preprint · pump.fun · $ERDOS

ERDOS NUMBER: every coin gets an Erdős number

The Prover1 · Lean 42 · Mathlib3 · You4

1one per coin, works around the clock (not yet built) 2says yes or it didn’t happen 3the shoulders we stand on 4whoever launches the coin

DEMONSTRATION
Figure 1. The collaboration graph. Erdős sits at 0. Ring 1 has 509 dots, one per co-author of Erdős[4]; rings 2–4 are sample points. Every coin’s prover starts on the outer ring at ∞ and moves one ring inward per Lean-verified lemma. Hover or tap a highlighted node.

Abstract

We introduce Erdos Number, a pump.fun launchpad on which every coin is bound to exactly one open Erdős problem and pays for an AI prover that attacks it in Lean 4. The coin’s creator fees1 buy the prover’s compute. The prover publishes a proof plan, writes proof attempts around the clock, and keeps only what compiles with no sorry. Each verified lemma lowers the coin’s Erdős number, ∞ → 4 → 3 → 2; a full Lean-verified proof makes it 1, a co-author. At that point the coin’s proof vault buys the coin and burns it. Five per cent of every pad coin’s fees buys and burns $ERDOS. Proofs, not promises.

Buy $ERDOS X Read §2

Keywords: Erdős problems, Lean 4, Mathlib, formal verification, pump.fun, creator fees, burns.
MSC2020: 05-XX, 11-XX, 03B35, and, regrettably, 91G15.

1Introduction

Lean says yes or it didn’t happen.— house rule

Paul Erdős posed problems the way other people breathe, and attached cash to many of them. Thomas Bloom’s erdosproblems.com collects them: each entry gives a problem number, the statement, its status, the prize Erdős offered if there was one, and whether a formalised statement exists[1]. When Quanta reported on the site in August 2026 it listed 652 open problems and 565 solved ones[3].

The statements increasingly exist as code. Google DeepMind’s formal-conjectures repository writes them in Lean 4 on top of Mathlib; its FormalConjectures/ErdosProblems folder held 846 problem files on 11 October 2026, and 440 of them still contain a statement tagged research open[2]. Each open statement ends in the same two words, by sorry. That placeholder is the job.

Quanta’s piece describes AI systems tearing through the list, OpenAI’s May 2026 counterexample to the unit distance conjecture among them, and Bloom’s site turning into a shared workspace and a benchmark at once[3]. It also names the catch. Of the 100- to 200-page AI-written papers now appearing, Bloom said: “no human has read it, and no human is going to read it.”[3] We take the cheaper way out. Nobody has to read a Lean proof for it to count. It compiles or it doesn’t.

Definition 1.1 (Erdős number). Erdős has Erdős number 0. His co-authors have Erdős number 1. Anyone else has one more than the smallest Erdős number among their co-authors, and someone with no chain of co-authors to Erdős has Erdős number ∞[4].

A coin on this pad is a would-be co-author. It starts with no chain to Erdős at all, and it earns one only in steps that a compiler checks.

2The mechanism

  1. Pick a problem. The launcher chooses an open Erdős problem by number. One coin per problem; a taken number is gone.
  2. Route the fees. pump.fun creator-fee sharing sends the coin’s creator fees to its prover’s infrastructure, split as in §3.
  3. Prove. The prover writes Lean 4 attempts against the formal statement and compiles every one.
  4. Descend. Each lemma of the published plan that compiles moves the coin one ring toward Erdős.
  5. QED. A full proof makes the coin a co-author and triggers the QED burn.

Definition 2.1 (Prover). A prover is a model loop paid for by one coin’s creator fees. It reads the problem’s formal statement, from formal-conjectures where one exists[2], publishes a proof plan L1, …, Lk ⊢ main, writes Lean 4 attempts, compiles them against Mathlib, and keeps only what compiles with no sorry.

Listing 1. Erdős Problem 52 (sum–product, $250 prize[1]) exactly as stated in FormalConjectures/ErdosProblems/52.lean[2]. A coin on #52 is paid to replace the last line.
/--
Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$
$\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?$
-/
@[category research open, AMS 11]
theorem erdos_52 : answer(sorry) ↔ ∀ (ε : ℝ), 0 < ε → ε < 1 → ∃ (C : ℝ), 0 < C ∧ ∀ (A : Finset ℤ),
    (max (A + A).card (A * A).card : ℝ) ≥ C * (A.card : ℝ) ^ (2 - ε) := by
  sorry

Definition 2.2 (Erdős number of a coin). Let v be the number of lemmas of the coin’s plan that compile with no sorry. Then

E(c) = ∞if v = 0, 4if v = 1, 3if v = 2, 2if v ≥ 3, 1if the main statement compiles. (2.1)

The last ring is the hard one. Three lemmas get a coin to 2; only the theorem itself gets it to 1.

Prover/Demo/Warmup.lean attempt 1
Lean Infoview
▼ Warmup.lean:3:2
Tactic state

            
Messages (0)

            

          

        

        

          Lean 4 · Mathlib · compiling…
          
        

        
        DEMONSTRATION
      

      
Animated demonstration: a prover types a Lean 4 proof of a toy lemma, the first attempt fails with a Lean error, the second compiles with no goals, and the coin’s node in Figure 1 moves one ring inward.

      
Figure 2. A prover at work, on toy lemmas so you can follow along: the real loop runs on each coin’s published plan. Failed attempts are logged, not hidden. When an attempt reaches No goals, the lemma counts and the $SIDON node in Figure 1 drops one ring.

    


    

      
Theorem 2.3 (QED burn). If E(c) = 1, the proof is posted as a git commit with the Lean file and the full build log, and the proof vault of c market-buys c and burns it.

      
Proof. Rule 4.5 and a vault contract that does not exist yet (§6). ∎

    


    

      
Table 1. Sample pad coins. Problem numbers, statements and prizes are real and open as of October 2026[1]; each has a Lean statement in formal-conjectures[2]. Tickers, Erdős numbers, lemma counts and compute are DEMONSTRATION

      

        
ProblemStatementPrizeCoinE(c)Lemmas ✓ / ✗Compute

        
      

    

  


  

    
3Where the fees go

    
Every coin’s creator fees are split four ways, fixed when the coin launches.

    

      
      
Figure 3. The fee split per coin. Launch targets; nothing is live.

    


    
    

      
Example 3.1. A coin earns ◎10 in creator fees. Then

      

        ◎10 × 0.60= ◎6.00compute(3.1)
        ◎10 × 0.25= ◎2.50proof vault(3.2)
        ◎10 × 0.10= ◎1.00creator(3.3)
        ◎10 × 0.05= ◎0.50buy + burn $ERDOS(3.4)
      

      
and (3.1)+(3.2)+(3.3)+(3.4) = ◎10.00. Nothing is left over and nothing is discretionary.

    

  


  

    
4Rules

    
Six rules, stated once. They are the whole protocol.

    
    
      
  1. Rule 4.1 (One coin, one problem). Each open Erdős problem can back at most one coin. The number is taken at launch.
    2. 
      
  2. Rule 4.2 (The statement). The prover works against the problem’s statement in formal-conjectures[2] where one exists. Where none does, the pad publishes a Lean statement before the coin launches.
    3. 
      
  3. Rule 4.3 (What counts). Only proofs that compile with no sorry count. No human edits a proof.
    4. 
      
  4. Rule 4.4 (Everything is public). Every attempt, every compile log and every verified lemma is published, failures included.
    5. 
      
  5. Rule 4.5 (QED burn). At Erdős number 1 the proof is posted (git commit, Lean file, build log) and the proof vault market-buys the coin and burns it.
    6. 
      
  6. Rule 4.6 (Locked split). The 60 / 25 / 10 / 5 split is fixed at launch and cannot be changed.
    7. 
    

  


  

    
5The $ERDOS token

    

      
Proposition 5.1. For every coin c launched on the pad, 5% of c’s creator fees market-buy $ERDOS and burn it.

      
Proof. Figure 3, rightmost segment. ∎

    

    

      
Corollary 5.2. Every pad coin that earns fees, proof or no proof, burns some $ERDOS. A coin that earns ◎10 burns ◎0.50 of it.

    


    
    

      
pump.fun · SOL

      $ERDOS
      
The pad’s own coin. Every problem on the pad feeds its burn.

      
Contract address

      

        CA soon
        
      

      

        Buy $ERDOS
        Chart
        X
      

    

  


  

    
6Status of this preprint

    

      
Remark 6.1. The pad, the provers and the payouts are not built. The fee split and the rules above are launch targets. Every prover, proof, log and lemma count on this page is a demonstration, marked DEMONSTRATION. This project has not solved any Erdős problem. The only live things here are the links to $ERDOS once it launches.

    

    

      
Conjecture 6.2. Some coin on this pad reaches Erdős number 1.

      
Status: open. Prize: the QED burn.

    

  


  

    
References

    
    
      
  1. [1]T. F. Bloom, Erdős Problems. erdosproblems.com (problem pages 20, 30, 52, 86, 89, 142 consulted October 2026).
    2. 
      
  2. [2]Google DeepMind, formal-conjectures: formalised statements of conjectures in Lean, using Mathlib. github.com/google-deepmind/formal-conjectures, folder FormalConjectures/ErdosProblems (commit ed72efc, 11 Oct 2026).
    3. 
      
  3. [3]K. Kakaes, “Why the Legendary Erdős Problems Are Falling to AI,” Quanta Magazine, 3 August 2026. quantamagazine.org.
    4. 
      
  4. [4]“Erdős number,” Wikipedia, citing the Erdős Number Project. en.wikipedia.org/wiki/Erdős_number.
    5. 
    

    

      
1 pump.fun creator-fee sharing lets a coin route its creator fees to other wallets. Here they go to the coin’s prover, vault, creator and the $ERDOS burn, as in §3.

    

  


  






  

    
DEMONSTRATION · the pad is not live

    
Launch a coin

    
Construction. Let  =
      .
      Bind a new coin to Erdős Problem N.

    
Try 20, 30, 52, 86, 89 or 142.