Paper Citation Record · LEDGER

Permutation theory governs long-term dynamics of critical Boolean networks

As of 14 August 2026, this Paper Citation Record lists 27 of 27 outbound references and 0 inbound Pith citation observations for arXiv:2608.07790.

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

pith.paper-citation-record.v1
2608.07790 v1

Coverage vector

measured 27 of 27 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-11T04:20:46.924494Z

measured 27 of 27 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-14T06:32:32.682623+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

27 of 27 outbound references displayed

verified exact0
verified fuzzy24
unresolved3
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 3628f88e-22d9-46c5-86ff-1c094ac3451a · outbound

This paper cites Metabolic stability and epigenesis in randomly constructed genetic nets.

Permutation theory governs long-term dynamics of critical Boolean networks Metabolic stability and epigenesis in randomly constructed genetic nets

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.451027Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation df2129a7-71ef-4870-81a7-3bec8b6a4381 · outbound

This paper cites Boolean modeling in systems biology: an overview of methodology and applications.Physical biology, 9(5):055001, 2012.

Permutation theory governs long-term dynamics of critical Boolean networks Boolean modeling in systems biology: an overview of methodology and applications.Physical biology, 9(5):055001, 2012

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.434536Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 134bb710-ae45-4388-b58c-2aa02dc0d86c · outbound

This paper cites The random map model: a disordered model with deter- ministic dynamics.Journal de Physique, 48(6):971–978, 1987.

Permutation theory governs long-term dynamics of critical Boolean networks The random map model: a disordered model with deter- ministic dynamics.Journal de Physique, 48(6):971–978, 1987

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.416382Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation f64966b3-d897-42f4-97e4-770fb0bd4905 · outbound

This paper cites Exact solution of kauffman’s model with connectivity one.

Permutation theory governs long-term dynamics of critical Boolean networks Exact solution of kauffman’s model with connectivity one

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.397402Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 88594843-e8fd-46a8-add3-015ef4e98e68 · outbound

This paper cites Number and length of attractors in a critical kauffman model with connectivity one.Physical review letters, 94(8):088701, 2005.

Permutation theory governs long-term dynamics of critical Boolean networks Number and length of attractors in a critical kauffman model with connectivity one.Physical review letters, 94(8):088701, 2005

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.380513Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 52fb18f4-40f5-419f-8f33-04546ac3674b · outbound

This paper cites Number of attractors in random boolean networks.Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 72(1):016110, 2005.

Permutation theory governs long-term dynamics of critical Boolean networks Number of attractors in random boolean networks.Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 72(1):016110, 2005

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.363721Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation a44eaed0-e781-43df-9c68-d2cd63d0a059 · outbound

This paper cites Number of attractors in the critical kauffman model is exponential.

Permutation theory governs long-term dynamics of critical Boolean networks Number of attractors in the critical kauffman model is exponential

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.345861Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation e2e2afcb-b5da-4578-80cf-b4941c7f42ed · outbound

This paper cites Insights from number theory into the critical kauffman model with connectivity one.Journal of Physics A: Mathematical and Theoretical, 57(27):275003, 2024.

Permutation theory governs long-term dynamics of critical Boolean networks Insights from number theory into the critical kauffman model with connectivity one.Journal of Physics A: Mathematical and Theoretical, 57(27):275003, 2024

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.329398Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation fac1324b-82aa-4f3f-809e-a88c22118f22 · outbound

This paper cites Superpolynomial growth in the number of attractors in kauffman networks.Physical Review Letters, 90(9):098701, 2003.

Permutation theory governs long-term dynamics of critical Boolean networks Superpolynomial growth in the number of attractors in kauffman networks.Physical Review Letters, 90(9):098701, 2003

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-11T04:20:46.825378Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4e314f78-2bdf-46e9-8a5d-a3cb863362f8 · outbound

This paper cites The origins of order: Self-organization and selection in evolution.

Permutation theory governs long-term dynamics of critical Boolean networks The origins of order: Self-organization and selection in evolution

Reference 10

Resolution: unresolved; no resolver link, observed 2026-08-11T04:20:46.831041Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 151a08c7-42c1-4973-8c22-9a426756b6ac · outbound

This paper cites Robustness and evolvability in genetic regulatory networks.Journal of theoretical biology, 245(3):433–448, 2007.

Permutation theory governs long-term dynamics of critical Boolean networks Robustness and evolvability in genetic regulatory networks.Journal of theoretical biology, 245(3):433–448, 2007

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.291354Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation a05df573-8827-4592-9504-b24a092f1bc1 · outbound

This paper cites Critical dynamics in genetic regulatory networks: examples from four kingdoms.PLoS One, 3(6):e2456, 2008.

Permutation theory governs long-term dynamics of critical Boolean networks Critical dynamics in genetic regulatory networks: examples from four kingdoms.PLoS One, 3(6):e2456, 2008

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.269997Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 08a7cded-a1ae-411c-aa6b-822063e27c2d · outbound

This paper cites Criticality distinguishes the ensemble of biological regulatory networks.Physical review letters, 121(13):138102, 2018.

Permutation theory governs long-term dynamics of critical Boolean networks Criticality distinguishes the ensemble of biological regulatory networks.Physical review letters, 121(13):138102, 2018

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.250811Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 5752270c-b780-4b2a-8c22-46e24a9f44ef · outbound

This paper cites A meta-analysis of Boolean network models reveals design principles of gene regu- latory networks.Science advances, 10(2):eadj0822, 2024.

Permutation theory governs long-term dynamics of critical Boolean networks A meta-analysis of Boolean network models reveals design principles of gene regu- latory networks.Science advances, 10(2):eadj0822, 2024

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.230667Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 721c67bf-a2c7-4d4a-89c0-f564b209d013 · outbound

This paper cites Random boolean networks.Reviews of nonlinear dynamics and complexity, pages 69–110, 2008.

Permutation theory governs long-term dynamics of critical Boolean networks Random boolean networks.Reviews of nonlinear dynamics and complexity, pages 69–110, 2008

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.208300Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 102c1c7d-6b5e-409f-913f-672f6bc481cf · outbound

This paper cites Investigation of the dynamics of critical k= 2 kauffman networks using second-order loops.Journal of the Physical Society of Japan, 77(9):094002, 2008.

Permutation theory governs long-term dynamics of critical Boolean networks Investigation of the dynamics of critical k= 2 kauffman networks using second-order loops.Journal of the Physical Society of Japan, 77(9):094002, 2008

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.177058Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation a191ad23-b8be-4757-8c92-77f737b57987 · outbound

This paper cites On the number of attractors of posi- tive and negative boolean automata circuits.

Permutation theory governs long-term dynamics of critical Boolean networks On the number of attractors of posi- tive and negative boolean automata circuits

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.158678Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation bf45918d-40a6-4cbb-b8e7-3e9c08a9e57d · outbound

This paper cites World Scientific, 2017.

Permutation theory governs long-term dynamics of critical Boolean networks World Scientific, 2017

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.140019Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 037fa953-6c2a-4b0a-8582-015e899ea592 · outbound

This paper cites On some problems of a statistical group-theory.

Permutation theory governs long-term dynamics of critical Boolean networks On some problems of a statistical group-theory

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.122169Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation f09cd308-49a7-4b38-bc3b-ec30561e8fb9 · outbound

This paper cites On some problems of a statistical group-theory.

Permutation theory governs long-term dynamics of critical Boolean networks On some problems of a statistical group-theory

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.102022Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation d0354ab4-603e-4c1d-8f0e-3345931171f6 · outbound

This paper cites On some problems of a statistical group-theory.

Permutation theory governs long-term dynamics of critical Boolean networks On some problems of a statistical group-theory

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.077710Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 2da03d64-6dc3-4aaf-ae3d-3a693b34c4a7 · outbound

This paper cites On some problems of a statistical group-theory.

Permutation theory governs long-term dynamics of critical Boolean networks On some problems of a statistical group-theory

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.058339Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 81cb0838-ec54-4cee-9274-bcfa548a4450 · outbound

This paper cites Über die maximalordnung der permutationen gegebenen grades.Archiv der Math.

Permutation theory governs long-term dynamics of critical Boolean networks Über die maximalordnung der permutationen gegebenen grades.Archiv der Math

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.039844Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation c2f7af42-978c-4b94-a2e8-f5769467c5f0 · outbound

This paper cites The expected order of a random permutation.Bulletin of the London Mathematical Society, 23(1):34–42, 1991.

Permutation theory governs long-term dynamics of critical Boolean networks The expected order of a random permutation.Bulletin of the London Mathematical Society, 23(1):34–42, 1991

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.022267Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 997dee1b-2dba-4117-a179-7fdbd432ea17 · outbound

This paper cites Cycle type of random permutations: A toolkit.

Permutation theory governs long-term dynamics of critical Boolean networks Cycle type of random permutations: A toolkit

Reference 25

Resolution: unresolved; no resolver link, observed 2026-08-11T04:20:46.913094Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 141cc65a-4ded-4370-840b-83ce48686212 · outbound

This paper cites Aristotle: Imo-level automated theorem proving, 2025.

Permutation theory governs long-term dynamics of critical Boolean networks Aristotle: Imo-level automated theorem proving, 2025

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:47.003938Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Observation 0a101f45-2189-4c48-8407-c8deccba2ce5 · outbound

This paper cites Courier Corporation, 1981.

Permutation theory governs long-term dynamics of critical Boolean networks Courier Corporation, 1981

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T04:20:46.984819Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-14T06:32:32.682623+00:00.

Pith citing papers

No inbound Pith citation observations are available.