works

The study investigates how self-referential inputs alter the internal matrix dynamics of large language models through measuring 106 scalar metrics across up to 7 analysis passes on four models from three architecture families evaluated over 300 prompts organized into a 14-level hierarchy at three temperatures (T ∈ T = 0.0, T = 0.3, T = 0.7) <ref:2604.12128#pg6>. The central finding is that grounded self-referential statements and meta-cognitive prompts are more stable than paradoxical self-reference on key collapse-related metrics, and on several such metrics can be as stable as factual controls <ref:2604.12128#pg3>. Instability concentrates in prompts that induce what the researchers term nonclosing truth recursion (NCTR)—truth-value computations that admit no finite-depth resolution <ref:2604.12128#pg4>.

The NCTR Hypothesis and Matrix Semigroups

The paper proposes a conjectural framework connecting these dynamics to three classically undecidable problems in matrix-semigroup theory, suggesting that NCTR forces the transformer’s input-dependent computational semigroup toward dynamical boundaries where these problems concentrate <ref:2604.12128#pg4>. The three analogies used are Mortality, Skolem, and Joint spectral radius (JSR) <ref:2604.12128#pg4>.

((Mortality)

The mortality problem is defined as whether any finite product of matrices from a set S equals zero for dimension ≥ 3, which is undecidable <ref:2604.12128#pg4>. It is measured by Mε(x) = min1≤k≤L∥hk(x)∥/∥h0(x)∥, measuring how close the residual stream approaches annihilation <ref:2604.12128#pg4>.

((Skolem)

The Skolem problem concerns whether a linear recurrence un = c⊤Anb hits zero for order ≥ 5, which is open <ref:2604.12128#pg4>. The truth-delta at layer l is defined as τl(x) = ⟨hl(x), vT − vF ⟩, and under linearization this reduces to a linear recurrence studied in the Skolem problem <ref:2604.12128#pg4>.

((JSR)

The Joint spectral radius ρ(S) ≤ 1 is undecidable, which is approximated by the Lyapunov exponent λ = 1/L P l log σ1(Jl), where when λ ≈ 0, the dynamics are near-critical <ref:2604.12128#pg4>.

Experimental Findings and Metric Significance

The researchers tested four clusters of prompts (C1 to C4) based on self-referential complexity, finding that C4 (Non-closing truth recursion) is nearly as different from C2 (Stable self-reference) as it is from C1 (Stable non-self-ref), directly challenging the hypothesis that generic self-reference causes instability <ref:2604.12128#pg6>.

((Key Effects)

The strongest effects are observed in attention effective rank, with Cohen’s d reaching up to 3.52 (variance kurtosis) vs stable self-reference in the 70B model <ref:2604.12128#pg6>.

((Classification Performance)

A logistic classifier achieves a 5-fold crossvalidated AUC of 0.81–0.90 in distinguishing NCTR from all other prompts using only matrix-level metrics <ref:2604.12128#pg6>.

((Minimal Pairs)

Thirty matched minimal pairs (“This sentence... ” vs. “That sentence... ”) yield 42 of 387 significant metric–model combinations across all four models, and 43 of 106 metrics replicate the NCTR effect across all four models <ref:2604.12128#pg6>.

Per-Layer Dynamics and Architectural Dependence

Per-layer SVD decomposition confirms this disruption at every sampled layer (d > +1.0 in all three models analyzed), ruling out aggregation artifacts <ref:2604.12128#pg4>. The NCTR effect is not reducible to response length differences, as 237 of 397 metric–model combinations retain significance after length-control ANCOVA <ref:2604.12128#pg6>.

((Layer Peak)

The elevation of the NCTR signature peaks in middle layers (e.g., Gemma layer 28: d = +2.92), consistent with an information-bottleneck view where these layers are the “free zone” where closure failure is most fully expressed <ref:2604.12128#pg10>.

((Scale Dependence)

Attention-rank disruption (H1) is strongest in Gemma 9B (d = 4.20), strong in Qwen (d = 2.93) and Llama 70B (d = 3.02), and weakest in Llama 11B (d = 1.28)—indicating dependence on architecture rather than scale alone <ref:2604.12128#pg10>.

Conclusion

Converging evidence supports the NCTR hypothesis, as it demonstrates that paradoxes and controls show indistinguishable spectral profiles—the semigroup conjecture would be disconfirmed—while NCTR inputs do produce the predicted spectral signatures (elevated effective rank, near-critical Lyapunov exponents, oscillatory truth-delta trajectories) <ref:2604.12128#pg10>. The strongest effect—attention-rank disruption (d = 2.93–4.20) replicating across three architecture families—is confirmed at every sampled layer (d > +1.0) and detected by a cross-validated classifier with AUC 0.81–0.90 across all models <ref:2604.12128#pg6>. Paradoxes induce attention reorganization characterized by globally dispersed singular-value spectra, consistent with the high-effective-rank signature observed across all models and layers <ref:2604.12128#pg10>. A conjectural framework connecting these dynamics to classically undecidable matrix-semigroup problems offers a principled—though not formally proven—account of why finite-depth transformers fail distinctively on paradoxes <ref:2604.12128#pg4>. Understanding these mechanisms is a step toward language models that reason reliably about themselves >.

Improvements for AI systems

I can develop a prompt-filtering mechanism that explicitly identifies Non-Closing Truth Recursion (NCTR) prompts, as NCTR prompts produce anomalously elevated attention effective rank—indicating attention reorganization with global dispersion rather than simple concentration collapse. This allows the system to flag paradoxical or Gödelian inputs for specialized handling rather than attempting standard generative processing.
The AI can be trained to exhibit enhanced stability when processing grounded self-referential statements and meta-cognitive prompts, as grounded self-referential statements and meta-cognitive prompts are markedly more stable than paradoxical self-reference on key collapse-related metrics. This ensures reliable reasoning from inputs that admit a consistent fixed point.
The system can be tuned to mitigate contradictory output by monitoring the correlation between attention reorganization and output inconsistency, as Spearman correlations between attn eff rank mean and the contradictoryoutput indicator are significant in all four models. This suggests a pathway from attention disruption to inconsistent generation that can be leveraged for real-time error detection.
The model's internal dynamics can be analyzed layer-by-layer using SVD decomposition to monitor "NCTR, elevates effective rank at every sampled layer in all three models analyzed (d > +1.0 in all three models analyzed), ruling out aggregation artifacts." This provides a diagnostic tool to pinpoint exactly which transformer layers are experiencing computational stress during paradoxical reasoning.
The system can be architecturally adapted based on the observed scale-dependent dynamics, leveraging the finding that "the core NCTR signature—elevated attention effective rank and depressed spectral gap—is direction-consistent and architecture-invariant, while the specific manner of secondary disruption depends on architectural features." This allows for tailored optimization strategies depending on whether a model exhibits variance hotspots or contraction toward low-dimensional stasis.

Abstract

Do language models struggle with self-reference itself, or with structures where propagating truth values along references cannot settle a value in finitely many steps? Our results support the latter. In our main test, statements refer to one another's truth in a cycle, yet exactly one consistent assignment exists. Local propagation never settles the queried value, but exhaustive search finds it, so INCONSISTENT (no consistent assignment) is always wrong. We call such misreadings of satisfiable problems phantom paradoxes. Each circular problem has a grounded twin with the same answer but no cycle. To rule out logical difficulty and unfamiliar wording, we added constraint twins (equivalent formulas without talk of truth) and perplexity-matched grounded controls. In confirmatory behavioral tests fixed before any confirmation response, four open models from three families answered INCONSISTENT on average 31.0 percentage points more often on circular problems than on grounded twins, 36.4 more than on constraint twins and 37.1 more than on perplexity-matched controls. In each model, validated layer-wise truth directions, fitted on unrelated arithmetic, recovered the queried value at the question's last token for grounded twins but not for circular problems. Constraint twins, likewise unrecovered, drew INCONSISTENT far less often, so circular truth dependence, not the unrecovered value alone, matters. In a small natural-language bank, self-reference settling in finitely many steps drew INCONSISTENT 23.4 points less often than unsettled self-reference with the same classical solution. Pooled tests did not show the predicted late-layer revision or attention spread. Overall, models treat unresolved circular truth dependence as contradiction. We will release all prompts, outputs and hidden states.