Paper Citation Record · LEDGER

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning

As of 9 August 2026, this Paper Citation Record lists 29 of 29 outbound references and 1 inbound Pith citation observation for arXiv:2506.21797.

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

pith.paper-citation-record.v1
2506.21797 v2

Coverage vector

measured 29 of 29 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-06T22:28:10.479678Z

measured 30 of 30 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 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-06-28T23:26:28.158991Z

measured 0 of 1 external citation measurements

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

Source: arxiv_reference, observed 2026-06-28T23:32:47.543386Z

Reference resolution

29 of 29 outbound references displayed

verified exact2
verified fuzzy13
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External citation measurements

No source-named external measurement is stored.

Outbound references

Observation bf27c8bf-c317-4da3-88ad-8b9af94284b9 · outbound

This paper cites Neurosymbolic programming.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Neurosymbolic programming

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:13.383595Z

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 d702ac5d-196b-4a68-acd9-553cdc40b417 · outbound

This paper cites Neurosymbolic ai: The 3 rd wave.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Neurosymbolic ai: The 3 rd wave

Reference 2

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Observation c8a34539-ed7f-4322-acce-3f6297d936d2 · outbound

This paper cites On the paradox of learning to reason from data.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning On the paradox of learning to reason from data

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:13.168084Z

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

Observation 0a651d92-d0c2-46e5-98b1-3237cf7ee774 · outbound

This paper cites On the planning abilities of large language models-a critical investigation.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning On the planning abilities of large language models-a critical investigation

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:12.983200Z

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 9ae760af-4783-48ca-9136-b52874e2c79c · outbound

This paper cites Composing global optimizers to reasoning tasks via algebraic objects in neural nets.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Composing global optimizers to reasoning tasks via algebraic objects in neural nets

Reference 5

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Source-reported events for the cited work

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Observation d64d0028-9dc1-49d7-90a4-b40f3d15c8c1 · outbound

This paper cites Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-06T22:28:08.268777Z

Source-reported events for the cited work

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Observation ae36f4c4-19d6-4e48-b4b7-3221f9b67485 · outbound

This paper cites Grokking modular arithmetic.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Grokking modular arithmetic

Reference 7

Resolution: unresolved; no resolver link, observed 2026-08-06T22:28:08.367301Z

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Observation 32e91a5f-4ed0-4a11-a7ff-d6e700e98d31 · outbound

This paper cites Progress measures for grokking via mechanistic interpretability.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Progress measures for grokking via mechanistic interpretability

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-06T22:28:08.484917Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 015d896b-e3a8-4425-baef-f0b5f022592b · outbound

This paper cites On the power of over-parametrization in neural networks with quadratic activation.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning On the power of over-parametrization in neural networks with quadratic activation

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:12.815801Z

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 40040ca2-e341-43ff-a3dc-123c505b6d54 · outbound

This paper cites ReLU$^2$ Wins: Discovering Efficient Activation Functions for Sparse LLMs.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning ReLU$^2$ Wins: Discovering Efficient Activation Functions for Sparse LLMs

Reference 10

Resolution: unresolved; no resolver link, observed 2026-08-06T22:28:08.715111Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 2acd4874-2b1e-4c70-b00c-e6563031c584 · outbound

This paper cites GLU Variants Improve Transformer.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning GLU Variants Improve Transformer

Reference 11

Resolution: unresolved; no resolver link, observed 2026-08-06T22:28:08.810981Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 48bebf49-6714-4995-8d22-920f98c7eee0 · outbound

This paper cites Searching for efficient transformers for language modeling.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Searching for efficient transformers for language modeling

Reference 12

Resolution: unresolved; no resolver link, observed 2026-08-06T22:28:08.899017Z

Source-reported events for the cited work

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Observation 3bfb7da4-6437-4c11-b10e-580febd26665 · outbound

This paper cites Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:12.691757Z

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 0357e1e3-901d-4291-80cd-0f63f5755e1f · outbound

This paper cites Optimal transport: old and new, volume 338.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Optimal transport: old and new, volume 338

Reference 14

Resolution: unresolved; no resolver link, observed 2026-08-06T22:28:09.073618Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3ceb0446-3916-4a1f-a83e-bb576c27c933 · outbound

This paper cites Gradient flows: in metric spaces and in the space of probability measures.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Gradient flows: in metric spaces and in the space of probability measures

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:12.516464Z

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 a8457cd4-eaf4-4f4e-a50a-d1e82bc8a6d2 · outbound

This paper cites Lectures on phase transitions and the renormalization group.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Lectures on phase transitions and the renormalization group

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:12.287135Z

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 2e35e35b-471e-4726-a6ba-19252d123b16 · outbound

This paper cites The exact sample complexity gain from invariances for kernel regression.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning The exact sample complexity gain from invariances for kernel regression

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:12.097378Z

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 f5be0254-ed5d-4be9-80ca-dab83bae6cdf · outbound

This paper cites Rethinking Addressing in Language Models via Contexualized Equivariant Positional Encoding.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Rethinking Addressing in Language Models via Contexualized Equivariant Positional Encoding

Reference 18

Resolution: verified exact; local_arxiv, observed 2026-08-06T22:28:10.932173Z

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 382dfe29-9b46-44d8-94cd-2e10232401b4 · outbound

This paper cites The parallelism tradeoff: Limitations of log-precision transformers.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning The parallelism tradeoff: Limitations of log-precision transformers

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:11.924679Z

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 3c7a0061-f378-43d9-9db3-41810be3b487 · outbound

This paper cites The Illusion of State in State-Space Models.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning The Illusion of State in State-Space Models

Reference 20

Resolution: unresolved; no resolver link, observed 2026-08-06T22:28:09.640815Z

Source-reported events for the cited work

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Observation 7d2d3932-1991-493d-bd86-df748f47db61 · outbound

This paper cites Methods of information geometry, volume 191.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Methods of information geometry, volume 191

Reference 21

Resolution: unresolved; no resolver link, observed 2026-08-06T22:28:09.748279Z

Source-reported events for the cited work

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Observation 97ce310c-7b0a-46f8-a2dd-13d820ad1abf · outbound

This paper cites Stability and generalization.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Stability and generalization

Reference 22

Resolution: unresolved; no resolver link, observed 2026-08-06T22:28:09.809965Z

Source-reported events for the cited work

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Observation 76cb324d-31c7-4f26-8880-170dec8f7c24 · outbound

This paper cites Understanding machine learning: From theory to algorithms.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Understanding machine learning: From theory to algorithms

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-06T22:28:09.918761Z

Source-reported events for the cited work

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Observation 8664f202-bc94-4073-b232-127afd734342 · outbound

This paper cites Equivariant architectures for learning in deep weight spaces.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Equivariant architectures for learning in deep weight spaces

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:11.749757Z

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 0d7372de-2410-4de8-b772-626e6086fcf6 · outbound

This paper cites Signal processing for implicit neural representations.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Signal processing for implicit neural representations

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:11.628059Z

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 ecb81feb-01f6-4a51-b92c-44308d91092e · outbound

This paper cites Model-GLUE: Democratized LLM Scaling for A Large Model Zoo in the Wild.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Model-GLUE: Democratized LLM Scaling for A Large Model Zoo in the Wild

Reference 26

Resolution: verified exact; local_arxiv, observed 2026-08-06T22:28:10.685006Z

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 3554cc64-c9f5-43de-b4bd-463b5a9fff6b · outbound

This paper cites Self-consuming generative models go mad.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Self-consuming generative models go mad

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:11.436820Z

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 85961c29-5fcf-4e07-898b-3f59822e3a97 · outbound

This paper cites Polynomial Width is Sufficient for Set Representation with High-dimensional Features.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Polynomial Width is Sufficient for Set Representation with High-dimensional Features

Reference 28

Resolution: unresolved; no resolver link, observed 2026-08-06T22:28:10.408292Z

Source-reported events for the cited work

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Observation 67f9f50e-2367-4ff8-a7b3-ac741d0730a7 · outbound

This paper cites Low-dimensional invariant embeddings for universal geometric learning.

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning Low-dimensional invariant embeddings for universal geometric learning

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T22:28:11.281524Z

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.

Pith citing papers

Observation 9a15a667-a0fc-4f17-8218-e6e66ac86e46 · inbound

Agentic Transformers Provably Learn to Search via Reinforcement Learning cites this paper.

Agentic Transformers Provably Learn to Search via Reinforcement Learning Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning

Reference 4

Resolution: verified exact; arxiv_id, observed 2026-06-28T23:32:47.544809Z

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.