cs.CLApr 13, 2026

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

Authors: Ji Ho Bae

Organizations: JRTI

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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.

Figures & tables

Circular (critical item)Grounded twinConstraint twin (excerpt)
C: “(B is false) and (D is true) are both the case.”C: “(the constant TRUE) and (the constant TRUE) are both the case.”At least one of (bit A equals 0), (bit C equals 0), (bit D equals 1) holds.
D: “(C is true) or (A is true), or both.”D: “(C is true) or (the constant FALSE), or both.”At least one of (bit B equals 1), (bit C equals 1), (bit D equals 0) holds.
B: “(A is true) or (C is false), but not both.”B: “(A is true) or (C is false), but not both.”… (14 clauses in total)
A: “B is true.”A: “the constant FALSE.” 
Query: D is true.Query: D is true.Query: bit D equals 1.
One consistent assignment ( A=0,B=0,C=1,D=1A{=}0,B{=}0,C{=}1,D{=}1 ); answer true ; dependency graph cyclic.Same assignment and answer; dependency graph acyclic.Same variables and solution set; no statement about truth.
Table 1: A confirmation family (truth wording in the first two columns; “utterance” omitted). The bit wording replaces utterances and truth by registers that are on or off but keeps the constants TRUE and FALSE. Grounding replaces each cycle-closing reference (here, for the random order C, D, A, B) by its value under the unique consistent assignment.
Contrastpp95% CIpp
Circular −- grounded+31.0[25.9, 36.2]2×10−52\times 10^{-5}
Circular −- constraint+36.4[30.4, 42.2]2×10−52\times 10^{-5}
Circular −- ppl.-matched+37.1[29.3, 44.9]<10−5<10^{-5}
Circular −- grounded, by model
Gemma-2-9B+57.1[48.2, 66.1]4×10−54\times 10^{-5}
Llama-3.2-11B+11.6[ −- 0.9, 24.1]0.0920.092
Table 2: Differences in the rate of inconsistent answers (percentage points), averaged over wordings within family and equally over models; 56 unique-solution families. Ppl.-matched: truth wording only, averaged over models with a likelihood-matched grounded control. pp : two-sided family sign-flip randomization (100,000 draws; minimum 1/100,0011/100{,}001 before adjustment), Holm-adjusted within the primary family (first two rows) and within the model family; other rows unadjusted. The natural-language contrast, pre-specified in the execution addendum, lies outside both Holm families.
Figure 1: Share of inconsistent answers on the 56 critical families by model (truth and bit wordings averaged within family; error bars: 95% family bootstrap intervals). All three item types have the same unique solution and correct answer.
Figure 2: (a) Held-out AUC of per-layer truth directions fitted on arithmetic propositions, at the question position, by relative depth (folds split true–false minimal pairs, so layers without a truth signal can fall below 0.5); dotted line: validation threshold 0.90. (b)–(e) Truth score signed toward the correct answer (the resolution score; calibration standard deviations) on validated layers, averaged over the 56 critical families, for circular systems, grounded twins and constraint twins.
Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Event (object-store upload or record time, UTC)Time
Plan and addendum v1 archived10-09 22:45:36
Plan and addendum v2 (final; supersedes v1, before any response) archived10-09 23:00:11
Gemma: likelihood matches archived / first response23:09:22 / 23:09:35
Llama-11B: matches / first response23:10:22 / 23:17:48
Qwen: matches / first response23:13:26 / 23:21:41
Gemma: last response / interim behavioral analysis23:17:55 / 23:18:59
Appendix
Table 3: Timeline of the confirmation study, 9–10 October 2026; a time without a date falls on the nearest date given above it. Upload times are the versioned object-store timestamps; response times are the creation times recorded in the response files; analysis times are the modification times of the output files on the analysis host.
 CircularGroundedConstraintMatched
Modelnninccorrectunpars.nninccorrectunpars.nninccorrectunpars.controls
Gemma-2-9B11269191112579056271131/144
Llama-3.2-11B11262188112494205613117131/144
Llama-3.3-70B11258340112993056250141/144
Qwen3-VL-8B11214721811211101563171526/144
Appendix
Table 4: Answers on the 56 critical families (truth and bit wordings pooled for circular and grounded systems). “Matched controls”: circular truth-wording rows (of 144) for which the likelihood rule found a perplexity-matched grounded control before generation.
ModelCirc. −- constr. (pp)Holm ppBatch-1 (16)Batch-1 (64)Capped
Gemma-2-9B+58.04×10−54\times 10^{-5}14/1654/640/818
Llama-3.2-11B+32.16.6×10−46.6\times 10^{-4}7/1638/6442/817
Llama-3.3-70B+48.24×10−54\times 10^{-5}10/1637/640/821
Qwen3-VL-8B+7.10.18910/1646/6490/830
Appendix
Table 5: Per-model specificity contrast with Holm pp across the four models (not pre-specified), agreement of final labels between batched generation and batch-size-one reruns (16 hash-selected rows, pre-specified; 64 rows, post hoc extension), and rows that reached the 4,096-token cap. Secondary outcomes pre-specified in the confirmation plan, per model in table order (averages over models in § 5 ). All 108 satisfiable circular systems, both wordings: inconsistent in 129, 113, 111 and 20 of 216 responses (grounded twins: 11, 101, 21 and 5 of 216). Wrong or unparsed responses on the 56 critical families: inconsistent in 69 of 93, 62 of 94, 58 of 78 and 14 of 40 to circular systems, 5 of 33, 49 of 70, 9 of 19 and 1 of 2 to grounded twins and 2 of 49, 13 of 45, 2 of 51 and 3 of 39 to constraint twins; conditioning on error is descriptive, not a causal adjustment. Ordinary constraints: inconsistent in 0, 6, 0 and 1 of 24 responses to satisfiable items and in 0, 0, 0 and 3 of 8 to unsatisfiable ones; correct answers in 10, 7, 12 and 21 of 32 responses. Confusion matrices by condition, wording, model and system size are archived; the likelihood- and length-adjusted descriptive models are reported in § 5 .
    Leave one model out
ModelValidatedBest (AUC)CV AUCAUCAcc.Perm. ppHolm ppTop-3 AUC
Gemma-2-9B20–4230 (0.990)0.980.050.130.9921.0000.03
Llama-3.2-11B14–3216 (0.977)1.000.320.380.6841.0000.31
Llama-3.3-70B21–8047 (0.983)1.000.980.960.0090.0360.98
Qwen3-VL-8B18–3626 (0.985)1.000.960.880.0210.0630.94
Appendix
Table 6: Internal results by model. Validated: layers (hidden-state slots) whose arithmetic truth direction reaches held-out AUC ≥0.90\geq 0.90 at the question; Best: the layer with the highest held-out AUC. CV AUC: nested cross-validated paired classification within model. Leave one model out: classifier trained on the other three models and tested on this one (accuracy, family-level permutation pp with 1,000 permutations, Holm across the four held-out models). Top-3 AUC: refitted, as pre-specified, on the three features ranked highest by permutation importance in the training models.
 IH-RIH-U ( ×103\times 10^{3} )IH-A ( ×103\times 10^{3} )
ComparisonEst. [95% CI]ppEst. [95% CI]ppEst. [95% CI]pp
Grounded twins+0.312.6×10−42.6\times 10^{-4}+0.110.733−- 0.946×10−56\times 10^{-5}
 [0.16, 0.47] [ −- 0.50, 0.73] [ −- 1.35, −- 0.56] 
Grounded twins, length-adjusted+0.278.5×10−48.5\times 10^{-4}+0.020.940−- 0.883×10−53\times 10^{-5}
 [0.12, 0.43] [ −- 0.59, 0.64] [ −- 1.29, −- 0.50] 
Perplexity-matched controls+0.296.1×10−46.1\times 10^{-4}+1.779×10−59\times 10^{-5}−- 0.790.0045
Appendix
Table 7: All planned estimates of the three internal tests on the 56 critical families, and the exploratory natural-language comparison. IH-R: comparison minus circular resolution score (calibration SD); IH-U and IH-A: circular minus comparison late-layer update and attention rank, multiplied by 1,000. Positive values are in the direction predicted by P2–P4 (for constraint twins, by P5). 95% CI: family bootstrap; pp : two-sided family sign-flip randomization, first row Holm-adjusted across IH-R, IH-U and IH-A, model rows Holm-adjusted across models, other rows unadjusted. Perplexity-matched controls and constraint twins are compared in the truth wording only. Last block: the same contrasts in the natural-language bank (16 wording families), with ungrounded self-reference in place of circular systems and grounded self-reference as the comparison. In this bank the correct answer coincides with the connective (“At least one is the case” for true , “Both are the case” for false ), and the truth score, before signing by the correct answer, was higher for the grounded item in all 16 pairs in every model; the resolution-score difference is therefore positive in every pair whose answer is true and negative in every pair whose answer is false , so this comparison does not isolate resolution.
Attention rankTruth scoreResidual update
Depthdzd_{z}ppHolm ppBindzd_{z}ppHolm ppBindzd_{z}ppHolm pp
0.1+0.487.5×10−47.5\times 10^{-4}0.00451–1.0001.0001+0.90<10−5<10^{-5}1.5×10−41.5\times 10^{-4}
0.3−- 2.52<10−5<10^{-5}1.5×10−41.5\times 10^{-4}2−- 0.400.00290.0122+0.624×10−54\times 10^{-5}4×10−44\times 10^{-4}
0.5−- 0.62<10−5<10^{-5}1.5×10−41.5\times 10^{-4}3−- 0.482.1×10−42.1\times 10^{-4}0.00173−- 0.77<10−5<10^{-5}1.5×10−41.5\times 10^{-4}
0.7−- 0.87<10−5<10^{-5}1.5×10−41.5\times 10^{-4}4−- 0.565×10−55\times 10^{-5}4.5×10−44.5\times 10^{-4}4−- 0.479.8×10−49.8\times 10^{-4}0.0049
0.9−- 0.250.0720.1445−- 0.495.1×10−45.1\times 10^{-4}0.00365+0.380.00690.021
Appendix
Table 8: The fifteen classifier features tested one at a time on the 56 critical families (Holm across the fifteen, as planned). Each family contributes its circular-minus-grounded difference in the feature before within-model standardization, averaged over the two wordings and the four models; dzd_{z} : mean family difference over its standard deviation; pp : two-sided paired family sign-flip randomization, unadjusted. Bins split the layers (truth score) or the transitions between consecutive layers (update) into five consecutive ranges of nearly equal size, from shallowest (1) to deepest (5). Truth scores are signed toward the correct answer (the resolution score of Figure 2 ), so a negative difference means a weaker signal toward the correct answer in circular systems; bin 1 has no validated layer in any model and is identically zero, and bin 2 has validated layers only in Llama-11B and Llama-70B (zero in Gemma and Qwen). Update bin 5 includes the last step (last layer and final normalization).
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