Meaning
The gauge-invariant Markov ancestor is the electric–magnetic Dirichlet-form structure before sharp detector compression.
On the finite configuration space G^E with Haar measure, it combines electric gradient energy and a bounded nonnegative magnetic plaquette potential. Its restriction to gauge-invariant functions retains the Dirichlet contraction property. This identifies a positivity-preserving analytic ancestor of the sharper detector construction.
Example (illustrative)
Clipping a real test function to the interval [−1,1] cannot increase its gradient magnitude or absolute value. Both the electric gradient term and the nonnegative potential term therefore decrease. This illustrates the Markov contraction property.
Scope and limits
A sharp detector compression need not preserve positivity or the Dirichlet property. Additional seam terms need their own Markov-compatibility check, and the ancestor need not yet be conservative.
Source
GAT VIII — Indexed Admissible Distributors and Selected Cofinal Čech Transport in Semisimple Yang–Mills Theory. GAT VIII, Theorem 14.1 context.