Rational Nondegeneracy Does Not Guarantee an Integral Betti Adjoint
Epistemic-graph event: 1352.
With pairing matrices P_G,P_H and coefficient pullback Q, Betti
adjunction is the integral matrix equation
P_H S=Q^T P_G.
An integral adjoint exists exactly when the right-hand columns lie in the
column lattice of P_H. Rational nondegeneracy is insufficient: for
P_H=[2], P_G=Q=[1], adjunction forces S=[1/2], so the rational adjoint
is unique but no integral pushforward exists.
This cokernel/lattice-index obstruction is distinct from pairing radical freedom. Smith divisibility tests decide it before computing any boundary commutator. The formal five-site delta pairing passes because its matrix is unimodular; the unavailable physical relative pairing must still be checked for both kernel and cokernel.
Research note:
research/grothendieck/integral-adjoint-pairing-lattice-obstruction.md.