Phase-Interference Energy and the Formal Structure of the PG1224 Prime Generation System
Description
This research note reformulates the elementary number-theoretic structure of natural numbers using the operational concept of phase-interference energy. In this model, a prime number is treated as a stable fixed point that contains no non-trivial internal interference.
The note defines interference energy as a measure of the non-trivial divisors of a natural number, clarifies the boundary condition excluding 0 and 1 through the domain condition 2 ≤ n, and sketches a Lean 4 formalization of the equivalence among Nat.Prime n, interferenceEnergy n = 0, and isStableFixedPoint n.
It also introduces PG1224, a prime-candidate generation system based on residue-class filtering modulo 12 and 24. The filter is interpreted as cancellation of trivial interference caused by divisibility by 2 and 3. The note does not claim a new definition of primality or computational superiority over existing sieve methods; rather, it reconstructs the standard notion of primality in the vocabulary of energy minimization, stable fixed points, formal verification, and AIKernel-style governance models.
The English manuscript is the canonical version. The Japanese manuscript is included as a companion translation.
Files
paper-en.pdf
Additional details
Additional titles
- Subtitle (English)
- A Lean 4 Formalization of Prime = Energy 0 = Stable Fixed Point
Related works
- Cites
- Conference paper: 10.1145/3372885.3373824 (DOI)
- Conference paper: 10.1007/978-3-319-21401-6_26 (DOI)
- Is supplemented by
- Other: https://github.com/AIKernel-NET/AIKernel.RH (URL)
- Other: https://aikernel.net/ (URL)
Software
- Repository URL
- https://github.com/AIKernel-NET/AIKernel.RH
- Programming language
- Lean
- Development Status
- Active
References
- The mathlib Community. (2020). The Lean mathematical library. Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, 367–381. https://doi.org/10.1145/3372885.3373824
- de Moura, L., Kong, S., Avigad, J., van Doorn, F., & von Raumer, J. (2015). The Lean Theorem Prover. Automated Deduction — CADE-25, Lecture Notes in Computer Science, 9195, 378–388. Springer. https://doi.org/10.1007/978-3-319-21401-6_26
- Lean FRO. (2026). Lean 4 Manual. Lean Focused Research Organization.
- Sogawa, T. (2026). AIKernel Trajectory Governance Model: A Kernel-Level Framework for Convergent Decision Control over Stochastic Language Model Inference. Zenodo.
- Sogawa, T. (2026). AIKernel Formal Foundations: Contract-Based Semantic Execution for Governed AI Systems. Zenodo.