Paper Citation Record · LEDGER

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition

As of 10 August 2026, this Paper Citation Record lists 19 of 19 outbound references and 0 inbound Pith citation observations for arXiv:2608.04451.

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

pith.paper-citation-record.v1
2608.04451 v2

Coverage vector

measured 19 of 19 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-08T18:34:18.350794Z

measured 19 of 19 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 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

19 of 19 outbound references displayed

verified exact0
verified fuzzy12
unresolved7
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 8b7d3b42-55bd-4357-9909-86ff0bf6e249 · outbound

This paper cites 2026 , eprint =.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition 2026 , eprint =

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:34:18.549938Z

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 60b52c77-6cda-4b0a-bcef-82a7bf0393a8 · outbound

This paper cites Progress measures for grokking via mechanistic interpretability.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition Progress measures for grokking via mechanistic interpretability

Reference 2

Resolution: unresolved; no resolver link, observed 2026-08-08T18:34:18.290455Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f5fb2138-ae31-48e8-b808-77703699bb94 · outbound

This paper cites 2026 , howpublished =.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition 2026 , howpublished =

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:34:18.539521Z

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 8e35f27a-80b0-4bbe-bc93-9dfd965c2300 · outbound

This paper cites an unresolved cited work.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition Unresolved cited work

Reference 4

Resolution: unresolved; raw_fallback, observed 2026-08-08T18:34:18.529751Z

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 7a8e1232-ca49-4eff-a07d-36004fbebea3 · outbound

This paper cites 2026 , howpublished =.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition 2026 , howpublished =

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:34:18.519796Z

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 033f9f2b-f5f0-4da5-aba8-594f0e9ea528 · outbound

This paper cites 2026 , howpublished =.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition 2026 , howpublished =

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:34:18.510067Z

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 5a621779-7eff-4e19-bb5d-889996c669fd · outbound

This paper cites and Hu, Wei and Lee, Jason D.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition and Hu, Wei and Lee, Jason D

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:34:18.500121Z

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 3d73aed7-d581-4578-99c2-f72136342bb0 · outbound

This paper cites , title =.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition , title =

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-08T18:34:18.313087Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 1a67a8cb-8488-4dc6-8bd0-b3a0d9590152 · outbound

This paper cites Tyrrell and Wets, Roger J.-B.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition Tyrrell and Wets, Roger J.-B

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:34:18.484323Z

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 6b82d8e4-7f84-476a-b5e0-2a64f89464e0 · outbound

This paper cites Journal of Convex Analysis , volume =.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition Journal of Convex Analysis , volume =

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:34:18.474668Z

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 0e2487fa-eb6a-45b7-87ab-3c7e0e240e6a · outbound

This paper cites Dying ReLU and Initialization: Theory and Numerical Examples.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition Dying ReLU and Initialization: Theory and Numerical Examples

Reference 11

Resolution: unresolved; no resolver link, observed 2026-08-08T18:34:18.323218Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 1f52de4d-770e-4054-ac2c-8c95cff68e85 · outbound

This paper cites Journal of Machine Learning Research , volume =.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition Journal of Machine Learning Research , volume =

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:34:18.464676Z

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 212b8338-795e-443a-915b-eb7d0dba9b6a · outbound

This paper cites Gradient Descent Maximizes the Margin of Homogeneous Neural Networks.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition Gradient Descent Maximizes the Margin of Homogeneous Neural Networks

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-08T18:34:18.330127Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 822b43eb-9bde-4315-b3eb-202750ed0ed1 · outbound

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

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition Advances in Neural Information Processing Systems , volume =

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:34:18.454265Z

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 53fa6d10-5b69-4f00-a15a-d54ac878683e · outbound

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

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition Advances in Neural Information Processing Systems , volume =

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:34:18.444272Z

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 a527d356-52f5-4b5b-983a-9aea061f19a7 · outbound

This paper cites Feature emergence via margin maximization: case studies in algebraic tasks.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition Feature emergence via margin maximization: case studies in algebraic tasks

Reference 16

Resolution: unresolved; no resolver link, observed 2026-08-08T18:34:18.340456Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f38cf238-182a-4137-a724-23edb9ba4999 · outbound

This paper cites Numerical Influence of.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition Numerical Influence of

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:34:18.435373Z

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 ce62102f-e464-43e0-a9d0-32d3366264e1 · outbound

This paper cites an unresolved cited work.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition Unresolved cited work

Reference 18

Resolution: unresolved; raw_fallback, observed 2026-08-08T18:34:18.426976Z

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 472eb659-d617-4c94-b4eb-2cd1767ceac5 · outbound

This paper cites 2026 , month = aug, note =.

A Counterexample to Fourier Alignment in Single-Neuron Modular Addition 2026 , month = aug, note =

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:34:18.417937Z

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

No inbound Pith citation observations are available.