The research behind the technology
Finite rules.
Larger arithmetic structure.
The lab’s development work starts with exact mathematical objects. These papers give their definitions, results and proofs.
From the collision count to Bellpair.
A collision measures agreement between digit bins before and after multiplication. The finite table describes that count without enumerating every residue. Its reflection law supplies Bellpair’s complementary roles.
Implemented in Bellpair
The Collision Periodic Table
A collision count over a coprime residue system reduces to a scale term and a finite table entry. The final two digits select the entry; reflection determines its counterpart. This is the exact relation used by the current receipt.
Research preprint · DOI 10.5281/zenodo.21853017
A direction for further development
The Collision Invariant
Finite determination and reflection extend to longer digit shifts. Each shift has a precise arithmetic resolution. Bellpair currently uses the one-shift case; the wider hierarchy suggests additional kinds of completion rule.
Research preprint · DOI 10.5281/zenodo.21855821
The wider arithmetic program.
The same collision structure leads to questions about prime sampling, spectral response and growth across bases. These results provide mathematical context; the receipt verifier relies on the finite pairing law.
The Centered Collision Sum
Centering, character structure and convergence for a specified weighted prime observable. The result depends on that observable’s sampling and weights.
Research preprint · DOI 10.5281/zenodo.21852556
The Collision Transform
The passage from collision tables to character responses and Dirichlet L-functions, including the analytic conditions needed for zero-detection statements.
Research preprint · DOI 10.5281/zenodo.21855983
The Cubic Law for Digit-Collision Energy
The leading growth of centered collision energy through prime bases and the limiting one-third/two-thirds division of that energy.
Research preprint · DOI 10.5281/zenodo.20547910
The Secondary Term in Digit-Collision Energy
The next arithmetic contribution beneath the cubic growth. It distinguishes the proved secondary term from stronger sampling estimates that remain open.
Research preprint · DOI 10.5281/zenodo.21875096
Each DOI leads to the paper and its version history. The research site also includes essays that develop the intuition alongside the mathematics.