Euler irrationality from explicit saddle estimates and arithmetic normalization

Assume the explicit oscillatory numerator limit and recurring noncancellation of the phase. Suppose nonzero real scalars c_n send P_(n+1),Q_(n+1) to integers p_n,q_n and c_n fModel_(n+1) tends to zero. Then Euler’s constant is irrational. The existence of such arithmetic scalars is not asserted: it is the missing arithmetic input, isolated from the analytic saddle limits.

Connection to the Euler tree (12 September 2026). The refexplicit p=2 approximation branch studies the size of rational approximation errors. The proved refconditional irrationality bridge shows that its numerator asymptotic, the proved refphase noncancellation, and a successful integer normalization would imply refirrationality of Euler’s constant. The required normalization consists of nonzero scalars c_n making both c_n P_(n+1) and c_n Q_(n+1) integers while c_n fModel_(n+1) tends to zero. No such scalars have been constructed. This is an explicit candidate route to the refvanishing integer linear forms leaf: one would select the infinitely many nonzero forms and normalize denominator signs. That connection is explanatory; it is not a submitted proof discharging the existence leaf. The mixed E/Gevrey lifting obligations in the reftranscendence tree remain open. Irrationality alone would not prove transcendence.