Paper Citation Record · LEDGER

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling

As of 5 August 2026, this Paper Citation Record lists 43 of 43 outbound references and 0 inbound Pith citation observations for arXiv:2606.13473.

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

pith.paper-citation-record.v1
2606.13473 v1

Coverage vector

measured 43 of 43 reference resolution

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Source: paper_references, paper_reference_links, observed 2026-06-27T07:37:18.384063Z

measured 43 of 43 standing notices

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Source: scholarly_work_events, retraction_status_cache, observed 2026-08-05T06:32:48.257954+00:00

measured 0 of 0 inbound itemization

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measured 0 of 1 external citation measurements

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Reference resolution

43 of 43 outbound references displayed

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Outbound references

Observation 4454efc7-7307-4ec0-8924-0d142a17a8c7 · outbound

This paper cites an unresolved cited work.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Unresolved cited work

Reference 1

Resolution: unresolved; no resolver link, observed 2026-06-27T07:37:18.384063Z

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Observation 0ee2a538-39db-4f16-81ba-7ef0aea5b082 · outbound

This paper cites an unresolved cited work.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Unresolved cited work

Reference 2

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Observation e032203e-8e1d-40c6-b4a3-65529bed80ca · outbound

This paper cites Base case𝑛=3.𝑆 3 ={(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)}.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Base case𝑛=3.𝑆 3 ={(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)}

Reference 3

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Observation 46ddb72e-bd5a-44fa-8134-e26f4f0621e6 · outbound

This paper cites Therefore the set of admissible𝑘is {0,1,3}.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Therefore the set of admissible𝑘is {0,1,3}

Reference 4

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Observation b9c4eba7-25f0-4821-9eeb-47bcf71589d5 · outbound

This paper cites Place𝑀=(0,0), 𝑁=(1,0)and let the radii be𝑟 < 𝑅.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Place𝑀=(0,0), 𝑁=(1,0)and let the radii be𝑟 < 𝑅

Reference 5

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Observation bdee7bb8-7009-4a97-9da3-947fe38673f8 · outbound

This paper cites Write𝑃=(𝑥 𝑃, 𝑦𝑃)with𝑦 𝑃 unknown.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Write𝑃=(𝑥 𝑃, 𝑦𝑃)with𝑦 𝑃 unknown

Reference 6

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Observation 8a428281-6623-46f2-8ee5-f8bcf6ce94d4 · outbound

This paper cites For a circle with centre𝑂and a point𝐴on it, the second intersection of the line𝐴+𝑡vwith the circle is obtained from|𝐴+𝑡v−𝑂| 2 =𝑅 2.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling For a circle with centre𝑂and a point𝐴on it, the second intersection of the line𝐴+𝑡vwith the circle is obtained from|𝐴+𝑡v−𝑂| 2 =𝑅 2

Reference 7

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Observation c1e4e14d-982c-48bc-96c0-1211a5940370 · outbound

This paper cites Inthecoordinate system with origin at𝐴, axes alongu(the𝑢-axis) andw(the𝑤-axis), we have 𝐴=(0,0), 𝐸=(𝑒,0), 𝐹=(𝑓 ,0), 𝐵=(𝑏, ℎ), where 𝑏=(𝐵−𝐴) ·u, ℎ=(𝐵−𝐴) ·w.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Inthecoordinate system with origin at𝐴, axes alongu(the𝑢-axis) andw(the𝑤-axis), we have 𝐴=(0,0), 𝐸=(𝑒,0), 𝐹=(𝑓 ,0), 𝐵=(𝑏, ℎ), where 𝑏=(𝐵−𝐴) ·u, ℎ=(𝐵−𝐴) ·w

Reference 8

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Observation 29dda361-f324-4fda-a2aa-bd30ba17eee8 · outbound

This paper cites The altitude from𝑀to𝑃𝑁has slope− 𝑥𝑃 −1 𝑦𝑃 and equation𝑦=− 𝑥𝑃 −1 𝑦𝑃 𝑥.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling The altitude from𝑀to𝑃𝑁has slope− 𝑥𝑃 −1 𝑦𝑃 and equation𝑦=− 𝑥𝑃 −1 𝑦𝑃 𝑥

Reference 9

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Observation 389b346f-18a4-4d0b-979f-e3e969fc85f7 · outbound

This paper cites Hence the circum- centre𝑂has coordinates(𝑆, 𝑘)for some𝑘.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Hence the circum- centre𝑂has coordinates(𝑆, 𝑘)for some𝑘

Reference 10

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Observation 87870cfb-5f49-45bf-a278-1aa024ee8300 · outbound

This paper cites Factorising the numerator as a difference of squares, 4𝑟2 − (1+𝑟 2 −𝑅 2)2 = 2𝑟− (1+𝑟 2 −𝑅 2) 2𝑟+ (1+𝑟 2 −𝑅 2).

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Factorising the numerator as a difference of squares, 4𝑟2 − (1+𝑟 2 −𝑅 2)2 = 2𝑟− (1+𝑟 2 −𝑅 2) 2𝑟+ (1+𝑟 2 −𝑅 2)

Reference 11

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Observation 163336dd-f3cf-4a1d-8abb-fe2c816ea3f9 · outbound

This paper cites The distance from the centre𝑂=(𝑆, 𝑘)to this line is|𝑘−𝐻 𝑤 |.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling The distance from the centre𝑂=(𝑆, 𝑘)to this line is|𝑘−𝐻 𝑤 |

Reference 12

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Observation b2dcdfb7-82c3-4c23-a695-f7d9894e8fcc · outbound

This paper cites - Taking𝑎=𝑏=1gives𝑓(1) |1−𝑓(1) 𝑓(1).

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling - Taking𝑎=𝑏=1gives𝑓(1) |1−𝑓(1) 𝑓(1)

Reference 13

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Observation 740ba3ba-32bc-4e63-907b-e2fa8a68624d · outbound

This paper cites As𝑝 𝑝 is a prime power, 𝑓(𝑝)=𝑝 𝑘 with0≤𝑘≤𝑝.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling As𝑝 𝑝 is a prime power, 𝑓(𝑝)=𝑝 𝑘 with0≤𝑘≤𝑝

Reference 14

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Observation 01d673a2-dcc7-4338-b594-3382f2af149f · outbound

This paper cites Substituting𝑎=𝑝into the bonza condition gives 𝑝𝑘 |𝑏 𝑝 −𝑓(𝑏) 𝑝𝑘 for all𝑏∈ℕ.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Substituting𝑎=𝑝into the bonza condition gives 𝑝𝑘 |𝑏 𝑝 −𝑓(𝑏) 𝑝𝑘 for all𝑏∈ℕ

Reference 15

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Observation b4c486a4-5194-4dbc-af58-c717cdbc4580 · outbound

This paper cites Therefore𝑓is the identity function.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Therefore𝑓is the identity function

Reference 16

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Observation 802d3055-c767-4cfe-bb90-f2d73bfee4d9 · outbound

This paper cites Fix an arbitrary𝑛∈ℕ.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Fix an arbitrary𝑛∈ℕ

Reference 17

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Observation b69b4859-71a0-4e12-bcb6-927bd9876297 · outbound

This paper cites - If𝑛is even, write𝑛=2 𝑗𝑚with𝑗≥1,𝑚odd; then𝑓(𝑛) ≤2 𝑗+2 ≤4𝑛.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling - If𝑛is even, write𝑛=2 𝑗𝑚with𝑗≥1,𝑚odd; then𝑓(𝑛) ≤2 𝑗+2 ≤4𝑛

Reference 18

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Observation 0e13ed0f-dc02-4bb0-ac33-0f07a0d5c7e2 · outbound

This paper cites (Equivalently, for even𝑛≠2, write𝑛=2 𝑗𝑚with𝑗≥1,𝑚odd, and set𝑓(𝑛)=2 𝑗+2.) We verify that this𝑓is bonza.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling (Equivalently, for even𝑛≠2, write𝑛=2 𝑗𝑚with𝑗≥1,𝑚odd, and set𝑓(𝑛)=2 𝑗+2.) We verify that this𝑓is bonza

Reference 19

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Observation 6a7f4f27-2a9a-43a9-aebf-7f27c3b89888 · outbound

This paper cites Therefore the smallest such real constant is 4.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Therefore the smallest such real constant is 4

Reference 20

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Observation 024c2b44-29f0-4618-be9d-0ca7bdec072c · outbound

This paper cites Thus 𝑒𝑛+1 =𝑒 𝑛 −2, 𝑓 𝑛+1 =𝑓 𝑛 −1, 𝑟 𝑛+1 =13𝑟 𝑛.(1).

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Thus 𝑒𝑛+1 =𝑒 𝑛 −2, 𝑓 𝑛+1 =𝑓 𝑛 −1, 𝑟 𝑛+1 =13𝑟 𝑛.(1)

Reference 21

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Observation 983d4065-4441-4a97-82bb-85ce1b90308e · outbound

This paper cites 47 MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling The factor2disappears, and the remaining product is odd.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling 47 MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling The factor2disappears, and the remaining product is odd

Reference 22

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Observation f39744ee-aba7-4da9-93d5-63b28f2ce600 · outbound

This paper cites Reaching a fixed point Suppose we start with a term𝑎 1 for which the sequence is infinite.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Reaching a fixed point Suppose we start with a term𝑎 1 for which the sequence is infinite

Reference 23

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Observation 7cfe2c4a-fb48-48cc-886f-eaa3f9c6e1c4 · outbound

This paper cites an unresolved cited work.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Unresolved cited work

Reference 24

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Observation 20264eb4-a716-4edd-a155-6ce31b1ce1d0 · outbound

This paper cites Thus 𝑎≤𝜆 < √ 2 2 < √ 2, so𝑎 2 <2and Bazza’s move𝑏𝑘+1 = √ 2−𝑎 2 is well defined.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Thus 𝑎≤𝜆 < √ 2 2 < √ 2, so𝑎 2 <2and Bazza’s move𝑏𝑘+1 = √ 2−𝑎 2 is well defined

Reference 25

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Observation d3696082-004f-40c6-9505-89ea8efc2b68 · outbound

This paper cites Now consider Alice’s next turn, number2(𝑘+1) +1=2𝑘+3.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Now consider Alice’s next turn, number2(𝑘+1) +1=2𝑘+3

Reference 26

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Observation d0315f51-5f93-4138-9034-fe046543ac37 · outbound

This paper cites Consequently Alice is always forced to choose𝑎≤ √ 2 2.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Consequently Alice is always forced to choose𝑎≤ √ 2 2

Reference 27

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Observation 7d69bbd1-2853-4055-b64c-44059f51c06c · outbound

This paper cites an unresolved cited work.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Unresolved cited work

Reference 28

Resolution: unresolved; no resolver link, observed 2026-06-27T07:37:18.384063Z

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Observation 483d5df6-878f-4795-821a-e65f86804816 · outbound

This paper cites Let𝑁, the midpoint of the minor arc𝐵𝐶, be(−1,0); the tangent at𝑁is then the line𝑥=−1.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Let𝑁, the midpoint of the minor arc𝐵𝐶, be(−1,0); the tangent at𝑁is then the line𝑥=−1

Reference 29

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Observation 34ded9b2-b521-4791-8c99-a149ff0cb170 · outbound

This paper cites * The equilateral triangleΔ𝐵 has its centre at the origin, so the midpoint of the two vertices different from 𝐵is− 𝐵.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling * The equilateral triangleΔ𝐵 has its centre at the origin, so the midpoint of the two vertices different from 𝐵is− 𝐵

Reference 30

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Observation 937a7ec2-cd49-4ffd-aad8-d5d4e6e601a5 · outbound

This paper cites ℓ𝐵 :𝑥cos𝜃+𝑦sin𝜃= 1 2.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling ℓ𝐵 :𝑥cos𝜃+𝑦sin𝜃= 1 2

Reference 31

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Observation 5e0956b9-93f2-4219-b6da-a74daa99fd76 · outbound

This paper cites Line𝐴𝐶meetsℓ 𝐵 at𝑌.Write𝑌=𝐴+𝑢(𝐶−𝐴).

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Line𝐴𝐶meetsℓ 𝐵 at𝑌.Write𝑌=𝐴+𝑢(𝐶−𝐴)

Reference 32

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Observation 8f813a66-ad44-46bc-86c6-0c4d4673426d · outbound

This paper cites Because𝐴lies on the unit circle,𝑎 2 +𝑏 2 =1, so 𝐹=−1−𝐷𝑎−𝐸𝑏,where𝑎=cos𝛼, 𝑏=sin𝛼.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Because𝐴lies on the unit circle,𝑎 2 +𝑏 2 =1, so 𝐹=−1−𝐷𝑎−𝐸𝑏,where𝑎=cos𝛼, 𝑏=sin𝛼

Reference 33

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Observation 077e18c1-8187-49da-ad76-4593d591771c · outbound

This paper cites Substituting𝐷, 𝐸 and simplifying yields 𝑂= − 𝑀(1−2 cos𝛼cos𝜃) cos𝛼+cos 3𝜃 , 2𝑀sin𝛼cos𝜃 cos𝛼+cos 3𝜃.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Substituting𝐷, 𝐸 and simplifying yields 𝑂= − 𝑀(1−2 cos𝛼cos𝜃) cos𝛼+cos 3𝜃 , 2𝑀sin𝛼cos𝜃 cos𝛼+cos 3𝜃

Reference 34

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Observation d4402d7e-2c44-451f-a7fe-bdc54f4caeef · outbound

This paper cites By symmetry its incenter lies on the𝑥-axis.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling By symmetry its incenter lies on the𝑥-axis

Reference 35

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Observation 7f6dd1e0-49b0-4ca8-89f1-d100de5dd0c6 · outbound

This paper cites an unresolved cited work.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Unresolved cited work

Reference 36

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Observation 8180b455-890b-436b-8bff-cc8e574c2dad · outbound

This paper cites Set𝑆=1+cos𝜃.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Set𝑆=1+cos𝜃

Reference 37

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Observation 2a893cf7-49ef-4b56-95fa-155abe6654b4 · outbound

This paper cites 62 MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Verification of (ii): From (i),𝑀2𝑃=2𝑁 𝑀 2, so 𝑀2𝑃−𝑁 2 =𝑁(2𝑀 2 −𝑁).

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling 62 MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Verification of (ii): From (i),𝑀2𝑃=2𝑁 𝑀 2, so 𝑀2𝑃−𝑁 2 =𝑁(2𝑀 2 −𝑁)

Reference 38

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Observation 9f694f54-9769-401f-9b2b-faa982e9fe0c · outbound

This paper cites Let𝑐𝑖 be the carry into the𝑖-th digit (𝑐0 =0), and let𝑛𝑖 be the𝑖-th digit of𝑛(𝑛0 units).

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Let𝑐𝑖 be the carry into the𝑖-th digit (𝑐0 =0), and let𝑛𝑖 be the𝑖-th digit of𝑛(𝑛0 units)

Reference 39

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Unavailable: canonical work link unavailable.

Observation e121e480-a77f-4bec-b279-a3892d679558 · outbound

This paper cites *𝑐 𝑖 =0:then𝑥=𝑛 𝑖 +10𝑐 𝑖+1.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling *𝑐 𝑖 =0:then𝑥=𝑛 𝑖 +10𝑐 𝑖+1

Reference 40

Resolution: unresolved; no resolver link, observed 2026-06-27T07:37:18.384063Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 125dbd31-33a9-4c16-b33d-6ca741c2b5dc · outbound

This paper cites The transitions are: *From𝐴:1↦→𝐵,9↦→𝐴,all other digits↦→𝐶.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling The transitions are: *From𝐴:1↦→𝐵,9↦→𝐴,all other digits↦→𝐶

Reference 41

Resolution: unresolved; no resolver link, observed 2026-06-27T07:37:18.384063Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 10e28d8d-c3e4-45de-9297-7de37774bde2 · outbound

This paper cites The accepting states are those containing0, i.e.𝐴and𝐶.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling The accepting states are those containing0, i.e.𝐴and𝐶

Reference 42

Resolution: unresolved; no resolver link, observed 2026-06-27T07:37:18.384063Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation fa7a0bd4-08aa-41aa-a4e8-04de329d93cf · outbound

This paper cites Hencethenumberofsolitarynumberswiththe1atposition𝑡is2 𝐿−𝑡−1.

MaxProof: Scaling Mathematical Proof with Generative-Verifier RL and Population-Level Test-Time Scaling Hencethenumberofsolitarynumberswiththe1atposition𝑡is2 𝐿−𝑡−1

Reference 43

Resolution: unresolved; no resolver link, observed 2026-06-27T07:37:18.384063Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

No inbound Pith citation observations are available.