Paper Citation Record · LEDGER

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism

As of 11 August 2026, this Paper Citation Record lists 21 of 21 outbound references and 0 inbound Pith citation observations for arXiv:2606.26811.

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

pith.paper-citation-record.v1
2606.26811 v1

Coverage vector

measured 21 of 21 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-06-26T02:03:16.507503Z

measured 21 of 21 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-11T06:34:44.6726+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

21 of 21 outbound references displayed

verified exact0
verified fuzzy0
unresolved20
parse uncertain0
malformed identifier1
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation dc73bbc8-e94c-408c-9a3d-0ddf1a026df9 · outbound

This paper cites 100 102 104 106 108 1010 1012 1014 N(N.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism 100 102 104 106 108 1010 1012 1014 N(N

Reference 1

Resolution: malformed identifier; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d857ed07-21f0-496c-ae16-414ad1373904 · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 2

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f2b39b1c-dd0f-42f9-9119-868e6ff537fa · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 3

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 59410c78-c4c3-406f-84cd-8204421af5cd · outbound

This paper cites Krasikov, How many numbers satisfy the 3x + 1 conjecture?, Internat.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Krasikov, How many numbers satisfy the 3x + 1 conjecture?, Internat

Reference 4

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a4c07acf-59f7-45bc-9a76-b512aa9cd9cd · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 5

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 935d8172-1bcc-4670-8eca-39574637556f · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 6

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 5890e122-c9f0-4da8-ac42-7aa611be70ac · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 7

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 89e3ca45-a18a-402d-bf13-f9ebd018aeea · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 8

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9f33c752-28e2-4372-924f-9a1fecbd4959 · outbound

This paper cites Akin, Why is the 3x+ 1 problem hard?, Contemporary Math- ematics356, 1 (2004).

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Akin, Why is the 3x+ 1 problem hard?, Contemporary Math- ematics356, 1 (2004)

Reference 9

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e02f60d1-6d8b-432c-b6d3-a955a637fcb4 · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 10

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 12937aae-ecbb-4a53-a6c0-39ae044903c9 · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 11

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 12a6a6bc-e61f-42ef-979d-aae9df496708 · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 12

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9a638c2d-0d08-4b42-86ac-fcae26c1117c · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 13

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 1690aa95-e47b-4a2d-b578-93a2943fe44f · outbound

This paper cites Terras, A stopping time problem on the positive integers, Acta Arithmetica30, 241 (1976).

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Terras, A stopping time problem on the positive integers, Acta Arithmetica30, 241 (1976)

Reference 14

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 71557634-89d7-4bae-acbc-70fee438229b · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 15

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a95b524e-5ddd-41af-b334-fc65b1b8840f · outbound

This paper cites Tao, Almost all orbits of the collatz map attain almost bounded values, Forum of Mathematics, Pi10, e12 (2022).

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Tao, Almost all orbits of the collatz map attain almost bounded values, Forum of Mathematics, Pi10, e12 (2022)

Reference 16

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8c9f4397-8731-4ed0-8e43-4a83de2eac50 · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 17

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 916af3ea-522b-4ccc-8543-3e990e6eddf4 · outbound

This paper cites Barina, Convergence verification of the collatz problem, J.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Barina, Convergence verification of the collatz problem, J

Reference 18

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9a533acd-f2f1-45af-85ec-c00c5b57ccca · outbound

This paper cites Barina, Improved verification limit for the convergence of the collatz conjecture, J.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Barina, Improved verification limit for the convergence of the collatz conjecture, J

Reference 19

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 12166500-1aa1-412b-b7c5-b5c1d70004d9 · outbound

This paper cites Fu and Y.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Fu and Y

Reference 20

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 37d4fbdb-b5ba-4431-8f2b-588c086723bb · outbound

This paper cites an unresolved cited work.

Emergence of Gamma-Type Upward-Phase Statistics in the Collatz Map: An Effective Poisson Process Mechanism Unresolved cited work

Reference 21

Resolution: unresolved; no resolver link, observed 2026-06-26T02:03:16.507503Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

No inbound Pith citation observations are available.