Paper Citation Record · LEDGER

Modular Arithmetic: Language Models Solve Math Digit by Digit

As of 10 August 2026, this Paper Citation Record lists 19 of 19 outbound references and 2 inbound Pith citation observations for arXiv:2508.02513.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2508.02513 v1

Coverage vector

measured 19 of 19 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-06T05:02:54.271729Z

measured 21 of 21 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-10T06:31:04.303077+00:00

measured 2 of 2 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-01T20:19:22.487288Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: arxiv_reference, observed 2026-05-13T18:28:06.490731Z

Reference resolution

19 of 19 outbound references displayed

verified exact0
verified fuzzy2
unresolved15
parse uncertain0
malformed identifier1
metadata mismatch1

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation b8a05790-13a5-4285-807c-045fcb787297 · outbound

This paper cites The Llama 3 Herd of Models.

Modular Arithmetic: Language Models Solve Math Digit by Digit The Llama 3 Herd of Models

Reference 5

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:52.989961Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 5c765efa-a25b-4f83-9f4d-a273a939d16d · outbound

This paper cites Language Models Use Trigonometry to Do Addition.

Modular Arithmetic: Language Models Solve Math Digit by Digit Language Models Use Trigonometry to Do Addition

Reference 7

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:53.111777Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6a1156e6-09dc-400e-85b5-ea639e3e2111 · outbound

This paper cites What is a Number, That a Large Language Model May Know It?.

Modular Arithmetic: Language Models Solve Math Digit by Digit What is a Number, That a Large Language Model May Know It?

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:53.251521Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8b57101c-0a3a-4c86-a3a5-29816eb55bd3 · outbound

This paper cites Language Models Encode Numbers Using Digit Representations in Base 10.

Modular Arithmetic: Language Models Solve Math Digit by Digit Language Models Encode Numbers Using Digit Representations in Base 10

Reference 10

Resolution: metadata mismatch; local_arxiv, observed 2026-08-06T05:02:54.483401Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.

Observation 78d1ca2e-4836-4621-a8af-61a6883d4a79 · outbound

This paper cites Locating and Editing Factual Associations in GPT.

Modular Arithmetic: Language Models Solve Math Digit by Digit Locating and Editing Factual Associations in GPT

Reference 11

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:53.369998Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3ca89a1b-ed40-404e-a907-2c4aabf782e7 · outbound

This paper cites Progress measures for grokking via mechanistic interpretability.

Modular Arithmetic: Language Models Solve Math Digit by Digit Progress measures for grokking via mechanistic interpretability

Reference 12

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:53.493786Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 2e160cd8-62b9-466b-b76a-3eb61f7d6ab7 · outbound

This paper cites Arithmetic Without Algorithms: Language Models Solve Math With a Bag of Heuristics.

Modular Arithmetic: Language Models Solve Math Digit by Digit Arithmetic Without Algorithms: Language Models Solve Math With a Bag of Heuristics

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:53.646766Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 795b420e-6fa3-47cd-acce-1ecff9f84a10 · outbound

This paper cites 2 OLMo 2 Furious.

Modular Arithmetic: Language Models Solve Math Digit by Digit 2 OLMo 2 Furious

Reference 14

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:53.778261Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3f677acd-e43e-4923-9a40-db45bfb51865 · outbound

This paper cites A Mechanistic Interpretation of Arithmetic Reasoning in Language Models using Causal Mediation Analysis.

Modular Arithmetic: Language Models Solve Math Digit by Digit A Mechanistic Interpretation of Arithmetic Reasoning in Language Models using Causal Mediation Analysis

Reference 15

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:53.886054Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e8a388ab-8440-4fb7-92da-8d8675183ebe · outbound

This paper cites an unresolved cited work.

Modular Arithmetic: Language Models Solve Math Digit by Digit Unresolved cited work

Reference 16

Resolution: unresolved; raw_fallback, observed 2026-08-06T05:02:55.172992Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.

Observation ba05d40a-c342-45ba-b7a6-785aef29896b · outbound

This paper cites Interpreting and Improving Large Language Models in Arithmetic Calculation.

Modular Arithmetic: Language Models Solve Math Digit by Digit Interpreting and Improving Large Language Models in Arithmetic Calculation

Reference 17

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:54.053545Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 579aac01-1df6-4128-9d10-723a59c6d0b5 · outbound

This paper cites Transformers Can Achieve Length Generalization But Not Robustly.

Modular Arithmetic: Language Models Solve Math Digit by Digit Transformers Can Achieve Length Generalization But Not Robustly

Reference 18

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:54.113287Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6cc321b0-015d-4fdd-bf7e-b3c48b47e002 · outbound

This paper cites 3”, “4”, “7.

Modular Arithmetic: Language Models Solve Math Digit by Digit 3”, “4”, “7

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T05:02:55.053471Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.

Observation 206c12e0-c4cf-41b4-af32-a8b97d28f54b · outbound

This paper cites Generalized Fisher Score for Feature Selection.

Modular Arithmetic: Language Models Solve Math Digit by Digit Generalized Fisher Score for Feature Selection

Reference 2012

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:53.050478Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 2c09e8c1-af5d-4909-b095-e1a0eb5b5f47 · outbound

This paper cites Transformer Circuits Thread.

Modular Arithmetic: Language Models Solve Math Digit by Digit Transformer Circuits Thread

Reference 2021

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T05:02:55.285287Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.

Observation f77a6226-41ca-4b14-be6c-f912467957cc · outbound

This paper cites Successor Heads: Recurring, Interpretable Attention Heads In The Wild.

Modular Arithmetic: Language Models Solve Math Digit by Digit Successor Heads: Recurring, Interpretable Attention Heads In The Wild

Reference 2023

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:52.940577Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 434a3628-1a9b-4d6a-805e-13e7682eb3cd · outbound

This paper cites arXiv preprint arXiv:2410.15580.

Modular Arithmetic: Language Models Solve Math Digit by Digit arXiv preprint arXiv:2410.15580

Reference 2024

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:52.737350Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 80262894-97a4-4c18-b517-872dd976d253 · outbound

This paper cites The Lookahead Limitation: Why Multi-Operand Addition is Hard for LLMs.

Modular Arithmetic: Language Models Solve Math Digit by Digit The Lookahead Limitation: Why Multi-Operand Addition is Hard for LLMs

Reference 2025

Resolution: unresolved; no resolver link, observed 2026-08-06T05:02:52.692795Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 53afdbf7-a350-4574-9759-198f3c51d630 · outbound

This paper cites 157 + 431 =.

Modular Arithmetic: Language Models Solve Math Digit by Digit 157 + 431 =

Reference 3584

Resolution: malformed identifier; raw_fallback, observed 2026-08-06T05:02:54.889836Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.

Pith citing papers

Observation 456a0331-11dd-4843-a539-7986f8acc19f · inbound

Automated Attention Pattern Discovery at Scale in Large Language Models cites this paper.

Automated Attention Pattern Discovery at Scale in Large Language Models Modular Arithmetic: Language Models Solve Math Digit by Digit

Reference 2

Resolution: verified exact; arxiv_id, observed 2026-05-13T18:28:06.492528Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.

Observation 58ee0d24-9f82-41c0-a3da-0b8fb3e9eb99 · inbound

Are Arithmetic Heuristic Neurons Form-Invariant? A Mechanistic Analysis of Symbols, Text, and Code in LLMs cites this paper.

Are Arithmetic Heuristic Neurons Form-Invariant? A Mechanistic Analysis of Symbols, Text, and Code in LLMs Modular Arithmetic: Language Models Solve Math Digit by Digit

Reference 48

Resolution: unresolved; no resolver link, observed 2026-08-01T20:19:22.487288Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.