Observer--Fragmentation--Exposure Tradeoffs: From Rectangular CFG Exposure to Ordered MCFG Scheduling

Abstract

We study finite resources governing exact positive reconstruction in fixed-observation CFG and MCFG learning. For an explicit rigid CFG family we compute safe observation size, internal residual fragmentation, and characteristic-data cost exactly. The result is a two-point Pareto frontier: after compulsory rigidity witnesses are fixed, exact reconstruction reduces to connectivity inside observer fibers, and the variable part of every minimum characteristic sample is a spanning forest of complete bipartite fiber graphs. For bounded-fan-out MCFG reconstruction, pure transition fragmentation multiplies across children. An explicit fan-out-two family X_{k,r} therefore has exact characteristic-data costs 2r[1+r(k-1)] and 2rk^r under two comparable observers. An integral lattice invariant yields an affine-span lower bound and a unimodularity test. In the binary-index subfamily, unimodularity suffices for minimum-cardinality samples through ranks two and three but not rank four. Two distinct obstructions then appear: an order-independent laminar support conflict, and an order-sensitive occurrence-scheduling conflict. For disjoint child requirements, the exact scheduling threshold is the largest monochromatic run count in the doubled reduced slot-colour word; in the two-colour case this is an alternation threshold. Horn nonlocking and Cartesian locking certificates make these constraints explicit. Hierarchical reuse can trade parent-root exposure for local fan-out: for the natural critical-module library of the mixed rank-four shapes, the exact width--anchor frontiers are {(2,1)}, {(2,2),(3,1)}, and {(4,1)} for separated, nested, and crossing orders. Thus observation, fragmentation, exposure, arithmetic span, laminar compatibility, ordered scheduling, and hierarchical reuse are genuinely distinct finite resources.