The Geometric Imbalance Functional, Probability Flux Conservation, and the Oscillation Matrix Approach to the Riemann Hypothesis

We introduce the oscillation matrix \(Δ_N\), the symmetric Toeplitz matrixwith entries \(Δ_N[i,j] = ψ(|i-j|)-|i-j|\) (where \(ψ\) is the Chebyshev function),and decompose it as \(Δ_N = Δ_N^prime+Δ_N^comp\).We prove three non-circular bounds:(i) the Large Sieve bound: \(E_θ|∑p≤ Nlog(p)eipθ|^2≤ 2Nlog N\)(proved without RH);(ii) the CRT independence bound: \(\|Δ_N^prime\|=O(√Nlog N)\)(proved from Mertens' theorem);(iii) the exact arithmetic identity \(∑n≤ N(ψ(n)-n)=E_ψ(N)+∫_2^N E_ψ(t)\,dt+O(√N)\)(proved by Abel summation).Numerically: \(\|Δ_N\|/√N≈ 0.63log^2 N\) for all \(N\) tested up to 500,consistent with RH. The gap between the proved average bound and the pointwise maximum isidentified as Gap G3 in oscillation language:\(|∑_ρ Nρ+1/(ρ(ρ+1))|=O(√Nlog^2 N)\).Nine Equivalent Reformulations of RH (§5).Combining the geometric and arithmetic approaches, we establish nine mutually equivalentconditions for \(σ_n=1/2\), each proved gap-free from first principles.geometric imbalance functional · probability flux · resonance-wave density · universal invariant ·oscillation matrix · large sieve · CRT independence · Mertens theorem · Chebyshev psi function ·Wheeler–DeWitt equation · Hartle–Hawking wave function · hyperbolic 3-space · Riemann Hypothesis ·Möbius function · prime/composite decomposition · Abel summation · Selberg–Dedekind factorisation ·Shannon entropy · nine equivalent conditions.