The Theory of Accessibility

The Theory of Accessibility introduces a mathematical framework in which accessibility is treated as a state-dependent property of an underlying structure. The foundational formulation is:Sλ(x) = A(x, λ)S(x)where S(x) represents the complete structure, A(x, λ) its accessibility in a given accessibility state, and Sλ(x) the structure accessible in that state. The work further introduces an asymptotic accessibility-stability criterion:C_A∞ = liminfₘ→∞ (Aₘ/A₀)which characterizes the absence of asymptotic accessibility collapse when C_A∞ > 0. The framework is tested against a range of major mathematical problems, including the Riemann Hypothesis, Yang–Mills and the mass gap, the Hodge Conjecture, the Birch–Swinnerton-Dyer Conjecture, the Collatz Conjecture, Goldbach's Conjecture, the Twin Prime Conjecture, Legendre's Conjecture, Landau's n² + 1 problem, the abc Conjecture, the Beal Conjecture, the odd perfect number problem, and Schanuel's Conjecture. The strongest direct result obtained in this testing concerns the Collatz process. With A(N) = 1/N, the criterion becomes:C_A∞ = liminfₘ→∞ (N₀/Nₘ)yielding the exact equivalence:C_A∞ > 0 ⇔ supₘ Nₘ < ∞This establishes an equivalence between asymptotic accessibility stability and boundedness of the corresponding Collatz trajectory, but does not by itself prove the Collatz Conjecture. The work also identifies limitations of a single scalar accessibility function for problems whose decisive structures depend on prime factorization, divisor structure, or transcendence. The present work therefore proposes a coherent general framework for structural accessibility and asymptotic accessibility stability rather than a universal proof mechanism for all open mathematical problems.