Complete Deficit and Equality Theory for Marked Rank-Four Seven-Column Configurations

In rank four, seven columns form the last universally Rayleigh column count under the known sharp degree/codegree boundary. We study the corresponding marked five-generator/two-probe configuration and its deficit, focusing on the equality structure that positivity alone does not determine. Using the known Gale-dual mechanism, we derive the native exact nonnegative deficit decomposition, including all degenerate and nondegenerate branches. We then classify the complete formal Gale zero set and separate formal equality from actual marked realizability through the rank-three Gale condition. Returning to the original marked configuration, we obtain the exhaustive all-rank equality classification. Finally, the pointwise actual classification lifts to positive finite atomizations and to finite positive Borel generating measures with finite first moments, with equality determined on every effective atom pair or almost everywhere, respectively. We call the resulting complete marked closure the Marked Rank-Four Seven-Column Deficit and Equality Theory (MRF7-DE), while treating seven-column Rayleigh positivity and the seven/eight-column threshold as known background.