Column-Matroid Backbone (Paper II)
A Structural Identity and Its Corollaries
fee(G) = corank_M(O): the backbone theorem with closed-form pairwise formula verified on 703 real-schema compositions
A composition where the backbone identity fee = corank_M(O) fails, or where the closed-form pairwise formula disagrees with the coboundary rank computation
Abstract
The coherence fee equals the corank of the observable column set in the linear matroid on the full coboundary columns. This structural identity yields a closed-form pairwise formula fee(A,B) = U(A) + U(B) + |shared_dims(A,B)| and proves additive decomposition as matroid direct-sum additivity. An empirical density law localizes the connecting homomorphism strictly in the interior of its attainable range on all 240 nontrivial compositions tested.