Factorized Representations of Concurrent Temporal Variables in Pretrained Language Models

Xinhe Zhang; Arnau Marin-Llobet

cs.LGOct 29, 2026

Factorized Representations of Concurrent Temporal Variables in Pretrained Language Models

Authors: Xinhe Zhang, Arnau Marin-Llobet

Abstract

Language models often need to maintain multiple contextual variables that evolve independently within a document. Prior work has identified structured representations of individual temporal or contextual quantities, but their joint organization remains less understood. We study this question using controlled documents with independently phased language-switch and topic-change timers. Across ten pretrained checkpoints, both timers are linearly decodable from the same hidden states. Removing a timer-associated centroid plane substantially reduces access to its associated variable while having little effect on the concurrent timer, although the planes are nonorthogonal and contain cross-timer information. Separate timer contributions also explain most reproducible variation among joint-state centroids. These signatures persist from 3 to 51 timer values and across Qwen3 checkpoints from 0.6B to 14B. Schedule position affects continuation probabilities, but the tested centroid displacement produces no consistent timer-specific output effect. Together, these results support approximate factorization based on selective accessibility and approximately additive joint-state geometry in overlapping subspaces.

Figures & tables

Figure 1: Selective ablation of concurrent timer representations. A , Independently phased language and topic schedules. B–C , Cross-fitted decoding after subspace removal and decoder retraining. Points represent models; bars indicate means.

Figure 2: Information overlap and approximately additive joint geometry. A–B , Decoding restricted to timer-associated planes. C , Joint-centroid interaction residuals across models. Pale bars indicate the mean

\pm

bootstrap standard error.

$H$	Values	$\rho$	Rank	Self	Cross	Random	Cross/self
2	3	0.91	1.75	84.9%	0.15%	0.18%	0.18%
5	6	0.89	2.50	75.3%	0.44%	0.31%	0.56%
10	11	0.87	3.25	79.9%	2.05%	0.53%	2.58%
20	21	0.87	3.48	79.8%	2.96%	0.69%	3.71%
50	51	0.86	3.98	84.6%	4.86%	0.76%	5.71%

Table 1: Cross-fitted horizon sweep across four models. Rank is the mean selected dimension; Self, Cross, and Random are percentage reductions in intact Spearman correlation. See Appendix D.3 for aggregation details.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Sent.	Text	Lang.	Topic	$y_{\mathrm{lang}}$	$y_{\mathrm{topic}}$
0	The lamp is bright.	EN	house	1	3 †
1	The mountain is high.	EN	field	2	0
2	Der Fluss ist breit.	DE	field	0	1
3	Der Stein ist grau.	DE	field	1	2
4	Der Wald ist dunkel.	DE	field	2	3 †
5	The chair is heavy.	EN	house	0	0

Appendix

Table 2: Excerpt from a family-A document with language and topic timer labels. † Topic-phase-3 rows are excluded from primary three-class topic analyses.

Paper name	Checkpoint	Layers	Analyzed
Llama-3.2-1B	meta-llama/Llama-3.2-1B	16	8
Mistral-7B-v0.3	mistralai/Mistral-7B-v0.3	32	16
Qwen3-1.7B	Qwen/Qwen3-1.7B	28	14
Qwen3-1.7B-Base	Qwen/Qwen3-1.7B-Base	28	14
Qwen3-4B	Qwen/Qwen3-4B	36	18
pythia-1.4b	EleutherAI/pythia-1.4b	24	12

Appendix

Table 3: Checkpoints, transformer-block counts, and analyzed layer indices.

	Language timer			Topic timer
Model	Intact	Own	Other	Intact	Own	Other
Llama-3.2-1B	0.891	0.423	0.886	0.460	0.390	0.459
Mistral-7B	0.917	0.433	0.909	0.513	0.403	0.506
Qwen3-1.7B	0.878	0.421	0.867	0.502	0.404	0.498
Qwen3-1.7B-Base	0.941	0.431	0.938	0.531	0.412	0.528
Qwen3-4B	0.864	0.428	0.851	0.522	0.414	0.512

Appendix

Table 4: Cross-fitted balanced accuracy before ablation (Intact), after own-plane removal (Own), and after concurrent-timer-plane removal (Other).

	Own, random	Own, phase-matched	Own, within-document	Other, random
Language	84.5%	84.0%	81.4%	1.6%
Topic	58.4%	63.0%	53.6%	2.3%

Appendix

Table 5: Percentage of above-chance decoding signal removed under each split condition, averaged over per-model removal fractions from a separate eight-draw experiment. Own and Other denote removal of the target timer’s plane and the concurrent timer’s plane, respectively.

Timer	Intact	Own, cross-fitted	Other, cross-fitted	Own removed	Other removed
Language	0.893	0.576	0.886	81%	1.7%
Topic	0.598	0.553	0.591	45%	6.3%

Appendix

Table 6: Balanced accuracy and percentage of above-chance signal removed in history-only decoding. Own and Other denote ablation of the target timer’s plane and the concurrent timer’s plane, respectively.

Transferred fraction $\alpha$	Cross-ablation	Self-ablation	Above-chance accuracy
0.00	0.6	16.5	+0.164
0.25	1.1	11.2	+0.109
0.50	4.4	8.1	+0.081
0.75	10.4	8.5	+0.084
1.00	11.0	10.8	+0.110

Appendix

Table 7: Ablation effects under imposed overlap, reported as balanced-accuracy decreases in percentage points. The final column reports intact balanced accuracy minus chance.

Model	Own plane	Concurrent plane	Placebo	Angle
SmolLM2-1.7B	+0.596	+0.136	+0.164	$66.6^{\circ}$
Qwen3-1.7B-Base	+0.537	+0.259	+0.149	$53.4^{\circ}$
OLMo-2-1B	+0.528	+0.188	+0.146	$52.4^{\circ}$
Mistral-7B-v0.3	+0.515	+0.206	+0.166	$53.1^{\circ}$
Llama-3.2-1B	+0.451	+0.153	+0.117	$55.4^{\circ}$
gemma-2-2b	+0.438	+0.217	+0.105	$41.6^{\circ}$

Appendix

Table 8: Above-chance language-timer decoding from the language plane (Own), topic plane (Concurrent), and placebo plane, with the smaller principal angle between the two timer planes.

	Interaction residual		Held-out variance explained
Model	Observed	Additive floor	Additive	Saturated
Llama-3.2-1B	9.1%	4.3%	0.839	0.833
Mistral-7B	6.8%	3.4%	0.876	0.852
Qwen3-1.7B	12.8%	6.0%	0.758	0.761
Qwen3-1.7B-Base	7.5%	3.1%	0.868	0.874
Qwen3-4B	12.8%	6.1%	0.767	0.753

Appendix

Table 9: Joint-centroid interaction residuals and held-out variance explained by additive and saturated nine-centroid models. The additive floor estimates the finite-sample residual under exactly additive population means.

Timer	Across	Split-half	Across / split-half	Shuffled
Language	0.855	0.889	96%	+0.005
Topic	0.331	0.587	54%	+0.005

Appendix

Table 10: Mean cosine similarity of timer-transition displacements across concurrent-timer values (Across), between document halves at a fixed concurrent-timer value (Split-half), and under shuffled labels. Across / split-half reports the mean of per-model ratios.

Figure 13

Model	Interaction $\pm$ SEM	Shuffled	Split-half angle (lang./topic)
Qwen3-0.6B	$12.1\pm 1.7\%$	50.4%	$17.2^{\circ}/58.7^{\circ}$
Qwen3-1.7B	$14.6\pm 2.1\%$	50.6%	$22.2^{\circ}/51.7^{\circ}$
Qwen3-4B	$15.9\pm 2.4\%$	49.9%	$27.0^{\circ}/51.7^{\circ}$
Qwen3-8B	$13.3\pm 2.1\%$	50.1%	$19.7^{\circ}/53.4^{\circ}$
Qwen3-14B	$12.7\pm 2.0\%$	50.3%	$20.2^{\circ}/41.1^{\circ}$

Appendix

Table 11: Interaction residuals and plane reproducibility across Qwen3 model sizes. Uncertainty is one bootstrap standard error estimated from 40 document resamples; Shuffled reports the shuffled-label interaction residual. Shuffled-label plane angles range from

84^{\circ}

86^{\circ}

Model	$A_{\mathrm{due}}$	$A_{\mathrm{control}}$	$D\pm$ SEM
pythia-1.4b	3.301	-0.837	$4.138\pm 0.658$
Qwen3-1.7B	2.147	-0.741	$2.888\pm 0.556$
Phi-3-mini	2.094	-0.750	$2.844\pm 0.456$
OLMo-2-1B	1.997	-0.767	$2.764\pm 0.569$
Llama-3.2-1B	2.262	-0.280	$2.542\pm 0.418$
SmolLM2-1.7B	1.962	-0.449	$2.411\pm 0.546$

Appendix

Table 12: Recency-matched continuation contrasts. Uncertainty is one standard error from 2000 document-bootstrap resamples. The final row reports a randomly initialized Qwen3-1.7B control.

Model	Target	Concurrent	Random	Target $-$ concurrent
Phi-3-mini	0.0249	0.0231	0.0010	0.0018
gemma-2-2b	0.0243	0.0066	-0.0070	0.0177
OLMo-2-1B	0.0173	0.0499	-0.0015	-0.0326
pythia-1.4b	0.0126	-0.0164	0.0037	0.0290
Mistral-7B-v0.3	0.0046	0.0215	-0.0036	-0.0169
SmolLM2-1.7B	0.0021	0.0324	-0.1014	-0.0303

Appendix

Table 13: Mean continuation log-odds change at strength

1.0

for language-timer (Target), concurrent-timer, and isotropic (Random) interventions.

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