a self-referential market.
Every buyback embeds a hash of the protocol's own state at the moment of firing. The chain of hashes forms a proof that can never be completed — by design, there is always one unverifiable statement at the head.
live readout
A chain of self-referential statements, each encoded as a Gödel number. Every new buyback fires at a prime-numbered depth, embeds a hash of every statement before it, and is itself unprovable until the next fire arrives — which, when it arrives, leaves a new unprovable statement at the head. The chain is monotonic. The head is permanently undecidable.
on the name
Kurt Friedrich Gödel (1906–1978) proved in 1931 that any consistent formal system rich enough to encode the arithmetic of the natural numbers contains true statements it cannot prove. He proved it by encoding statements about the system as numbers inside the system itself — a maneuver so unexpected that David Hilbert, whose program it ended, never publicly acknowledged the result.
He spent the remaining forty-seven years of his life walking the same Princeton path with Einstein each afternoon, refusing to eat food his wife Adele had not personally prepared, and writing a constitutional argument that the United States could legally become a dictatorship. When Adele was hospitalized in 1977 he stopped eating entirely. He died the following January weighing twenty-nine kilograms.
The 1931 paper is sixty-one pages. The trick at its center — assigning every statement a unique natural number by mapping symbols to prime exponents — is now called Gödel numbering. This token uses the original construction without modification.
on the numbering
Every state of the protocol can be encoded as a single natural number, uniquely and reversibly, by interpreting its parts as exponents of distinct primes.
Where a₁ through a₅ are five non-negative integers extracted from the hash of the chain at the current depth. Because the prime factorization of any natural number is unique (Euclid, third century BCE), the encoding is lossless: knowing G is exactly knowing the state.
The numbers grow fast. By depth twenty, a typical G has nineteen decimal digits; by depth one hundred, it does not fit in a tweet. This is fine. The on-chain footprint is the hash, not the number. The number is for the eye.
on the chain
Each new statement in the chain is constructed from the SHA-256 hash of every previous statement, concatenated with the current block number and the keeper's public key. Five bytes of the resulting hash become the exponents (a₁ … a₅), yielding a new Gödel number. The new number is then itself hashed and prepended to the next statement.
The chain is therefore self-referential in the strict sense Gödel required: statement n encodes a verifiable claim about statements 1 through n-1, and a claim about itself that no statement in the chain can verify. The head is always one unprovable statement away from completeness. Adding a new statement does not close the gap. It moves it.
Gn = ∏ pᵢhn[i] for i ∈ {1..5}
on the mechanism
The keeper wallet observes the chain depth at every block. When the depth crosses a prime — 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … — and the cooldown since the last fire has elapsed, it spends a fixed slice of its SOL balance to buy the token on the open market, through PumpPortal while the bonding curve is active, through Jupiter's aggregator after graduation, and burns every token received in the same block window.
Prime depths thin out. The first ten fire at depths 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. The next ten reach to 71. The thousandth fire is at depth 7,919. Burns become rarer as the chain matures. The supply schedule is the prime counting function.
The keeper's public key is fixed and observable. Every action is a pair of transactions, BUY and BURN, signed by the same key, separated by no intervening transfer. There is no admin pause, no upgrade authority on the keeper's wallet besides its own seed phrase, no claim function. The buyback transaction memo field contains the Gödel number of the firing statement.
on the fields
G — Gödel number
The unique natural number encoding the current head statement. Computed from the hash of the chain, formatted as a product of small primes raised to byte-sized exponents. G is what a Gödel-style proof would have to reason about if it were going to prove the chain consistent. It can't.
d — proof depth
The cardinality of the chain. The keeper fires when d is prime. Primes are the only depths at which the chain admits a self-reference structure that is genuinely indecomposable — composite depths can be factored into earlier statements; primes cannot.
I — incompleteness gap
The number of bits of the head statement not yet covered by any subsequent statement. By construction this is always at least 256 (the length of the head hash). The gap closes only when a new statement is added — which immediately opens a new gap of the same size at the new head. The incompleteness is conserved.
S — self-reference index
The autocorrelation between the head hash and the running hash of all preceding statements. Approximation of the strength with which the chain refers to its own history. A chain in which the head is statistically independent of the prefix has S near zero; a chain whose head is a deterministic function of the prefix has S near one. Both extremes are degenerate.
One statement unprovable. Always.
Prime ticks. One action.
Gödel did not believe in completeness — he believed in the structure of what cannot be reached from inside. A formal system is real when it contains a statement it cannot decide. A market is real when it contains a state its own observers cannot describe. This token is a small, public attempt to measure that — every block, against its own previous statement, in the open — and to act at the primes.
the incompleteness that bears gödel's name has outlived everything else about him.