We introduce the oscillation matrix \ (N\), the symmetric Toeplitz matrixwith entries \ (Ni, j = (|i-j|) -|i-j|\) (where \ (\) is the Chebyshev function), and decompose it as \ (N = Nᵖrime+Nᶜomp\). We prove three non-circular bounds: (i) the Large Sieve bound: \ (E_| ₍ (p) e^ip|² 2N N\) (proved without RH) ; (ii) the CRT independence bound: \ (\|Nᵖrime\|=O (N N) \) (proved from Mertens' theorem) ; (iii) the exact arithmetic identity \ (₍ ₍ ( (n) -n) =E_ (N) +₂N E_ (t) \, dt+O (N) \) (proved by Abel summation). Numerically: \ (\|N\|/N 0. 63² 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: \ (|_ N^+1/ ( (+1) ) |=O (N² N) \). Nine Equivalent Reformulations of RH (§5). Combining the geometric and arithmetic approaches, we establish nine mutually equivalentconditions for \ (ₙ=12\), 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.
Islam Emad El-Gammal (Thu,) studied this question.