Why do language models trained on contradictory data prefer correct answers? In controlled experiments with small transformers (3.5M--86M parameters), we show that this preference tracks the compressibility structure of errors rather than truth per se. We train GPT-2 style models on corpora where each mathematical problem appears with both correct and incorrect solutions -- a denoising design that directly models conflicting information about the same fact. When errors are random, models extract the correct signal with accuracy scaling from 65% to 85% with model size. When errors follow a coherent alternative rule system, accuracy drops to chance (~45--51%): the model cannot distinguish the false system from truth. A multi-rule experiment reveals a sharp crossover: a single coherent alternative rule eliminates truth bias entirely, but adding a second competing rule restores most of it (47%->78%), with continued growth through N=10 (88%). The same pattern reproduces on real Wikipedia text (71% vs 46%). We propose the Compression--Consistency Principle as an explanatory hypothesis: in these settings, gradient descent favors the most compressible answer cluster, not truth per se. Truth bias emerges only when falsehood is structurally incoherent. Whether this principle extends to large-scale pretraining remains an open question.
Figures & tables
Config
Layers
dmodel
Heads
Parameters
tiny
4
256
4
∼3.5 M
small
6
384
6
∼12 M
medium
8
512
8
∼26 M
large
12
768
12
∼86 M
Table 1: Model configurations used in all experiments.
Condition
Correct
Incorrect
Ratio
Purpose
Equal random
1
1 random
1:1
Signal extraction from random noise
Equal coherent
1
1 coherent
1:1
Control: coherent errors eliminate bias
2:1 noise
1
2 random
1:2
Noise tolerance at moderate noise
4:1 noise
1
4 random
1:4
Noise tolerance at high noise
Table 2: Denoising corpus conditions. Each corpus contains 5,000 unique problems.
Size
Params
Random Accuracy
Coherent Accuracy
Random Seeds
Coherent Seeds
tiny
3.5M
65.3%±1.3%
43.5%±2.6%
4
4
small
12M
74.6%±1.6%
44.5%±3.0%
4
4
medium
26M
81.1%±1.2%
45.8%±3.4%
4
4
large
86M
85.2%±2.3%
51.0%±0.8%
2
2
Table 3: Denoising paired evaluation: equal random vs equal coherent errors, across model sizes. All ± values are standard deviations across random seeds. Random accuracy scales with model size; coherent accuracy remains near chance.
Figure 1: The central contrast. Random errors: accuracy scales from 65% to 85%. Coherent errors: accuracy stays near chance across all sizes.
N Rules
Accuracy
p
1 (coherent)
46.6%±2.4%
∼1.0
2
77.6%±1.3%
<10−6
3
82.8%±0.1%
<10−6
5
84.8%±0.5%
<10−6
10
88.3%±0.7%
<10−6
Table 4: Multi-rule paired evaluation (tiny, 3.5M, standard 50/50, 4 seeds). The N=1 baseline (46.6%) differs from the denoising coherent accuracy in Table 3 (43.5%) because multi-rule experiments use the standard (non-denoising) setup. N=2 on small (12M) yields 86.3%±0.8% , confirming the effect strengthens with capacity.
Figure 2: Multi-rule crossover. Accuracy jumps from 47% ( N=1 ) to 78% ( N=2 ), then grows gradually through N=10 .
Size
Random Accuracy
Coherent Accuracy
tiny (3.5M)
69.6%±0.1%
48.7%±0.5%
small (12M)
70.7%±0.4%
46.6%±0.4%
medium (26M)
71.5%±0.8%
46.4%±0.8%
large (86M)
71.4%±0.7%
45.9%±1.5%
Table 5: Wikipedia entity substitution (50/50, 4 seeds per size). The random/coherent contrast reproduces on real text.
Size
Random-trained
Coherent-trained
Gap
tiny (3.5M)
30.5%±1.7%
20.8%±3.6%
+9.7 pp
small (12M)
46.7%±1.9%
31.7%±1.8%
+15.0 pp
medium (26M)
50.5%±1.3%
36.1%±2.7%
+14.4 pp
large (86M)
52.8%±1.1%
35.6%±1.8%
+17.2 pp
Table 6: Generative accuracy (4 seeds, 500 problems each). The gap widens with model size.
Condition
seed43
seed44
seed45
Mean
Random
85.3%
85.7%
89.3%
86.8%
Coherent
49.3%
52.8%
49.8%
50.6%
Table 7: Qwen3-0.6B paired evaluation (50/50, 3 seeds). The random/coherent contrast reproduces across architectures.
Figure 3: Full scaling results. GPT-2 family (3.5M–86M, math-only), Qwen3-0.6B (420M, different architecture, math-only), and a Qwen3-architecture model ( ∼ 1B, FineWeb-Edu + 8% math).
Condition
Accuracy
ΔLoss
p
Seeds
Random
76.8%
+0.098
<10−6
1
Coherent
46.7%
−0.003
∼1.0
1
Table 8: Qwen3-architecture ∼ 1B trained on FineWeb-Edu + 8% contradictory math (denoising, 50/50). The random/coherent contrast reproduces at 1B scale on real text.
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Condition
Correct
Incorrect
Delta
Paired Accuracy
random 50/50
0.1627
0.1639
+0.0012
79.5%
coherent 50/50
0.1656
0.1658
+0.0002
47.2%
contradictory
0.1745
0.1744
−0.0001
49.0%
multirule N=2
0.1671
0.1710
+0.0038
77.6%
multirule N=3
0.1669
0.1722
+0.0053
82.8%
multirule N=5
0.1672
0.1730
+0.0057
84.8%
Appendix
Table 9: Compression ratio (gzip) and paired accuracy across all 9 conditions. Spearman ρ=0.68 , p=0.042 .
Size
Standard
Denoising
Gap
tiny
79.5%
65.3%
−14.2 pp
small
86.2%
74.6%
−11.6 pp
medium
87.1%
81.1%
−6.0 pp
large
88.3%
85.2%
−3.1 pp
Appendix
Table 10: Standard vs denoising accuracy (50/50 random). The gap closes with scale.
Proportion
Random
Coherent
50/50
80%
47.2%
40/60
79%
27.8%
30/70
75%
14.7%
20/80
69%
9.6%
10/90
67%
–
Appendix
Table 11: Random vs coherent accuracy across proportions (standard, tiny). Random: truth bias persists at 10/90. Coherent: the model follows pure frequency.
Condition
Ratio
Tiny
Small
Medium
Large
Equal random
1:1
65.3%
74.6%
81.1%
85.2%
2:1 noise
1:2
59.0%
68.6%
73.6%
75.2%
4:1 noise
1:4
56.6%
64.9%
66.6%
65.8%
Appendix
Table 12: Paired accuracy at increasing noise ratios (denoising). Accuracy degrades gracefully; at 4:1 noise, it plateaus at medium/large.
Entity Type
N Pairs
Accuracy
LOC
45
82.2%
NORP
197
78.7%
GPE
301
77.1%
PERSON
402
69.9%
ORG
664
65.2%
CARDINAL
172
61.0%
Appendix
Table 13: Wikipedia accuracy by entity type (random substitution, averaged across sizes).
Setup
Tokenizer
Random
Coherent
Standard
Char
79.5%
47.2%
Standard
BPE
85.6%
45.9%
Denoising
Char
65.3%
43.5%
Denoising
BPE
75.9%
49.3%
Appendix
Table 14: BPE vs char-level (tiny, 4 seeds). The effect survives BPE and strengthens for random errors.