Large negative subspaces for positive Euler deformations

Abstract

We construct positive Euler products on the ordinary primes whose prime-shift quadratic forms are negative on subspaces of growing dimension. The prime exponents can be uniformly as close to one as prescribed, agree with one on any prescribed finite prime prefix, and remain the same at every power of each prime. The corresponding logarithmic derivatives satisfy a classical subexponential prime number theorem remainder. For one fixed deformation and every sufficiently large support radius L, the negative index is at least cL exp(-C sqrt(log L)), with an exponentially large negative Rayleigh bound. The test functions are odd and satisfy two polar-cancellation conditions. The same construction works against each fixed Hermitian comparison form controlled by a finite Sobolev norm, with an onset depending on that form. Finite linear constraints and corrections are treated by dimension bounds; for general Hermitian repairs, only their positive index on the constructed space reduces the guarantee. The proof combines classical oscillatory templates with progression-free packet centers, then controls every mixed interaction and all higher prime powers. The result quantifies how large a signed-positivity obstruction can coexist with positive Euler structure, finite-prefix agreement and prime-counting control.