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arxiv: 2511.17699 · v2 · submitted 2025-11-21 · 💻 cs.CV · cs.AI

Understanding Counting Mechanisms in Large Language and Vision-Language Models

Hosein Hasani , Amirmohammad Izadi , Fatemeh Askari , Mobin Bagherian , Sadegh Mohammadian , Mohammad Izadi , Mahdieh Soleymani Baghshah This is my paper

Pith reviewed 2026-05-17 20:06 UTC · model grok-4.3

classification 💻 cs.CV cs.AI
keywords counting mechanismslarge language modelsvision-language modelsmechanistic interpretabilitynumerical representationslayerwise analysisinternal countercausal mediation
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The pith

Large language and vision-language models maintain counts through an internal counter that updates with each item and stores the total mainly in the final token or region.

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

The paper examines how LLMs and LVLMs handle counting by running controlled tests with repeated text or image items. Behavioral tests combined with observational and causal analyses show that tokens or visual features carry hidden position-based count signals that can move between settings. These signals build gradually across layers, with early layers handling small numbers and later layers handling bigger ones. The work identifies a dedicated counter process that adds one for each new item and keeps the running total in the last position. In vision-language models the same signals also move between background and foreground parts of an image depending on layout, and models often lean on separators or other structural markers as quick ways to track totals.

Core claim

Counting emerges as a structured, layerwise process in which an internal counter mechanism updates with each successive item. The count value is encoded in latent positional information carried by individual tokens or visual features and is stored primarily in the final token or image region. Lower layers represent small counts while higher layers represent larger counts. In vision-language models numerical information also appears in visual embeddings and shifts between background and foreground regions according to spatial arrangement. Models further exploit structural cues such as text separators as shortcuts that strongly affect numerical accuracy.

What carries the argument

The internal counter mechanism, which increments with each new item and holds the running total chiefly in the final token or region, carries the numerical state through the model.

If this is right

  • Count information encoded in single tokens can be read out and reused in new contexts.
  • Lower layers handle small counts while higher layers handle large counts, creating a progressive buildup of numerical precision.
  • In vision-language models the same numerical signals move between background and foreground image regions based on scene layout.
  • Models rely on separators and other structural markers in text as shortcuts that can boost or hurt counting accuracy.
  • The same general counting pattern appears in both language-only and vision-language models.

Where Pith is reading between the lines

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

  • Targeting the final token with small activation edits might improve counting performance on tasks where models currently fail.
  • The layerwise shift from small to large counts suggests that deeper layers could be the right place to add explicit numerical modules in future architectures.
  • If similar counters exist for other sequential operations, the same analysis approach could map how models track order or time.
  • Models that depend heavily on separators may underperform on counting problems that lack clear textual boundaries.

Load-bearing premise

The observational and causal analyses truly detect real counting processes inside the models rather than side effects created by the repeated-item test prompts or the analysis tool.

What would settle it

A direct test would intervene on the final token's activations and check whether the model's reported count changes while other capabilities stay intact.

Figures

Figures reproduced from arXiv: 2511.17699 by Amirmohammad Izadi, Fatemeh Askari, Hosein Hasani, Mahdieh Soleymani Baghshah, Mobin Bagherian, Mohammad Izadi, Sadegh Mohammadian.

Figure 1
Figure 1. Figure 1: A minimal graphical abstract illustrating how [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Representational behavior of embeddings in a selected [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Ground-truth (total count) probability of visual objects [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Per-item latent count encoding, decoded by CountScope. Each heatmap shows the probability of decoding numbers (1–9) across [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Type-specific counter behavior revealed by CountScope. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Layerwise decoding of latent counts using online patch [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Per-item latent count for separators, decoded by [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Example images from the visual dataset. Monotypic [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Mean ground-truth probability across layer windows un [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Layer-wise PCA of Qwen2.5 input-token representations. PCA trajectories across layers for (a) element tokens and (b) separator tokens in the monotypic, question-first setting. 6 [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Layer-wise PCA of Qwen2.5 output representations. PCA embeddings across layers for generated numerical responses, colored by (a) predicted count and (b) item type, in the monotypic, question-first setting. 7 [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Layer-wise PCA of Qwen2.5VL embeddings. PCA trajectories across layers for (a) input item embeddings and (b) generated output embeddings in the monotypic setting. 8 [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Layer-wise cosine similarity of Qwen2.5 representations. Cosine similarity matrices across layers for (a) element tokens and (b) separator tokens in the monotypic, question-first setting. Cosine similarities are computed across different tasks with different item types and then averaged over the dataset. 9 [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
read the original abstract

Counting is one of the fundamental abilities of large language models (LLMs) and large vision-language models (LVLMs). This paper examines how these foundation models represent and compute numerical information in counting tasks. We use controlled experiments with repeated textual and visual items and analyze counting in LLMs and LVLMs through a set of behavioral, observational, and causal mediation analyses. To this end, we design a specialized tool, CountScope, for the mechanistic interpretability of numerical content. Results show that individual tokens or visual features encode latent positional count information that can be extracted and transferred across contexts. Layerwise analyses reveal a progressive emergence of numerical representations, with lower layers encoding small counts and higher layers representing larger ones. We identify an internal counter mechanism that updates with each item, stored mainly in the final token or region. In LVLMs, numerical information also appears in visual embeddings, shifting between background and foreground regions depending on spatial composition. We further reveal that models rely on structural cues such as separators in text, which act as shortcuts for tracking item counts and strongly influence the accuracy of numerical predictions. Overall, counting emerges as a structured, layerwise process in LLMs and follows the same general pattern in LVLMs, shaped by the properties of the vision encoder.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper examines counting in LLMs and LVLMs via behavioral, observational, and causal mediation analyses on repeated textual and visual items. It introduces CountScope for mechanistic interpretability and claims to locate an internal counter that updates per item (primarily in the final token/region), with layerwise progressive emergence of numerical representations (lower layers for small counts, higher layers for larger ones). Numerical information also appears in LVLMs' visual embeddings (shifting by spatial composition), while models rely on structural cues such as separators as shortcuts that strongly affect prediction accuracy.

Significance. If the mediation analyses and CountScope tool successfully isolate genuine internal mechanisms rather than prompt artifacts, the work would advance mechanistic interpretability of numerical reasoning in foundation models and supply a reusable tool for probing count representations. The layerwise and cross-modal findings could inform architecture design for better numerical generalization.

major comments (2)
  1. [Causal mediation analyses and CountScope results] The central claim that an internal counter mechanism (stored mainly in the final token/region) has been identified rests on repeated-item prompts; the abstract itself states that models rely on structural cues such as separators as shortcuts that strongly influence numerical predictions. No explicit test or ablation is described that rules out the possibility that CountScope and the mediation analyses are extracting these prompt-induced positional/separator signals rather than a general counting procedure.
  2. [Layerwise analyses] Layerwise analyses claim progressive emergence (lower layers encode small counts, higher layers larger ones), yet the abstract and high-level description provide no quantitative metrics, error bars, statistical tests, or ablation controls to support the layer-specific encoding distinction or its load-bearing role in the counting mechanism.
minor comments (2)
  1. [Abstract] The abstract summarizes methods and high-level results but omits key quantitative findings, sample sizes, or performance numbers that would allow readers to assess effect sizes.
  2. [Methods] Clarify the precise input format and output of CountScope (e.g., how it extracts and transfers latent positional count information) to support reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address each major comment below and describe the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Causal mediation analyses and CountScope results] The central claim that an internal counter mechanism (stored mainly in the final token/region) has been identified rests on repeated-item prompts; the abstract itself states that models rely on structural cues such as separators as shortcuts that strongly influence numerical predictions. No explicit test or ablation is described that rules out the possibility that CountScope and the mediation analyses are extracting these prompt-induced positional/separator signals rather than a general counting procedure.

    Authors: We agree that it is important to distinguish prompt artifacts from a general internal counting procedure. The causal mediation analyses intervene on activations within the final token while holding the input prompt (including separators) fixed, and demonstrate that these interventions causally alter the model's numerical output. CountScope similarly extracts and transfers count information from internal states across contexts. Nevertheless, we acknowledge that an explicit ablation varying or removing separators was not included. We will add this experiment in the revised manuscript, showing that the internal counter mechanism remains operative even when structural cues are minimized. revision: yes

  2. Referee: [Layerwise analyses] Layerwise analyses claim progressive emergence (lower layers encode small counts, higher layers larger ones), yet the abstract and high-level description provide no quantitative metrics, error bars, statistical tests, or ablation controls to support the layer-specific encoding distinction or its load-bearing role in the counting mechanism.

    Authors: The quantitative support for progressive emergence, including per-layer accuracy curves, error bars from multiple random seeds, and statistical comparisons, appears in Section 4.2 and the associated figures. We will revise the abstract and introduction to explicitly report key metrics (e.g., the layer thresholds at which accuracy for count k exceeds baseline) and briefly summarize the ablation controls that establish the functional role of these layer-specific representations. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical mediation and observational analyses

full rationale

The paper conducts behavioral experiments, observational analyses, and causal mediation interventions via the CountScope tool on repeated-item prompts in LLMs and LVLMs. No first-principles derivation, mathematical model, or predictive equation is presented whose output is shown to equal its inputs by construction. Layerwise emergence claims and internal-counter localization rest on direct measurements and interventions rather than any self-definitional loop, fitted-parameter renaming, or self-citation chain that would render the central result tautological. The work is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard assumptions from mechanistic interpretability research; the abstract does not introduce or fit new free parameters or postulate new entities.

axioms (1)
  • domain assumption Causal mediation analysis can isolate the causal contribution of specific internal representations to counting behavior
    Invoked when the paper states it uses causal mediation analyses to study numerical content.

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Forward citations

Cited by 1 Pith paper

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    CounterCount shows VLMs perform well on factual counting images but degrade on counterfactual edits, revealing reliance on object priors, and introduces an attention reweighting method that improves accuracy by up to 8%.

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