Smith Factors Exactly Decide an Integral Mackey Trace
Epistemic-graph event: 1356.
For an integral pushforward S:Z^n->Z^m, an integral trace satisfying
S T=d I_m
exists exactly when S has full row rank and every Smith invariant factor
s_i divides d. Equivalently, coker(S) must be killed by the prescribed
norm degree. The minimal possible scalar is lcm_i(s_i).
The hostile rank-one case S=2,d=3 has no integral trace, while S=2,d=4
admits T=2. Existence does not give canonicity: all traces differ by maps
into ker(S).
For a physical five-site branch of degree 2^k, every Smith factor of the
relative pushforward must divide 2^k. Any odd factor or excessive two-adic
valuation falsifies the integral Mackey norm before boundary compatibility is
considered.
Research note:
research/grothendieck/integral-mackey-trace-smith-criterion.md.