formal-methods

Published on May 25, 2026 10:01 AM GMT TL;DR: Applications are now open for the Secure Program Synthesis Fellowship , powered by Apart Research and Atlas computing . Apply by Sunday the 31st of May . This fellowship offers part-time research opportunities on mentor-led projects at the intersection of formal methods, AI systems, and security . Participants work in small teams to tackle challenging…

We prove that E7 is the unique simple Lie algebra g for which the quantity dim(g) + max_mark(Ĝ) is simultaneously prime, congruent to 1 (mod 4), and a Gaussian prime norm on the cascade tower of z = 2+i. The proof is by exhaustive enumeration over all simple Lie algebras—the four infinite classical families (An, Bn, Cn, Dn) and the five exceptional algebras (G2, F4, E6, E7, E8)—using a three-laye…

We study the expression complexity of the Riemann explicit formula from the perspective of grammar-based symbolic regression. Starting from the EML operator eml(a,b) = exp(a) − log(b), we show that this operator fails to produce oscillatory terms and therefore cannot encode the explicit-formula structure. We then derive a replacement operator eml_zeta(g, t) = (cos(γt) + 2γ sin(γt)) / (¼ + γ²) dir…

This paper develops interpolations for continuous tetration beyond positive integer heights. Using algebraic manipulation, telescoping sums, and the standard recursive definition of tetration, the paper derives extensions for both positive and negative non-integer heights.

The introduction of quantum-secure cryptography in iMessage marked the start of a significant security transition to protect Apple users from threats posed by future quantum computers. Deploying this new generation of algorithms at scale across all Apple platforms requires high assurance, so we developed rigorous new formal verification methods to prove the mathematical correctness of our impleme…

This paper introduces a combinatorial scalability framework designed to extend a 5x5 ternary palindromic matrix template into a macrostructural 10x10 grid. By applying discrete axial reflections and mirroring techniques to four quadrant-aligned 5x5 sub-matrices, a highly symmetrical, periodic tiling system is generated. This framework preserves the fundamental phonotactic constraints—specifically…

This paper presents a numerically stable framework for solving real-coefficient nonic equations (degree 9). The method is based on a cubic Tschirnhaus transformation y = x^3 + \alpha x^2 + \beta x + \gamma that approximately eliminates the y^8, y^7, and y^6 terms. The parameters (\alpha, \beta, \gamma) are determined by solving a 3\times3 nonlinear system using Newton's method with an LU/SVD fall…

In this paper, we investigate the geometric properties of metric projections in reflexive and strictly convex Banach spaces. We prove that the structural linearity of a metric projection operator PV onto any closed subspace V fundamentally forces the underlying space to be an inner product space. This condition establishes a strict isometry, rather than a mere topological isomorphism, demanding t…

Version 3.0. Added information-theoretic derivation of V=N/D from S/N ratio. Proximal-distal knowledge structure applied. Concrete behavioral evidence of inheritance from Masatada added. Scope Declaration included.