Transformers Learn the Mestre-Nagao Heuristic

Pranav Venkata Konda

cs.LGJun 13, 2026

Transformers Learn the Mestre-Nagao Heuristic

Authors: Pranav Venkata Konda

Abstract

We train a two-layer transformer encoder to classify rational elliptic curves $E/\mathbb{Q}$ of conductor $\leq 10000$ as either rank 0 or rank 1 from the first 128 normalized Frobenius traces. We achieve >99% accuracy on both classes, and accuracy is essentially unchanged on test curves with no isogeny or quadratic-twist relative in the training set. We then apply techniques from mechanistic interpretability such as attention analysis, linear probing, activation patching, logit attribution, and neuron-level circuit analysis to reverse-engineer the algorithm the (centroid in function space) model learned. We find that a sparse circuit of 20 out of 512 layer-1 MLP neurons is sufficient for rank prediction under a linear probe with an AUROC of 0.992 at plateau, implementing a push-pull detector architecture of rank-0 and rank-1 detectors with a one-sided readout. However, we notice that the model has sub-optimal readout problems indicating a mismatch in rank-order between the readout pathway and the discriminative circuit. Critically, the learned input weights of the top discriminating neuron match the Mestre-Nagao sum heuristic weights $\log(p)/(p\cdot \log{B})$ with a Spearman coefficient $r = 0.997$ and Pearson coefficient $r = 0.952$ : the model has learnt a result from analytic number theory from the Frobenius trace data alone. We additionally find that all 50 independently trained models concentrate CLS attention on prime positions at 2-50 $\times$ the rate of composite positions. The CLS embedding encodes $\log{L(E,1)}$ with $R^2 = 0.962\pm 0.011$ across the 50 models (after controlling for the conductor). Activation patching analysis reveals that attention weights are dissociated from causal information flow. Additionally, the 50 solutions from training are near-identical in function space (with pairwise agreement $>$ 98.8%) despite large weight space barriers.

Explore similar work

Jul 25, 2026cs.LG

The Entropic Bound for Transformers: Why Static Rank Fails and Attention-Native Rank Recovers

Neural scaling laws describe how loss decreases as models, data, and compute grow, but they do not answer a prior question: for a fixed task, what is the minimum model capacity required to solve it? We study this through the Entropic Bound, a spectral notion of task-intrinsic capacity for Transformers. We first prove that, in a linear attention surrogate, the intrinsic rank

r^*

of the token-mixing operator is a tight lower bound: any rank-deficient model incurs unavoidable excess risk, and the bound is achievable at

r^*

. We further show that gradient descent recovers this rank under standard low-rank implicit-bias assumptions, confirm all three properties empirically, and show

r^*

is recoverable from data before training. We then ask whether this transfers to real attention. A naive transfer fails, and a controlled interpolation ladder localizes the cause precisely: it is not softmax and not a rank constraint, but the input-conditioned nature of attention's mixing operator, which a static weight kernel cannot summarize. Motivated by this, we introduce an attention-native intrinsic rank -- the minimum query-key kernel rank realizing the task within the attention class -- and show that under this definition the full Entropic Bound structure (deficiency, achievability, recovery) is restored for both linear and softmax attention, with the energy effective rank as the estimator robust to softmax distortion. Finally, we map the boundary of data-only predictability:

r^*

is exactly recoverable for linear QK attention, even without the value map at scale, while softmax attention admits only partial pre-training recovery due to nonlinear inversion and kernel-value identifiability effects. Our results reframe the Entropic Bound from a post-hoc descriptor into an attention-native capacity measure with a precisely characterized predictability frontier.

Byeong Hoon Yoon

Low-Rank Factorization Intrinsic Dimensionality Ranking Distortion

Ask PDF

Jul 8, 2026cs.LG

Multiplication Beyond Groups: Stratified Fourier Mechanisms in Transformer Circuits

Transformers have demonstrated a remarkable ability to learn algorithmic reasoning, yet mechanistic analyses have mostly focused on globally invertible operations such as cyclic addition and group composition. In this work, we investigate how small transformers learn modular integer multiplication over composite moduli, a fundamentally non-invertible operation due to the presence of zero-divisors. We propose the monoid extension: a localized generalization of Group Composition via Representation (GCR) that suggests the learned computation does not rely on a single global representation space. Instead, the model partitions the input space into local hierarchical algebraic regions, where group-like structure survives and Fourier mechanisms can be applied. In transformers trained on square-free modular multiplication, we find that embeddings organize around these regions, attention exhibits class-sensitive routing and low-rank write directions, and local character features explain a large fraction of the model's output logits. Our results suggest that representation-theoretic mechanisms previously identified for group operations can extend beyond groups to more general structures.

Zitong Andrew Chen, Junaid Hasan, Akhil Srinivasan +2

Modular Addition Fourier Multilayer Perceptrons Small Transformers Mechanical Response

Ask PDF

May 21, 2026cs.LG

Represented Is Not Computed: A Causal Test of Candidate Algorithmic Intermediates in a Transformer

Structured prompts require integrating components according to task-relevant relations. How a network implements this integration is often hard to judge in language or vision, where those relations are rarely specified precisely enough to define a candidate internal algorithm. Arithmetic offers a cleaner setting. We study a Transformer trained on base-digit extraction: given

N

B

, and

D

, it must report the coefficient of

B^D

in the base-

B

expansion of

N

. The closed-form solution,

\lfloor N/B^D \rfloor \bmod B

, provides explicit candidate algorithmic intermediates. Across three seeds, the model reaches 99.83% exact-answer accuracy on held-out number-base intersections, establishing reliable task competence. Linear probes decode the intermediates, making staged arithmetic computation plausible. Causal tests then separate representation from use: within the localized route from the stream with

D

as input to the output positions, behavior depends on early

D

-selective communication, independent of

N

and

B

. Relatedly, a sparse circuit search finds mostly separate

N

B

, and

D

routes that combine late rather than the staged route suggested by the probes. Thus, the model represents the intermediates that make the closed-form solution plausible, but the identified localized causal route does not transmit them to the output stream. This case shows that probe-based conclusions can diverge sharply from causal observations, even when explicit algorithmic hypotheses are available.

Ishita Darade, Sushrut Thorat

Numerical Reasoning Task-Relevant Evidence

Ask PDF