FINITE ADAPTABILITY OF COUNTEREXAMPLE LIBRARIES

How many fixed noise witnesses cover every measurement batch compatible with a pair of model predictions? Under diagonal error, requests and witnesses share a knapsack constraint. Identical best coverage and demand sizes within a factor of two can still yield exponentially different library sizes. We separate informa- tion lost by grouping requests from information lost by grouping completions. An input-level test computes the unique coarsest partition preserving compati- ble demand, constructing groups within a cap or certifying its insufficiency. Op- timistic and robust type programs then bracket the full fractional cover with a computable loss guarantee; partial-demand counts supply lower certificates when stability fails. Distinct-weight families isolate joint information beyond optimal packing and strengthened counting, and quantify unavoidable aggregation loss. On all 164 small instances, canonical grouping expands eight-group bracket appli- cability from 39 to 157; 107 of these 157 brackets have ratios at most 1.25. All 48 optimized witnesses require more than eight stable groups, exposing a substantial transfer boundary. Checked greedy libraries and matched-time counting/packing controls distinguish stronger certificates from changed integer decisions and from runtime gains.