The Arithmetic Skeleton of Ramanujan's Correction

This paper uncovers an unexpected rigidity hidden inside Ramanujan's correction to Euler's constant. By formally inverting the Euler–Maclaurin expansion, a new sequence of coefficients emerges whose arithmetic behaviour is far more structured than the series from which it originates. Every coefficient carries an exact power of two in its denominator, decreasing linearly with the index, so that the 2 adic valuation satisfies v2(cn) = −n with no exceptions. At the same time, each odd prime p ≥ 5 leaves its first trace in the sequence at a predictable position, and remarkably the residue at that point is always the same, giving p cp−3 ≡ 4 mod p. Together these results show that the inversion process acts as a sharp arithmetic filter. The complexity of the Bernoulli numbers collapses into two simple invariants, a perfect dyadic descent and the universal appearance of the integer 4. What begins as a classical asymptotic expansion thus reveals a rigid p adic skeleton, suggesting that Ramanujan's approximation is only the visible surface of a much deeper arithmetic structure.