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

Liuyuan Wen, Xun Zhu, Lihao Huang, Wenbin Li, Yang Gao

arXiv:2606.03645v1cs.LGcs.AI

TL;DR

LLMs remain fragile on multi-digit addition, and the mechanism linking internal representations to output errors is unclear. This paper analyzes residual-stream geometry, proposes a noisy quantization account, and shows that dual-stream consistency correction reaches 89.56% accuracy while revealing retained latent arithmetic information.

  • Problem

    The mechanism linking LLM activation geometries to specific multi-digit arithmetic output failures remains unclear.

  • Method

    The paper analyzes residual-stream activations at every digit position, identifies IRST geometry, and models arithmetic as noisy quantization of a continuous Carry Potential.

  • Results

    89.56% accuracy: dual-stream consistency correction achieves the highest accuracy among evaluated baselines and supports IRST as a causal geometric account.

  • Takeaways & Limitations

    Arithmetic errors often preserve correct internal mathematical components but arise when neural noise causes local quantization shifts during token selection.

  • Takeaways & Limitations

    Evidence combines UMAP visualization and probe decoding rather than full circuit-level localization, and focuses mainly on addition.

Abstract

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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.

1. Introduction

The paper investigates why LLMs that solve complex mathematics remain fragile on multi-digit addition, despite encoding correct arithmetic information internally. It introduces a geometric account linking residual-stream representations, quantization errors, probe versatility, and inference-time correction.

  • Motivation: LLMs achieve strong complex-mathematics performance yet frequently make off-by-one errors in multi-digit addition, indicating a disconnect between internal computation and discrete output.This fragility becomes more apparent as the number of operands increases.
  • Method: The study analyzes residual-stream activations at every generated digit while adding three or more 10-digit integers, rather than focusing only on the first token.It uses lightweight probes, including logistic regression and MLPs, to examine arithmetic variables at each digit position.
  • Geometric Structure: Iso-Raw-Sum Trajectories organize representations as continuous constant-raw-sum fibers passing through hierarchical digit basins with input-carry texture.Arithmetic errors arise when continuous representational drift crosses decision boundaries between adjacent digit basins.
  • Mechanism: The Noisy Quantization Model treats arithmetic failures as noise-induced threshold crossings when a continuous Carry Potential is discretized, predicting the observed periodic bathtub error distribution.The same geometric framework explains probe versatility as structurally determined by the manifold’s topology.
  • Inference-Time Correction: A dual-stream consistency check aligns decoded local Raw Sum and global Carry Potential signals during inference, recovering performance and confirming retention of correct latent arithmetic signals.The method intervenes when final token selection is erroneous.

2. Preliminaries

This section formalizes multi-operand addition and separates mathematical arithmetic states from the model’s predicted and latent internal states. It also defines the probing framework for decoding these states from hidden representations, including discrete digits, carries, raw sums, and Carry Potential.

  • Mathematical setup: The task adds n integers, each represented by m decimal digits, with ai,k denoting the digit of Ai at position 10k.Each addend is expressed as Ai = Pm−1 k=0 ai,k · 10k.
  • Representation geometry: Representations within each digit basin are further stratified into distinct fibers according to the input carry state cp.The arithmetic manifold has macroscopic digit Anchors ranging from 0–9, with correct and erroneous samples occupying corresponding regions.
  • Mathematical setup: At generation position p, the arithmetic state comprises raw sum rp, input carry cp, and ground-truth digit sp, satisfying sp ≡ (rp + cp) (mod 10).The position p maps autoregressive generation order to the corresponding mathematical digit significance.
  • Model state formalization: The model’s state is represented by the predicted digit ˆsp and latent predicted raw sum ˆrp and input carry ˆcp, distinguishing internal representations from mathematical truths.These variables are aggregated into a state tuple for each generation position.
  • Probing framework: Lightweight probes decode ground-truth and predicted digits, input carry, raw sum, and the continuous Carry Potential Φ from hidden states, primarily at the final layer L.The probes are trained on extracted hidden states h(l)p, while Carry Potential is defined later as a scalar modeling accumulated arithmetic value from context.
  • Latent-state inference: Because ˆcp and ˆrp are latent, the framework infers them from observed output errors while assuming local raw-sum computation is generally correct.Under this assumption, predicted-digit errors primarily arise from incorrect carry states.

3. Representational Geometry

The model’s last-layer activations form a geometric manifold that separates output-digit identity from carry-dependent arithmetic state. Correct computations occupy stable fiber centers, while errors concentrate in sparse transitions between digit basins and carry fibers.

  • Experimental Setup: Using Qwen3-4B on 10,000 three-term additions of three 10-digit integers, the analysis examines final-layer hidden states at interior digit positions.The model has L = 36 transformer layers, and the input carry cp ∈ {0, 1, 2}.
  • Global Semantic Structure: UMAP reveals ten activation basins centered on digit anchors, establishing a macroscopic backbone organized by output-token identity.Cosine distance is used, with digit-token unembedding vectors serving as fixed semantic landmarks.
  • Local Arithmetic Structure: Within each digit basin, representations form three parallel carry-specific fibers corresponding to cp ∈ {0, 1, 2}.This fine-grained texture indicates that arithmetic state is encoded locally within each digit-identity basin.
  • Prediction Reliability: Correct predictions cluster near the centers of carry fibers, whereas errors concentrate in sparse transition zones between digit basins or carry fibers.Incorrect samples often lie on the fringes of their ground-truth digit basins.

4. Geometric Analysis of Internal States

The geometric analysis proposes Iso-Raw-Sum Trajectories (IRSTs) as continuous manifolds that connect arithmetic states sharing a raw sum while crossing digit basins as carry varies. Arithmetic errors are Geometric Slippages along unstable IRST segments near anchor decision boundaries, where ambiguous carry representations mix signals and confuse the unembedding layer.

  • Iso-Raw-Sum Trajectories: IRSTs are continuous manifolds connecting internal states with the same raw sum, traversing adjacent digit basins as the carry varies.The trajectory follows fixed raw-sum fibers rather than remaining within one digit basin.
  • Iso-Raw-Sum Trajectories: Model transitions between successive output digits occur by sliding along fixed raw-sum fibers as the internal carry state shifts.The states are represented with tuples (sp, ˆsp, cp, ˆcp).
  • Geometric Slippages: Arithmetic errors are Geometric Slippages in which activation vectors drift onto unstable segments connecting stable nodes instead of aligning with the nodes.This frames errors as structured movements along IRSTs rather than random noise.
  • Geometric Slippages: On T1, the hallucination state (1, 2, 0, 1) confuses carry 0 with carry 1 by pushing the representation from Anchor 1 toward Anchor 2.The ground truth requires c = 0, while the model predicts ˆc = 1.
  • Geometric Slippages: On T0, hallucination state (1, 2, 1, 2) jumps to ˆc = 2 despite c = 1, while leakage state (2, 1, 2, 1) retreats to ˆc = 1.These errors reflect confusion between carries 1 and 2.
  • Geometric Slippages: Correct predictions cluster near stable nodes, whereas errors lie on sparse connecting paths near anchor decision boundaries, leaving unembedding with mixed signals.Raw-sum continuity conflicts with an ambiguous carry state in these transitional regions.

5. Formal Modeling of Carry Dynamics

The Noisy Quantization Model formalizes carry as a continuous latent potential that is discretized into integer carries, with neural noise causing threshold-crossing arithmetic errors. Empirical results support this mechanism through smooth potential gradients, periodic boundary-sensitive error rates, and an estimated noise level of σ ≈0.05.

  • 5. Formal Modeling of Carry Dynamics: The Noisy Quantization Model treats arithmetic representations as continuous manifolds connecting discrete digit anchors through a latent carry-pressure variable.This formulation explains why internal arithmetic need not be purely symbolic.
  • 5.1. Carry Potential (Φ): Carry Potential Φp is the continuous accumulation of numerical value flowing from lower-order positions into position p.It is derived from a weighted sum of raw sums in the right context, unlike the discrete ground-truth carry cp.
  • 5.1. Carry Potential (Φ): The discrete carry is cp = ⌊Φp⌋, so Φ = 1.51 quantizes to cp = 1.The example decomposes the potential as 1.4 + 0.11 before flooring.
  • 5.1. Carry Potential (Φ): Activation manifolds exhibit a smooth Φp gradient, placing high-risk states near the next carry basin.States with Φp ≈0.9 lie spatially closer to that basin than states with Φp ≈0.1.
  • 5.2. Noisy Quantization Hypothesis: The model represents perceived potential as the true Φp corrupted by additive Gaussian noise, then floors it to produce predicted carry ˆcp.The noise variance is σ2, and σ is termed the Cognitive Noise Level.
  • 5.2. Noisy Quantization Hypothesis: Errors occur when neural noise pushes perceived potential ˆΦp across an integer quantization boundary.This threshold-crossing mechanism is the proposed source of arithmetic failures.
  • 5.3. Bathtub Error Rate: Error rates form a periodic bathtub pattern, peaking near integer potentials and reaching minima near half-integer values.The model separates off-by-one errors into underestimation (Leakage) and overestimation (Hallucination), with metastability near integer thresholds and robust plateaus near i + 0.5.
  • 5.3. Bathtub Error Rate: σ ≈0.05 is extracted by fitting the theoretical curve to the empirical error distribution.The close theory–observation alignment supports internal noise crossing quantization boundaries as the mechanism determining arithmetic errors.

6. Trajectory-Level Validation and Causal Steering

Trajectory-level interventions support IRST’s claim that arithmetic states are ordered by a continuous carry coordinate that causally controls generated digits. On representative trajectory T3, boundary-adjacent errors switched under smaller perturbations than stable states, supporting a carry-like continuous decision direction.

  • Trajectory-Level Validation: The validation defines a steering direction between adjacent stable centroids on the same IRST to test whether moving representations along the carry coordinate changes generated digits.The intervention targets a final-layer pre-norm activation h.
  • Causal Steering: The steering coefficient α moves states toward higher-carry basins when positive and suppresses the carry signal when negative.The prediction is that boundary-near samples flip under smaller perturbations than deep in-basin states.
  • Trajectory-Level Validation: On trajectory T3 at p = 4, off-by-one errors concentrated in transition regions of the predicted V-shaped Carry Potential progression rather than forming a separate cluster.The trajectory includes stable states (3, 3, 0), (4, 4, 1), and (5, 5, 2), with steering instantiated between µ(4,4,1) and µ(5,5,2).
  • Causal Steering: α ≈−0.1 and α ≈0.3 were sufficient to switch boundary-adjacent states, versus α ≈−0.5 and α ≈0.5 for stable states.The boundary-adjacent states were (4, 5, 1) and (5, 4, 2), while the stable states were (5, 5, 2) and (4, 4, 1).

7. Geometric Origins of Probing Phenomenon

Linear-probe accuracy reflects geometric separability within arithmetic representations. The results support IRST trajectories with continuous carry structure, while errors and correctness occupy smooth transition geometries rather than discrete boundaries.

  • Model Output vs. Ground Truth: 98.81% Model Output probe accuracy contrasts with 94.85% Ground Truth accuracy, indicating predicted anchors are easier to identify than drifted activations’ underlying answers.The Model Output probe effectively mimics the unembedding matrix, whereas recovering ground truth requires mapping incorrect Voronoi basins back to the correct basin.
  • Correctness as Boundary Detection: 82.41% Correctness probe accuracy shows that correctness is a smooth stability gradient across basin interiors and transition zones, not a sharp discrete boundary.The probe distinguishes samples deep within a basin from likely-error transition states, but its moderate performance reflects continuous manifold structure.
  • Decoding the IRST Structure: 98.60% Raw Sum probe accuracy supports distinct IRSTs, while off-by-one carry errors remain on the same trajectory and preserve raw-sum separability.The representation slides along an IRST rather than jumping to another one during most arithmetic errors.
  • Decoding the IRST Structure: 96.84% Input Carry probe accuracy shows that probes can disentangle parallel carry fibers, including carry 0, 1, and 2, within shared trajectories.This supports the interpretation that IRSTs contain organized parallel fibers corresponding to discrete input-carry states.
  • Continuous Potential vs. Discrete Carry: 92.08% Carry Potential probe accuracy after quantization shows that continuous trajectory position can be mapped to a discrete carry with high but reduced precision.Its lower performance than the discrete carry probe reflects the complexity of quantizing a continuous position along an IRST; the correlation validates continuous magnitude organization.

8. Inference-Time Self-Correction

The section introduces a dual-stream geometric consistency check that uses raw-sum digits and continuous carry potentials to detect and correct arithmetic outputs. Its performance and ablations support IRST as a recoverable latent structure whose noisy carry representation is the main failure bottleneck.

  • Inference-time intervention: Inconsistent outputs are treated as geometric divergence and corrected by overriding logits with the raw sum plus the estimated integer carry modulo 10.This intervention projects the representation back onto the valid IRST manifold fiber and tests whether the arithmetic signal remains recoverable.
  • Dual-stream decoding: Two orthogonal lightweight probes decode the local raw-sum digit and global continuous carry potential from the final-layer activation.The raw-sum probe captures immediate column-wise arithmetic, while the carry probe estimates cumulative context information.
  • Consistency check: The robust consistency check accepts an output when its raw sum and digit satisfy the modular identity for any carry in a potential-derived plausible set.Using a δ-neighborhood avoids false positives caused by strict integer carries near quantization boundaries.
  • Correction results: 89.56% accuracy is achieved by the proposed method at δ = 0.1, exceeding the evaluated re-prompting, linear-steering, and hard-replacement baselines.The authors interpret this as causal validation of IRST geometry because internal consistency corrects errors without requiring external ground truth.
  • Tolerance sensitivity: 44.39% TP Corr is obtained at δ = 0, but δ = 0.1 provides more balanced performance by avoiding aggressive overwriting of valid ambiguous states.This sensitivity supports the Noisy Quantization Model and its predicted bathtub-like stability under stochastic boundary noise.
  • Ablation study: 96.0% token accuracy with R+TC versus 90.5% with TR+C indicates robust latent local computation and identifies noisy carry potential as the main arithmetic bottleneck.R+TC uses the raw-sum probe with true carry, whereas TR+C uses true raw sum with the carry probe.

9. Related Works

Related work frames LLM arithmetic through symbolic, geometric, and modular-encoding perspectives, while this paper reinterprets arithmetic generation as noisy quantization of a Continuous Carry Potential. This framing also explains why linear probes can disentangle coexisting arithmetic signals.

  • Arithmetic Mechanisms: Prior work treats arithmetic as discrete symbolic manipulation, mapping inputs to categorical sum types, whereas this paper models generation as noisy quantization of a Continuous Carry Potential.The proposed reinterpretation provides a mechanistic account of arithmetic behavior and its failures.
  • Arithmetic Mechanisms: The Continuous Carry Potential framework provides a mechanistic explanation for why linear probes can successfully disentangle arithmetic signals.The claim connects the proposed noisy-quantization view to probe-based signal separation.
  • Geometric Perspectives: Broader geometric studies provide foundational context, but deciphering arithmetic requires analyzing specific algebraic structures.Related approaches hypothesize modular arithmetic through rotational dynamics on circular manifolds, with refinements proposing high-dimensional spirals or trigonometric encodings.

10. Conclusion and Discussion … B.1. Quantitative Evaluation

The paper frames arithmetic errors as geometric slippages caused by noisy quantization of latent carry states, and evaluates its activation-space visualization pipeline using deterministic Qwen3 experiments and embedding comparisons.

  • 10. Conclusion and Discussion: The IRST describes arithmetic representations as semantic-digit trajectories modulated by continuous carry states, with neural noise causing predominantly off-by-one errors at quantization boundaries.The Noisy Quantization Model posits that models can maintain a correct continuous Carry Potential internally but fail during discrete token selection.
  • 10. Conclusion and Discussion: The methodology relies on models whose tokenizers represent each decimal digit as a single token, while BPE-based models remain a future research direction because carry signals may be entangled in multi-digit embeddings.The single-digit tokenization enables direct alignment between visual digits and internal arithmetic states.
  • A.1. Model Configuration and Inference: All main-text experiments use Qwen3-4B with greedy decoding at temperature T = 0, making generations deterministic.This setup is intended to ensure that internal representations reflect a deterministic arithmetic reasoning path.
  • A.2. Dataset Construction: The dataset contains N = 10, 000 distinct three-term addition problems, with each addend represented as a 10-digit integer.Each sample has the form A0 + A1 + A2 = S.
  • A.2. Dataset Construction: A forced-prefix prompt appends the arithmetic expression and equals sign to the chat-template instruction, suppressing conversational filler and enforcing numeric output.The example explicitly requests only a number without commas.
  • A.2. Dataset Construction: Generation stops at the first incorrect digit, so every analyzed activation follows a strictly correct history before the observed arithmetic divergence.The protocol isolates the precise origin of errors by terminating immediately after the first deviation.
  • A.3. Activation Extraction: Activations are analyzed position-wise from the final layer after normalization and immediately before unembedding, with the primary target at p = 4 corresponding to the 5th generated token.The indexing proceeds from the most significant digit toward the least significant digit.
  • A.4. Dimensionality Reduction Settings / B. Comparison of Different Dimensionality Reductions / B.1. Quantitative Evaluation: UMAP projects 2560-dimensional activations into 2D using cosine distance, n neighbors=300, and min dist=0.3, while PCA and t-SNE provide robustness comparisons.Trustworthiness with n neighbors=50 evaluates local-neighborhood preservation, and UMAP and t-SNE are reported as outperforming PCA.

B.2. Analysis of Visualizations … C.3. Native-Space Validation Across Trajectories

The visualizations and native-space analyses support IRST as an intrinsic, nonlinear residual-stream geometry rather than a dimensionality-reduction artifact. Trajectory-level steering and native-space profiles further show that arithmetic decisions track Carry Potential across stable states and threshold errors.

  • B.2. Analysis of Visualizations: t-SNE recovers IRST’s macroscopic backbone and microscopic texture, while PCA collapses the fine-grained fibers because its first two components capture only 9.76% and 8.22% variance.The PCA projection shows only macroscopic digit clustering.
  • B.2. Analysis of Visualizations: The first two principal components explain approximately 17.98% of total variance, indicating that relevant arithmetic information is distributed across a higher-dimensional, nonlinear manifold.Nonlinear methods are more trustworthy because local-neighborhood preservation is essential for interpreting the model.
  • B.2. Analysis of Visualizations: t-SNE shows digit basins and stratified carry fibers consistent with UMAP, supporting IRST as intrinsic; UMAP’s trustworthiness is 0.9953 versus t-SNE’s 0.9907.UMAP was selected for better global continuity across multi-digit trajectories.
  • C. Geometric Validation and Causal Steering: The appendix restores the full trajectory-level interpretation of the T3 case study before presenting broader native-space ablations and intrinsic-dimensionality measurements.The main text already includes T3’s representative trajectory-level and causal-steering results.
  • C.1. Detailed Case Study on T3: For T3, centroids of stable clusters (3, 3, 0), (4, 4, 1), and (5, 5, 2) are computed in the last-layer p = 4 representation space.The detailed case study visualizes this trajectory’s latent geometry.
  • C.1. Detailed Case Study on T3: T3 samples form a V-shaped progression connecting low-, central-, and high-potential clusters in cosine distance relative to the central centroid and analytical Carry Potential Φ.Figure 6 projects all T3 samples using cosine distance to (4, 4, 1) and Φ.
  • C.2. Detailed Causal Steering Dynamics: Steering thresholds show that stable (5, 5, 2) and (4, 4, 1) states require α ≈−0.5 and α ≈0.5, whereas nearby hallucination and leakage errors require α ≈−0.1 and α ≈0.3.The threshold ordering quantitatively confirms that arithmetic decisions are determined by projection onto the Carry Potential direction.
  • C.3. Native-Space Validation Across Trajectories: Native-space cosine-distance profiles preserve the same V-shaped correlation with Φ across T0 … T9, showing that the ordering already exists in R2560 rather than depending on a 2D UMAP layout.Figure 8 provides the projection-choice ablation in the original residual space.

C.4. Intrinsic-Dimensionality Analysis … J. Details for Inference-Time Correction Method

The paper presents arithmetic representations as compact, trajectory-specific geometric structures governed by latent carry states and noisy quantization. It validates this framework across dimensions, task complexities, model families, and an inference-time consistency correction method.

  • C.4. Intrinsic-Dimensionality Analysis: Trajectory-conditioned IRST subsets remain low-dimensional in native space, whereas their pooled union has a larger effective dimension.Layer-wise estimates are broadly similar before layer 23; T5 has slightly lower nonlinear ID than All between layers 23–31, then the trend reverses after layer 31.
  • D. Validation of the Raw Sum Assumption: 89.35% Accuracy and 99.59% AUC for raw-sum prediction, versus an Accuracy ∼3.6% random baseline, support predominantly correct local summation despite erroneous outputs.The probe was evaluated on a balanced dataset containing equal numbers of correct and incorrect generations, while imperfect performance leaves room for genuine local calculation failures.
  • E. Derivation of Latent Carry States: Assuming the raw sum is recovered correctly, output underestimation implies predicted carry ˆc_p = c_p − 1, while overestimation implies ˆc_p = c_p + 1.These mappings identify underestimation as Leakage of one carry unit and overestimation as Hallucination of an additional carry unit.
  • F. Derivation of the Noisy Quantization Error Rate: The Noisy Quantization Model treats carry as floor-quantized noisy potential, with leakage below the integer floor and hallucination above the next integer ceiling.The model defines ˆΦ = Φ + ϵ with ϵ ∼ N(0, σ^2), discrete carry ˆc = ⌊ˆΦ⌋, and total error as leakage plus hallucination risk when non-unit errors are negligible.
  • G. Scaling Dynamics of Cognitive Noise: Noise grows nonlinearly with operand count, from σ ≈0.05 at n = 3 to σ ≈0.25 at n = 5, while boundary-focused profiles differ from interior steady-state behavior.Operand count, digit length, and boundary position contribute to cognitive noise; the periodic bathtub structure persists for 4-term and 5-term addition.
  • H. Generalize to Other LLMs; H.1. IRSTs; H.2. Noisy Quantization Dynamics: Qwen3-8B and Gemma-3-4B-IT retain digit-basin and IRST geometry, while Qwen3-8B, Gemma-3-4B-IT, and specialized arithmetic models reproduce threshold-centered error patterns.The fitted noise levels are σ ≈0.05 for Qwen-3-8B and σ ≈0.07 for Gemma-3-4B-IT, with R2 ≥0.70 fits for both models.
  • J. Details for Inference-Time Correction Method: Dual-stream correction decodes raw sum and carry potential, accepts the original digit when a plausible carry is consistent, and otherwise intervenes with the reconstructed digit.The method forms K_p(δ) from the δ-neighborhood of ˆΦ_p and checks whether ˆs_p ≡ (ˆr_p + c) (mod 10) for any c in that set.

J.1. Experiment Settings … L.1. Analysis of the Decoding Gap

Across experiments and analyses, arithmetic information emerges sharply around Layer 24 as a latent geometric state that probes can decode before it is expressed in vocabulary space. Tolerance-based consistency correction improves accuracy by preserving plausible carry states, while the latent computation remains largely invisible to the Logit Lens.

  • J.1. Experiment Settings: Experiments use Qwen3-4B, Qwen3-8B, and Gemma-3-4B-IT on 10,000 three-number addition problems with 10-digit operands and an 0.8:0.1:0.1 split.The main metrics include Token Accuracy and True Positive Correction.
  • J.2. Further Research on Tolerance: At δ = 0.12, token accuracy reaches 0.8973 and question accuracy reaches 0.3300, versus 0.8727 and 0.2320 at δ = 0.Both metrics rise initially and then decline as tolerance increases from 0 to 0.2.
  • J.2. Further Research on Tolerance: The plausible-carry set acts as a confidence interval that preserves ambiguous but plausible internal states, while 93.19% of original errors and 88.9% of corrected errors are off-by-one.This targets the boundary-crossing failure mode emphasized by the dual-stream intervention.
  • J.3. Generalization Analysis: The Dual-Stream Probe generalizes to Gemma-3-4B-IT with 10-digit addition and Qwen3-8B with 12-digit addition, maintaining competitive performance despite sometimes trailing strict Replacement.These results are presented as causal validation that correct latent information persists internally even when the output is wrong.
  • K. Layer-Wise Analysis of Internal Activations / K.1. Probe Analysis: Layer-wise probing and representation analysis identify Layer 24 as the transition where arithmetic variables and carry potential become sharply decodable, marking the emergence of explicit arithmetic logic.Across Raw Sum, Ground Truth, and Model Output, early-layer accuracy remains near random before a sudden informational surge at Layer 24.
  • K.1. Probe Analysis: Attention outputs show sharper stepwise gains than FFN outputs, suggesting that carry information is first consolidated by attention and then refined through feed-forward processing.The analysis separately probes attention and FFN block outputs to localize carry-signal writing.
  • K.2. UMAP Visualization: Aligned UMAP shows stable 10 × 10 hierarchical clusters through Layers 1–23, collapse into an elliptical manifold at Layer 24, and later dissociation into IRST curves.The disjoint late-layer IRSTs are identified as an Aligned UMAP artifact caused by path-dependent temporal regularization.
  • L. Probe vs. Logit Lens: Evidence of Latent Arithmetic States / L.1. Analysis of the Decoding Gap: Linear probes decode arithmetic states above 90% during Layers 24–29 while the Logit Lens remains low, then both converge during Layers 30–36 as the state enters vocabulary space.This decoding gap motivates trained probes for analyzing latent arithmetic and geometric slippages.
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