Smith Divisibility Gives the Minimal Localization for a Betti Adjoint
Epistemic-graph event: 1354.
For U P_H V=diag(d_i) and B'=U Q^T P_G, the unique rational adjoint has
reduced entry denominators
e_ij=d_i/gcd(d_i,B'_ij).
Their least common multiple L is the exact denominator index: an integral
adjoint exists iff L=1, and localization admits the adjoint precisely after
inverting every prime dividing L.
The control P_H=[6], B=[4] has L=3, whereas P_H=[6], B=[6] has L=1.
Thus determinant primes alone overestimate the obstruction. Pairing-lattice
denominators form an arithmetic support independent of deck norm degree and
monodromy resonance.
A physical five-site packet must supply this Smith data degreewise; the formal unimodular delta pairing cannot stand in for it.
Research note:
research/grothendieck/minimal-localization-for-integral-adjoint.md.