Paper Citation Record · LEDGER

The Discrete-Log Clock: How a Transformer Learns Modular Multiplication

As of 9 August 2026, this Paper Citation Record lists 8 of 8 outbound references and 0 inbound Pith citation observations for arXiv:2606.17399.

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

pith.paper-citation-record.v1
2606.17399 v1

Coverage vector

measured 8 of 8 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-06-27T02:10:10.610990Z

measured 8 of 8 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-09T06:31:02.800959+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

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

Source: cited_works

Reference resolution

8 of 8 outbound references displayed

verified exact1
verified fuzzy0
unresolved7
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 4a0bfc1a-21a1-4c5f-a0d9-5f2999504053 · outbound

This paper cites ICLR 2022 Workshop on MATH-AI , year=.

The Discrete-Log Clock: How a Transformer Learns Modular Multiplication ICLR 2022 Workshop on MATH-AI , year=

Reference 1

Resolution: unresolved; no resolver link, observed 2026-06-27T02:10:10.610990Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 18eae4b2-b16c-4f12-8abe-f914a480544b · outbound

This paper cites International Conference on Learning Representations , year=.

The Discrete-Log Clock: How a Transformer Learns Modular Multiplication International Conference on Learning Representations , year=

Reference 2

Resolution: unresolved; no resolver link, observed 2026-06-27T02:10:10.610990Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 5c7cffc4-9985-4cb4-89c3-642e40be5406 · outbound

This paper cites Advances in Neural Information Processing Systems , year=.

The Discrete-Log Clock: How a Transformer Learns Modular Multiplication Advances in Neural Information Processing Systems , year=

Reference 3

Resolution: unresolved; no resolver link, observed 2026-06-27T02:10:10.610990Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c8df9e15-8d35-4647-b3e5-ef2f6e7c77f5 · outbound

This paper cites Grokking Modular Polynomials.

The Discrete-Log Clock: How a Transformer Learns Modular Multiplication Grokking Modular Polynomials

Reference 4

Resolution: verified exact; arxiv_id, observed 2026-07-03T19:08:49.920050Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation a1a6ec6d-afed-4fc4-856f-9078ab4da774 · outbound

This paper cites Transactions on Machine Learning Research , year=.

The Discrete-Log Clock: How a Transformer Learns Modular Multiplication Transactions on Machine Learning Research , year=

Reference 5

Resolution: unresolved; no resolver link, observed 2026-06-27T02:10:10.610990Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a2ef61df-fc1e-4153-810e-2000fffbe665 · outbound

This paper cites International Conference on Machine Learning , year=.

The Discrete-Log Clock: How a Transformer Learns Modular Multiplication International Conference on Machine Learning , year=

Reference 6

Resolution: unresolved; no resolver link, observed 2026-06-27T02:10:10.610990Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d57ca1a7-5e31-4c77-85a5-569bccb6929e · outbound

This paper cites Advances in Neural Information Processing Systems , year=.

The Discrete-Log Clock: How a Transformer Learns Modular Multiplication Advances in Neural Information Processing Systems , year=

Reference 7

Resolution: unresolved; no resolver link, observed 2026-06-27T02:10:10.610990Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 68b28a25-a033-494a-9628-de8c5cb15078 · outbound

This paper cites International Conference on Machine Learning , year=.

The Discrete-Log Clock: How a Transformer Learns Modular Multiplication International Conference on Machine Learning , year=

Reference 8

Resolution: unresolved; no resolver link, observed 2026-06-27T02:10:10.610990Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

No inbound Pith citation observations are available.