An Inverse Apéry Audit for e + π: Positivity-Smallness Separation

We study a finite inverse Apéry program for the open irrationality problem of e + π. The construction starts from an exact integral identity that converts each integer polynomial R in Z[x] into an integer linear form R(1)(e + π) − R(0). We audit whether natural finite search classes can simultaneously provide the main ingredients of an Apéry certificate: integrality, smallness, nonzero certification, and an infinite arithmetic structure. Residual-based lattice searches produce small non-continued-fraction linear forms and small kernels, but only sparse sign dominance. A structured Hermite–Padé restart shows the opposite behavior: sampled one-sign kernels become abundant, while endpoint errors and kernel norms grow enormously. An extended coupled postprocess over 2411 merged candidates still finds no row combining endpoint smallness, kernel smallness, and sign dominance. The tested classes therefore exhibit a positivity–smallness separation: smallness and sign control can be forced separately, but not together within a recognizable Apéry-type structure.