The Apéry Silence in Operator Geometry

We report a striking empirical phenomenon in the learned latent geometry of polynomial continued fractions (PCFs): small perturbations to the quadratic coefficient a_2 of the numerator polynomial — an Apéry-sensitive coordinate in the experimental parameterization — produce operator displacements that are 38-127 times smaller than those induced by the elementary index-shift operator, depending on the latent dimension k. We call this effect the Apéry silence. Using the OGRE (Operator Geometry Research Engine) framework and a trained GOTv3 geometric autoencoder, we show that the Apéry perturbation operators T_A+ and T_A- produce near-zero mean displacement vectors, have vanishing or near-vanishing commutators with most tested operators, and behave as epsilon-central elements of the empirical operator algebra at epsilon = 0.01. We interpret the result cautiously: it is evidence for latent flatness in the a_2 direction under the current encoder and perturbation scale, not a proof of an Apéry-specific arithmetic theorem. The paper formulates direct numerical and symbolic tests needed to determine whether this flatness reflects a genuine mathematical invariant of PCF space.