Bellpair: Arithmetic Structure for Verifiable Work

Finite structure at arbitrary scale

A prime’s digit-collision count runs over an entire residue system. Once the prime exceeds the chosen base, the count’s deviation from a known scale term is determined exactly by its final two digits in that base. The prime can grow; the fingerprint table stays fixed.

Reflection pairs those fingerprints with exactly opposite centered weights. Bellpair turns that relation into a completion rule: two independently found prime contributions must fill matching roles in a fresh, request-bound receipt. The paper explains the mathematical reduction before specifying the protocol that uses it.

Computation before admission

A service can spend substantial compute answering a request that was cheap to submit. Bellpair’s first proposed application puts a work exchange at that boundary: issue a challenge, check the requester’s receipt, consume it once, then admit the expensive action.

The local C/GMP prototype already runs the exchange across bases 2–36. The paper follows a real receipt through search, independent verification and one-use acceptance, with measured costs and the exact challenge bytes needed to reproduce the check.

A family of arithmetic contracts

The underlying collision theorem extends to longer digit shifts, each with its own finite residue address. That gives a concrete research direction beyond the first pair: completion rules built from several compatible arithmetic observations. The paper connects this wider structure to what the prototype implements today.

From receipt to service

The exact fingerprint and reflection laws are established mathematics; the receipts and recorded measurements come from the working implementation. A practical advantage over ordinary hash puzzles remains to be demonstrated. The next application test is a protected endpoint with a separate requester-side worker and persistent redemption, compared at matched solving cost.

Receipts and measurements