A Structural Reconstruction and Extension of Maynard's Prime Gap Work via the Theory of Differential Algebraic Finite Representations

This monograph develops a unified structural theory of prime distribution, grounded in first principles and centered on the method of differential algebra, supplemented by modern analysis, algebraic geometry, and dynamical systems. We reconstruct the discrete combinatorial weight system of Maynard’s sieve as the discrete sampling of a continuous differential-algebraic system, and by applying the theory of differential algebraic finite representations, we transform the problem of finding optimal weights into an optimization problem on function spaces governed by differential-algebraic equations. We prove: (1) The optimal Maynard weight functions and their squares satisfy explicit algebraic differential equations and admit finite representations whose complexity (degree and height) is directly linked to the dimension k; (2) The sieve inequality can be rigorously interpreted as a spectral problem for a corresponding integral operator, with the limit k → ∞ corresponding to the divergence of representation complexity; (3) For the twin prime conjecture, we construct a system of coupled differential equations whose solution defines a weight system that overcomes the classical barrier ρ = 1, thereby providing a formal reduction of the twin prime conjecture to a concrete solvability problem within the differential-algebraic framework; (4) We derive complete asymptotic formulae in high-dimensional cases, including combinatorial correction terms and branch selections arising from representation theory, with explicit numerical verification for k = 3. Our model not only reproduces all classical sieve results (as special cases) but unifies and surpasses existing methods, offering a completely new continuous dynamical perspective on the distribution of primes.