Witness Geometry Beyond Scalar Fee
Repair Entropy, Operational Sparsity, and the Decision Theory of Disclosure
β(G) = ∏|C_{d,j}| with sharp bounds φ+1 ≤ β ≤ 2^φ; every basis realizable under some cost vector; a worked motif pair shows fee alone underspecifies repair structure
A composition in the DFD + CHP regime where the basis count disagrees with the component-size product, or where the sharp bounds are violated
Abstract
The coherence fee counts blind spots but says nothing about how to fix them. This paper develops the geometry that lives inside the fee. The witness Gram K(G) determines all repair combinatorics: under the DFD + CHP regime, the number of minimum disclosure bases is β(G) = ∏|C_{d,j}|, a product over connected components of K. Repair entropy H_repair = log β decomposes additively. Sharp bounds φ+1 ≤ β ≤ 2^φ hold at fixed fee φ. A full realizability theorem shows every basis is the unique optimum under some cost vector — operational sparsity is a property of the cost family, not the matroid. Three projections of one object: fee (obligation count), repair entropy (structural flexibility), stability ratio (operational flexibility under cost constraints).