Galilean Reeh--Schlieder Obstruction

We prove that the standard Galilean Haag--Kastler axioms, augmented by Bargmann mass superselection, are inconsistent with the Reeh--Schlieder property: no such net admits a vacuum that is cyclic and separating for every local field algebra. Two ingredients combine: Galilean Schr\"odinger fields annihilate the Fock vacuum, and Bargmann mass superselection forbids the Hermitian-combination evasion that keeps relativistic axioms consistent. The result extends beyond the Fock-representation hypothesis: any Galilean Haag--Kastler net whose canonical fields carry definite Bargmann mass charges and admit time-zero restrictions on a field-algebra-stable common dense domain is incompatible with Reeh--Schlieder. The bounded-below mass spectrum and the vacuum-at-spectral-minimum, usually imposed as separate axioms, are derived consequences -- of positive-energy boost positivity and a Bose-CCR algebraic descent, respectively. The Tomita--Takesaki modular flow is consequently unavailable on Galilean local field algebras. We identify the Reeh--Schlieder property as the precise structural ingredient distinguishing relativistic from Galilean algebraic quantum field theory: relativistic AQFT has it as a theorem, Galilean AQFT cannot.