The Prime Matrix Framework

For $n\ge 2$, let $p(n)$ be the largest prime $\le n$ and $r(n)=n-p(n)$.The diagonal embedding $M_2(n)=diag(p(n),r(n))$ with $Tr M_2=n$ serves as a useful algorithmic state cache but is spectrally trivial. We present a unified, full-scope matrix-theoretic framework extending $M_2(n)$ in three directions. First, local extensions introduce $\tilde M_2(n)=\begin{pmatrix} p & r \\ r & g \end{pmatrix}$ coupling remainder to prime gaps $g(n)$, and a tridiagonal Jacobi matrix $J_k$ whose lower trace moments encode gap autocorrelations[cite: 1]. Globally, prime packing matrices $P_k^c=diag(p_1,\dots,p_k)$ link operator log-traces to Chebyshev's function $\theta(N)$, while its unbounded $\ell^2$ limit $P_\infty$ yields the prime zeta function $P(s)$ via zeta regularization. Under Cram'er's probabilistic model, we formalize variance bounds and restate the Riemann Hypothesis in operator-theoretic language. Finally, we formulate the empirical gap estimator $G_{\text{ML}}$ using a 10-dimensional causal feature space (logarithmic drift, local moving averages, and primorial wheel obstructions mod 30) and show how it directly parameters the affine prime predictor matrix $M^*_{\text{ML}}$. Detailed parameter weights, baseline comparisons, discretization projections, and complexity analysis for Goldbach $r$-skip searches complete the framework. *Author DisclaimerAll underlying concepts, frameworks, and core research directions are the original work of the author. Generative AI tools were used strictly for assistance in manuscript drafting, code generation, and figure preparation. All theoretical formulations and mathematical proofs included herein are unverified AI-generated drafts shared as preliminary proposals for discussion and community review.