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arxiv: 2606.03645 · v1 · pith:OBKCHHNCnew · submitted 2026-05-29 · 💻 cs.LG · cs.AI

The Shape of Addition: Geometric Structures of Arithmetic in Large Language Models

Liuyuan Wen , Xun Zhu , Lihao Huang , Wenbin Li , Yang Gao This is my paper

Pith reviewed 2026-06-28 23:42 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords large language modelsarithmeticresidual stream geometrycarry potentialquantizationgeometric slippageerror correction
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The pith

Large language models represent multi-operand addition as an Iso-Raw-Sum Trajectory in residual streams, anchored by semantic digits and modulated by continuous carry fibers, with errors as geometric slippages from noisy quantization.

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

The paper examines the geometry of the residual stream in large language models while they perform addition with multiple operands. It identifies a trajectory in which semantic digits fix the base position of representations while continuous carry fibers adjust values that exceed single-digit ranges. The authors introduce the Noisy Quantization Model, which treats arithmetic mistakes as slips that occur when internal neural noise moves a latent carry potential across discrete thresholds. This same account explains why lightweight probes can separate coexisting signals, such as correct values and hallucinations, from one activation vector. The framework also yields a geometric consistency check that detects and corrects these failures during inference.

Core claim

By analyzing the residual stream geometry during multi-operand addition, the authors identify the Iso-Raw-Sum Trajectory (IRST), a geometric structure where representations are anchored by semantic digits and modulated by continuous carry fibers. They propose the Noisy Quantization Model to explain this geometry, framing arithmetic errors as Geometric Slippages caused by internal neural noise pushing a continuous, latent Carry Potential across quantization thresholds. This geometric framework elucidates Probe Versatility, explaining how lightweight probes can disentangle coexisting latent signals from a single activation vector, and validates the insights through a geometric consistency chec

What carries the argument

The Iso-Raw-Sum Trajectory (IRST), a geometric structure in the residual stream where representations are anchored by semantic digits and modulated by continuous carry fibers, together with the Noisy Quantization Model that attributes errors to noise-driven crossings of quantization thresholds.

Load-bearing premise

The observed trajectories and error patterns are produced by a continuous carry potential that is quantized at discrete thresholds rather than by other mechanisms such as attention patterns, token embeddings, or training data statistics.

What would settle it

An experiment that clamps or suppresses the continuous carry dimension in the residual stream during addition and checks whether the specific patterns of geometric slippage errors disappear; persistence of those error patterns would falsify the model.

Figures

Figures reproduced from arXiv: 2606.03645 by Lihao Huang, Liuyuan Wen, Wenbin Li, Xun Zhu, Yang Gao.

Figure 1
Figure 1. Figure 1: Overview of our probing framework. (Left) The LLM performs multi-operand addition (e.g., 123 + 392 + 136) in an autoregressive manner. At each generation step (e.g., p = 1, corresponding to the tens digit), we extract the hidden state vectors h (l) p (mainly focusing on the final layer L). (Right) We train versatile probes on these activation vectors to decode several critical arithmetic variables, includi… view at source ↗
Figure 2
Figure 2. Figure 2: 2D UMAP visualization of the arithmetic manifold. (Left) Macroscopic Backbone: Global geometry of h (L) p (p = 4) organized around digit Anchors (0–9). Blue points denote correct samples labeled as sˆp; red points denote errors labeled as sˆp(sp). The inset highlights high-error transition zones between digit basins. (Right) Microscopic Texture: Magnified view around Anchor 1 labeled with (sp, sˆp, cp), sh… view at source ↗
Figure 3
Figure 3. Figure 3: The Iso-Raw-Sum Trajectory (IRST) framework of the arithmetic manifold. (Left) Magnified UMAP projection around digit Anchors 1, 2, and 3. Points are labeled with (ˆsp, cp). The geometry reveals distinct IRSTs (T0 ∼ T3) that act as continuous “threads” piercing through adjacent digit basins. For instance, T1 (where rp mod 10 = 1) connects stable nodes (1, 0) ↔ (2, 1) ↔ (3, 2) as the input carry increases. … view at source ↗
Figure 5
Figure 5. Figure 5: Empirical validation of the Noisy Quantization Model. (Top) The distribution of Carry Potential Φ across all generated positions p in the dataset. Green vertical lines indicate integer quantization thresholds (1.0, 2.0, . . . ). (Bottom) The conditional error rate as a function of Φ. The empirical data (red bars, off-by￾one errors only) exhibits a distinct periodic bathtub shape, spiking near integer bound… view at source ↗
Figure 6
Figure 6. Figure 6: Representative trajectory-level validation on T3. The markers are labeled as (sp, sˆp, cp). Circles denote correct predictions and squares denote errors. (Left) Empirical projection of last-layer activations for samples in T3. The x-axis shows the cosine distance from the central centroid (4, 4, 1). The manifold exhibits a clear V-shaped progression connecting stable basins. Crucially, error states such as… view at source ↗
Figure 7
Figure 7. Figure 7: presents the visualizations for PCA and t-SNE. 0.6 0.4 0.2 0.0 0.2 0.4 0.6 PC1 (9.76% variance) 0.6 0.4 0.2 0.0 0.2 0.4 PC2 (8.22% variance) 8 9 1 3 7 2 2(3) 1 6 3 4(5) 2 0 3(4) 9 2 9 6 2 3 0 5 5 5 0 5 0 7 1 1 1 9 1(0) 3 5(6) 5 3(4) 4 6 9 0 3 8 3 9 6 5 4 2 7 9 5(4) 6 7 1 6(8) 9 1 5 55 3 7 8 2 4 4(3) 0 1 0 1 8 4 1 2 1 8 3 3(4) 3 7(5) 3 8 7 4 8 9 1 3 3 6 5 1 1 7 2 9 2 1 6(7) 9 5 0 1 8 9 2 0 8 6 2 9 5 6(5) 1 … view at source ↗
Figure 8
Figure 8. Figure 8: Projection-independent validation of the IRSTs in the native representation space. Cosine distance to anchor states is evaluated directly in R 2560 across multiple trajectories. The characteristic V-shaped correlation between native-space distance and the continuous carry potential Φ generalizes across T0 . . . T9, supporting the claim that the IRST organization is not solely a 2D projection artifact. The … view at source ↗
Figure 9
Figure 9. Figure 9: Intrinsic dimensionality across IRSTs. We report the participation ratio, TWO-NN, and Levina–Bickel MLE estimates for T0 . . . T9 and their pooled union. The trajectory-conditioned subsets remain stably low-dimensional in native space, while the pooled set exhibits a larger linear effective dimension. D. Validation of the Raw Sum Assumption In the main context, we posit that the model’s arithmetic errors a… view at source ↗
Figure 10
Figure 10. Figure 10: Layer-wise evolution of intrinsic dimensionality. We compare the nonlinear and linear intrinsic-dimension estimates for T5 and the pooled set of all trajectories across layers. The two remain broadly similar in early layers, while later layers develop more trajectory-specific local structure. an incorrect output digit sˆp. This suggests that arithmetic errors are largely geometrically constrained: the rep… view at source ↗
Figure 11
Figure 11. Figure 11: Geometric signatures of carry-based and non-carry errors. Error modes are decoupled using the raw-sum probe. Carry￾based errors, where the local raw sum is still recovered correctly, remain concentrated near adjacent decision boundaries. In contrast, non-carry errors scatter across more distant regions and do not follow a single continuous trajectory, indicating a distinct failure mode beyond the dominant… view at source ↗
Figure 12
Figure 12. Figure 12: Scaling of cognitive noise with task complexity. We extend the analysis of [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Boundary-position bathtub profiles. Frequency and conditional-error distributions at the most significant digit (p = 0, top row) and the least significant digit (p = 9, bottom row) for 3-term 10-digit addition. Unlike the interior columns summarized in the main text, these boundary positions show structurally different profiles, highlighting that the steady-state bathtub regime is primarily an interior-po… view at source ↗
Figure 14
Figure 14. Figure 14: Generalization of the IRST geometry across different models. (Left) UMAP visualization for Qwen3-8B on a 12-digit addition task. The manifold structure is highly consistent with the 4B model, featuring a sequential arrangement of digit basins (0–9) connected by clear trajectories (e.g., T0, T1, T2). (Right) UMAP visualization for Gemma-3-4B-IT on a 10-digit addition task. Despite architectural differences… view at source ↗
Figure 15
Figure 15. Figure 15: Generalization of the Noisy Quantization hypothesis. Validation on Qwen3-8B (12-digit addition, top) and Gemma-3-4B-IT (10-digit addition, bottom). The Left panels show the sample frequency distribution relative to Carry Potential Φ, indicating that the dataset covers the entire potential space. The Right panels display the conditional error rates (red bars) overlaid with our theoretical fit (dashed blue … view at source ↗
Figure 16
Figure 16. Figure 16: Additional evidence from specialized arithmetic transformers. (Top) UMAP projections from the under-converged and fully converged single-task addition models reported by Quirke et al. (2025). The under-converged model exhibits continuous inter-basin geometry similar to our main setting, while the fully converged model forms more isolated basins. (Bottom) Conditional error-rate curves for the under-converg… view at source ↗
Figure 17
Figure 17. Figure 17: IRST analysis for 4-term addition (cp ∈ {0, 1, 2, 3}). (Left) 2D UMAP projection of activations at p = 4. While the digit backbone (0-9) persists, the increased density of carry fibers causes visual entanglement of trajectories in 2D space. However 3D plots can resolve these overlaps. (Right) Schematic of the expanded geometry. Between any adjacent digit basins (e.g., 0 and 1), there are now three distinc… view at source ↗
Figure 18
Figure 18. Figure 18: Impact of Tolerance δ. Token accuracy (left axis, blue) and Question accuracy (right axis, orange) under different values of δ. The peak at δ ≈ 0.12 validates the existence of a noise margin in the model’s carry potential estimation. context: ”That step looks incorrect. Let’s re-do just this step: {expression} = {current output}”. The model is then forced to regenerate the current digit based on this augm… view at source ↗
Figure 19
Figure 19. Figure 19: Layer-wise performance of different probes trained on Qwen3-4B combing all positions. (nostalgebraist, 2020). The Logit Lens applies the final layer’s unembedding matrix WU to intermediate hidden states h (l) p to project them directly into the vocabulary space. We evaluated both methods on identifying the Ground Truth Digit (GT) and the Model’s Final Output Digit (Pred) across all 36 layers. The results … view at source ↗
Figure 20
Figure 20. Figure 20: Layer-wise input-carry decoding accuracy for attention and FFN outputs. Attention blocks exhibit sharper stepwise gains, while FFN outputs largely follow these updates, suggesting that carry information is consolidated through a staged pipeline across layers. 2 4 0 6(7) 1 9 9 1 6 8 4 2(1) 3 6 3 4 0 6(5) 2 6 1 6 1(2) 9 7(5) 1 2 9 1(0) 9 9 1 2 4(3) 7 6 8 5 2(1) 1 1 7(8) 0 7 4 4 7 9 6(3) 5 2 5(4) 4 6 3 3 7 5… view at source ↗
Figure 21
Figure 21. Figure 21: Layer-wise Alighed UMAP visualization. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Layer-wise Performance Comparison: Linear Probes vs. Logit Lens. The Green and Red lines represent the accuracy of linear probes trained on h (l) p for the Ground Truth (GT) and Model Prediction (Pred), respectively. The Blue and Orange lines represent the accuracy of the Logit Lens (applying the unembedding matrix directly). Key Observation: There is a significant decoding lag. Probes successfully decode… view at source ↗
read the original abstract

Large Language Models exhibit paradoxical fragility in fundamental arithmetic, implying a disconnect between internal computation and discrete output. By analyzing the residual stream geometry during multi-operand addition, we identify the Iso-Raw-Sum Trajectory (IRST), a geometric structure where representations are anchored by semantic digits and modulated by continuous carry fibers. We propose the Noisy Quantization Model to explain this geometry, framing arithmetic errors as Geometric Slippages caused by internal neural noise pushing a continuous, latent Carry Potential across quantization thresholds. This geometric framework further elucidates Probe Versatility, explaining how lightweight probes can disentangle coexisting latent signals (such as ground truth versus hallucination) from a single activation vector. Finally, we validate these insights through a geometric consistency check method that effectively detects and corrects these quantization failures during inference. Our code is available at https://github.com/RL-MIND/Shape-of-Addition.

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 / 1 minor

Summary. The paper analyzes residual stream geometry in LLMs during multi-operand addition tasks. It identifies an Iso-Raw-Sum Trajectory (IRST) in which representations are anchored by semantic digits and modulated by continuous carry fibers. The authors propose a Noisy Quantization Model that attributes arithmetic errors to geometric slippages arising when internal neural noise drives a latent continuous Carry Potential across discrete quantization thresholds. The framework is also used to explain probe versatility for disentangling coexisting signals (e.g., ground truth vs. hallucination) and is validated via a geometric consistency check that detects and corrects quantization failures at inference time. Code is released.

Significance. If the IRST geometry and the attribution of errors specifically to a continuous Carry Potential quantized at thresholds can be substantiated, the work would offer a mechanistic account of arithmetic fragility in LLMs and a practical inference-time correction method. The public code release is a clear strength that supports reproducibility and further testing of the geometric claims.

major comments (2)
  1. [Abstract] Abstract and introduction: the central claim that arithmetic errors arise from geometric slippages of a continuous latent Carry Potential across quantization thresholds is not accompanied by described controls or ablation experiments that would distinguish this mechanism from alternatives such as discrete attention patterns, token embedding statistics, or training-data regularities. Without such disambiguation the attribution remains unsecured.
  2. [Abstract] Abstract: the validation of the Noisy Quantization Model and the geometric consistency check is described only at a high level; no quantitative metrics, error bars, or statistical tests are mentioned that would allow assessment of whether the observed trajectories and error patterns are better explained by the proposed model than by simpler alternatives.
minor comments (1)
  1. [Abstract] The abstract introduces several new terms (IRST, Noisy Quantization Model, Carry Potential, Geometric Slippages) without immediate definitions or pointers to the sections where they are formalized.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for your constructive feedback. We value the emphasis on rigorous disambiguation of mechanisms and quantitative validation of the Noisy Quantization Model. We will revise the manuscript to incorporate explicit controls, ablations, and quantitative metrics as outlined below. These additions will strengthen the attribution of errors to geometric slippages of the continuous Carry Potential while preserving the core geometric findings on the IRST.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: the central claim that arithmetic errors arise from geometric slippages of a continuous latent Carry Potential across quantization thresholds is not accompanied by described controls or ablation experiments that would distinguish this mechanism from alternatives such as discrete attention patterns, token embedding statistics, or training-data regularities. Without such disambiguation the attribution remains unsecured.

    Authors: We agree that the abstract does not detail explicit controls. The manuscript's geometric consistency check functions as an implicit disambiguation by demonstrating that interventions aligned with the continuous carry dimension correct errors in a manner not predicted by discrete attention patterns or static embedding statistics. However, to directly address the concern, the revision will add a dedicated ablation section comparing the Noisy Quantization Model against alternatives, including attention-head ablations and training-data regularity baselines, with quantitative error-prediction comparisons. revision: yes

  2. Referee: [Abstract] Abstract: the validation of the Noisy Quantization Model and the geometric consistency check is described only at a high level; no quantitative metrics, error bars, or statistical tests are mentioned that would allow assessment of whether the observed trajectories and error patterns are better explained by the proposed model than by simpler alternatives.

    Authors: The current manuscript emphasizes qualitative trajectory visualizations and the functional success of the consistency check. We acknowledge the absence of explicit quantitative metrics in the abstract and high-level description. The revision will add quantitative results, including detection accuracy with error bars across multiple seeds, statistical significance tests against baseline models, and tables comparing slippage prediction performance to simpler alternatives. revision: yes

Circularity Check

0 steps flagged

No circularity in observational geometry analysis

full rationale

The paper presents observational analysis of residual stream geometry during addition, identifying structures such as the Iso-Raw-Sum Trajectory and proposing the Noisy Quantization Model to frame errors as geometric slippages. No load-bearing derivations, equations, or results are shown to reduce by construction to fitted inputs, self-citations, or self-definitional loops. The central claims remain descriptive and model-proposing without the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 3 invented entities

The central claims rest on the assumption that residual-stream geometry directly reflects computational mechanisms and that the proposed quantization thresholds exist as latent continuous variables; no free parameters are named in the abstract, but the model introduces new descriptive entities without independent falsification criteria beyond the observed trajectories.

axioms (1)
  • domain assumption Residual stream activations during addition contain linearly readable semantic and carry information
    Invoked when the authors state that representations are anchored by semantic digits and modulated by carry fibers.
invented entities (3)
  • Iso-Raw-Sum Trajectory (IRST) no independent evidence
    purpose: Describes the observed geometric path of activations during addition
    New descriptive structure introduced to organize the activation patterns
  • Noisy Quantization Model no independent evidence
    purpose: Explains arithmetic errors as slips of a continuous carry potential across thresholds
    New explanatory model proposed in the abstract
  • Carry Potential no independent evidence
    purpose: Continuous latent variable that is quantized to produce carry decisions
    Postulated continuous signal whose noise produces observed errors

pith-pipeline@v0.9.1-grok · 5687 in / 1566 out tokens · 21786 ms · 2026-06-28T23:42:33.275720+00:00 · methodology

discussion (0)

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