Arithmetic Obstructions to Exact Quantum Survival Laws

Can golden hierarchy weights become the probabilities of successive recorded events? For the fixed native recorded-current instrument of Recognition Science, we derive an obstruction: no initial full-field state and no fixed choice of continuation outcomes produces exact reciprocal-golden survival. The operation has a finite rational transfer even with all quantum coherences and the unrestricted integer field retained. An exact golden rate would force its algebraic conjugate −φ into the spectrum, contradicting bounded powers. More generally, every exact geometric output of a finite power-bounded operation with rational characteristic polynomial has all its algebraic conjugates in the unit disk. Finite memory and periodic control retain the obstruction when the enlarged transfer is rational. Finite certificates locate a failed equality; arbitrarily accurate finite-horizon approximations remain possible. The result identifies a necessary constraint on realizing RS hierarchy weights as physical event probabilities.