Research Methodology

We introduce the **spectral information state** as a canonical mode-level state object for spectral inference. After eigendecomposition, each mode $k$ carries an estimated coefficient $\hat{A}_k$ and a residual uncertainty $\sigma_k^2$ induced by finite sample size, noise level, and eigenvalue strength.

We show that Bayesian and frequentist statistics admit a common spectral formulation: after eigendecomposition, both frameworks act on the same mode-level evidence and therefore agree on the central inferential question of model complexity.

We develop the carry arithmetic of the Syracuse map $S(x) = (3x+1)/2^{v_2(3x+1)}$ from first principles, establishing exact identities that govern popcount evolution under iteration.

We introduce a structural decomposition of partial differential equations into a triple $(D, C, P)$ of dissipation, coupling and constraint operators and prove that the solvability behavior of the system is determined by a single dimensionless quanti

We introduce the **PDE Tensor Algebra**, a framework that represents any system of partial differential equations as a triple $(D, C, P)$ of tensors encoding dissipation, coupling, and constraints.

We study the Gevrey regularity of solutions to the 3D incompressible Navier-Stokes equations on the torus \(\mathbb{T}^3\).