Large negative subspaces for positive Euler deformations

Abstract

We construct positive Euler products on the ordinary primes whose associated prime-shift quadratic forms have large negative subspaces despite strong prime-counting control. The prime exponents can be uniformly as close to one as prescribed, agree with one on any prescribed finite prime prefix, and use the same exponent at every power of a prime. Their logarithmic derivatives satisfy a classical subexponential prime number theorem remainder. For one fixed deformation, the negative index on odd polar-free tests is at least cL exp(-C sqrt(log L)) at every sufficiently large support radius L, with an exponentially negative Rayleigh bound on the constructed spaces. The same construction applies to every fixed Hermitian comparison form bounded by a finite Sobolev norm, and survives finite-rank corrections and finitely many additional linear constraints. The proof combines classical positive oscillatory templates with progression-free packet centers. Progression-freeness removes the sum-midpoint modulation exactly, while normalization exponentially suppresses every mixed difference-shift term, including interactions with earlier blocks. Complete matrix estimates and an operator bound for all higher prime powers transfer the negative subspaces to the ordinary-prime model. The resulting index bound quantifies the obstruction to recovering signed form positivity from positive Euler structure, a finite prime prefix and a prime number theorem remainder.